# Categorical absorptions of singularities and degenerations

Alexander Kuznetsov  Algebraic Geometry Section, Steklov Mathematical
Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow 119991 Russia
Laboratory of Algebraic Geometry, National Research University Higher School
of Economics, Russian Federation
<EMAIL_ADDRESS>and  Evgeny Shinder  School of Mathematics and Statistics,
University of Sheffield, Hounsfield Road, S3 7RH, UK, and Hausdorff Center for
Mathematics at the University of Bonn, Endenicher Allee 60, 53115.
<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of categorical absorption of singularities: an
operation that removes from the derived category of a singular variety a small
admissible subcategory responsible for singularity and leaves a smooth and
proper category. We construct (under appropriate assumptions) a categorical
absorption for a projective variety $X$ with isolated ordinary double points.
We further show that for any smoothing ${\mathcal{X}}/B$ of $X$ over a smooth
curve $B$, the smooth part of the derived category of $X$ extends to a smooth
and proper over $B$ family of triangulated subcategories in the fibers of
${\mathcal{X}}$.

A.K. was partially supported by the HSE University Basic Research Program.

E.S. was partially supported by the EPSRC grant EP/T019379/1 “Derived
categories and algebraic K-theory of singularities”, and by the ERC Synergy
grant “Modern Aspects of Geometry: Categories, Cycles and Cohomology of
Hyperkähler Varieties”.

###### Contents

  1. 1 Introduction
  2. 2 Categorical ordinary double points and ${\mathbb{P}}^{\infty}$-objects
    1. 2.1 Categorical ordinary double points
    2. 2.2 ${\mathbb{P}}^{\infty}$-objects
  3. 3 Kronecker quiver and categorical ordinary double points
    1. 3.1 The graded Kronecker quiver
    2. 3.2 Localization and adherence
  4. 4 Adherence and categorical absorption of singularities
    1. 4.1 From adherence to categorical absorption
    2. 4.2 Absorptions and deformation absorptions
  5. 5 Crepant categorical resolutions for nodal varieties
    1. 5.1 Bondal–Orlov localization
    2. 5.2 Crepant localization for ordinary double points
  6. 6 Absorption of singularities for nodal varieties
    1. 6.1 Absorption for nodal varieties
    2. 6.2 Obstructions
    3. 6.3 Curves and threefolds

##  1\. Introduction

Resolution of singularities is a very important instrument in algebraic
geometry, that allows one to replace complicated geometry of singular schemes
by much more tractable geometry of smooth varieties. Its categorical version —
categorical resolution of singularities — has been defined in [Kuz08, KL15].

Let $X$ be a singular proper scheme over a field ${\Bbbk}$. One associates
with it the pair
$({\mathbf{D}}^{\mathrm{perf}}(X),{\mathbf{D}}^{\mathrm{b}}(X))$ of
triangulated categories, where ${\mathbf{D}}^{\mathrm{perf}}(X)$ is the
category of perfect complexes on $X$ (this category is proper but not smooth),
and ${\mathbf{D}}^{\mathrm{b}}(X)$ is the bounded derived category of coherent
sheaves on $X$ (this category is smooth but not proper). Categorical
resolution allows one to replace this pair by a single _smooth and proper_
triangulated category ${\mathcal{D}}$.

Such a category ${\mathcal{D}}$ is “larger” than
${\mathbf{D}}^{\mathrm{b}}(X)$ and ${\mathbf{D}}^{\mathrm{perf}}(X)$ in the
sense that ${\mathbf{D}}^{\mathrm{b}}(X)$ is (expected to be) a localization
of ${\mathcal{D}}$ and ${\mathbf{D}}^{\mathrm{perf}}(X)$ is a subcategory of
${\mathcal{D}}$. In this paper we look at the problem from a different angle
suggesting to replace (when possible) the categories
${\mathbf{D}}^{\mathrm{b}}(X)$ and ${\mathbf{D}}^{\mathrm{perf}}(X)$ by a
“smaller” smooth and proper triangulated category in the following sense.

######  Definition 1.1.

We say that a triangulated subcategory
${\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$ absorbs singularities of
$X$ if ${\mathcal{P}}$ is admissible in ${\mathbf{D}}^{\mathrm{b}}(X)$ and
both orthogonals ${\mathcal{P}}^{\perp}$ and ${}^{\perp}{\mathcal{P}}$ are
smooth and proper.

Note that since we assume ${\mathcal{P}}$ to be admissible, we have
${\mathcal{P}}^{\perp}\simeq{}^{\perp}{\mathcal{P}}$, hence in the definition
of absorption it suffices to assume that either of the orthogonals is smooth
and proper. Furthermore, if ${\mathcal{P}}$ absorbs singularities of a
Gorenstein scheme $X$, it is easy to check (see Lemma 4.3) that the smooth and
proper category ${\mathcal{D}}\coloneqq{}^{\perp}{\mathcal{P}}$ (which is
equivalent to ${\mathcal{P}}^{\perp}$) is contained in
${\mathbf{D}}^{\mathrm{perf}}(X)$ and admissible in both
${\mathbf{D}}^{\mathrm{perf}}(X)$ and ${\mathbf{D}}^{\mathrm{b}}(X)$; in this
sense the operation replacing ${\mathbf{D}}^{\mathrm{b}}(X)$ and
${\mathbf{D}}^{\mathrm{perf}}(X)$ by ${\mathcal{D}}$ is “opposite” to the
categorical resolution operation.

There is a trivial example ${\mathcal{P}}={\mathbf{D}}^{\mathrm{b}}(X)$, so
that ${\mathcal{D}}=0$, which of course is not interesting. Therefore, the
idea is to make ${\mathcal{P}}$ as small as possible, so that ${\mathcal{D}}$
maximally reflects the geometry of $X$.

The following geometric example of absorption illustrates the idea. Let
$\upsigma\colon X\to Y$ be an “antiresolution”: a proper morphism, where $X$
is singular, $Y$ is smooth, and both $X$ and $Y$ are proper, and assume that
$\upsigma_{*}({\mathcal{O}}_{X}){}\cong{}{\mathcal{O}}_{Y}$, where
$\upsigma_{*}\colon{\mathbf{D}}^{\mathrm{b}}(X)\to{\mathbf{D}}^{\mathrm{b}}(Y)$
is the derived pushforward functor. Let

${\mathcal{P}}\coloneqq\operatorname{\mathrm{Ker}}(\upsigma_{*})\subset{\mathbf{D}}^{\mathrm{b}}(X).$

We claim that ${\mathcal{P}}$ absorbs the singularities of $X$. Indeed the two
adjoint functors
$\upsigma^{*}\colon{\mathbf{D}}^{\mathrm{b}}(Y)\to{\mathbf{D}}^{\mathrm{b}}(X)$
and
$\upsigma^{!}\colon{\mathbf{D}}^{\mathrm{b}}(Y)\to{\mathbf{D}}^{\mathrm{b}}(X)$
of $\upsigma_{*}$ are fully faithful and provide semiorthogonal decompositions

${\mathbf{D}}^{\mathrm{b}}(X)=\langle{\mathcal{P}},\upsigma^{*}({\mathbf{D}}^{\mathrm{b}}(Y))\rangle=\langle\upsigma^{!}({\mathbf{D}}^{\mathrm{b}}(Y)),{\mathcal{P}}\rangle,$

in particular, the category
${\mathcal{P}}^{\perp}\simeq{\mathbf{D}}^{\mathrm{b}}(Y)\simeq{}^{\perp}{\mathcal{P}}$
is smooth and proper.

######  Example 1.2.

Let us further consider two special cases of this situation:

  1. (a)

$Y$ is a smooth and proper surface,
$Z=\operatorname{\mathrm{Spec}}({\Bbbk}[\upepsilon_{0}]/\upepsilon_{0}^{2})\subset
Y$, $X=\operatorname{\mathrm{Bl}}_{Z}(Y)$, and $\upsigma\colon X\to Y$ is the
blowup morphism. Let $E\cong{\mathbb{P}}^{1}\subset X$ be the exceptional
locus taken with the reduced scheme structure (notice that the exceptional
divisor $2E$ is Cartier, while $E$ is only a Weil divisor).

  2. (b)

$Y={\mathbb{P}}^{1}$, $X=Y\cup_{y_{0}}E$, where $E\cong{\mathbb{P}}^{1}$
intersects $Y$ transversely at a point $y_{0}$, and $\upsigma\colon X\to Y$ is
the contraction of $E$ to $\\{y_{0}\\}$, so that $E$ is again, in a sense, the
exceptional locus of $\upsigma$.

A simple computation shows that in both cases
${\mathcal{P}}=\langle{\mathcal{O}}_{E}(-1)\rangle$, but while in case (a) we
have

(1)
${\mathcal{P}}\simeq{\mathbf{D}}^{\mathrm{b}}({\Bbbk}[\upepsilon_{0}]/\upepsilon_{0}^{2})\simeq{\mathbf{D}}^{\mathrm{perf}}({\Bbbk}[\uptheta_{1}])\hphantom{,}$

by the blowup formula (note that $Z$ is a local complete intersection), in
case (b) we have

(2)
${\mathcal{P}}\simeq{\mathbf{D}}^{\mathrm{b}}({\Bbbk}[\upepsilon_{1}]/\upepsilon_{1}^{2})\simeq{\mathbf{D}}^{\mathrm{perf}}({\Bbbk}[\uptheta_{2}]),$

where $\upepsilon_{p}$ is a variable of degree $-p$, $\uptheta_{q}$ is a
variable of degree $q$, and the second equivalences in (1) and (2) are given
by the Koszul duality (see Proposition 2.2 for details). The difference in the
structure of ${\mathcal{P}}$ in these two cases leads to a completely
different behaviour with respect to smoothings of $X$, where we use the
following

######  Definition 1.3.

We say that a flat projective morphism $f\colon{\mathcal{X}}\to B$ to a smooth
pointed curve $(B,o)$ is a smoothing of $X$ if:

  * •

the central scheme fiber of $f$ is isomorphic to $X$, i.e.,
${\mathcal{X}}_{o}\cong X$;

  * •

the morphism $f$ is smooth away from the central fiber; and

  * •

the total space ${\mathcal{X}}$ is smooth.

In the case of Example 1.2(a), a smoothing of $X$ can be obtained as follows.
Let ${\mathcal{Y}}\to B$ be any smooth deformation of $Y$ (e.g.,
${\mathcal{Y}}=Y\times B$), let ${\mathcal{Z}}\to B$ be a double covering
ramified over the central point $o\in B$ (so that the central fiber
${\mathcal{Z}}_{o}$ is isomorphic to $Z$) with smooth ${\mathcal{Z}}$, and let
${\mathcal{Z}}\hookrightarrow{\mathcal{Y}}$ be a closed embedding over $B$
(coinciding with the embedding $Z\hookrightarrow Y$ over the point $o$). Then
${\mathcal{X}}\coloneqq\operatorname{\mathrm{Bl}}_{\mathcal{Z}}({\mathcal{Y}})$
is a smoothing of $X$. The blowup formula gives in this case a $B$-linear
semiorthogonal decomposition

$\displaystyle{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$
$\displaystyle=\langle{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}),{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}})\rangle,$
and for each point $b\neq o$ in $B$ we obtain by base change (see [Kuz11]) a
semiorthogonal decomposition
$\displaystyle{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{b})$
$\displaystyle=\langle{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{b}),{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}}_{b})\rangle.$

So, in this case we see that both components
${\mathcal{P}}={\mathbf{D}}^{\mathrm{b}}(Z)$ and
${}^{\perp}{\mathcal{P}}={\mathbf{D}}^{\mathrm{b}}(Y)$ of
${\mathbf{D}}^{\mathrm{b}}(X)$ deform simultaneously to a pair of smooth and
proper categories; thus, in a sense, the smoothing of $X$ is achieved through
smoothing of the singular component ${\mathcal{P}}$.

In the case of Example 1.2(b) the situation is quite different. In this case
$X$ is a singular conic, hence any smoothing of $X$ is a conic bundle
${\mathcal{X}}\to B$ with $X={\mathcal{X}}_{o}$ being the only singular fiber.
Note that $E\subset{\mathcal{X}}$ is a smooth rational curve (one of the
components of the central fiber) and a standard computation shows that its
normal bundle is ${\mathcal{O}}_{E}(-1)$. Therefore, the curve
$E{}\subset{\mathcal{X}}$ can be contracted, i.e, there is a smooth surface
${\mathcal{Y}}$ with a point $y_{0}\in{\mathcal{Y}}$ such that
${\mathcal{X}}=\operatorname{\mathrm{Bl}}_{y_{0}}({\mathcal{Y}})$ with $E$
being the exceptional divisor. Then ${\mathcal{Y}}\to B$ is a smooth
${\mathbb{P}}^{1}$-fibration (with ${\mathcal{Y}}_{o}\cong Y$) and the blowup
formula gives a $B$-linear semiorthogonal decomposition

(3)
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle{\mathcal{O}}_{E}(-1),{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}})\rangle.$

Its first component is generated by the exceptional sheaf
${\mathcal{O}}_{E}(-1)$ supported in the central fiber of the conic bundle,
therefore it disappears after base change to any point $b\neq o$ in $B$, and
we obtain an equivalence

${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{b})\simeq{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}}_{b}).$

So, in this case the component ${\mathcal{P}}$ does not deform away from the
central fiber; thus a smoothing of $X$ is achieved by dropping “the singular
part” ${\mathcal{P}}$ of ${\mathbf{D}}^{\mathrm{b}}(X)$ and deforming its
“smooth part”
${}^{\perp}{\mathcal{P}}{}\simeq{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}}_{o})$.

We consider the latter case as more interesting. Axiomatizing it, we introduce
the following notion. For a set of objects $S$ in a triangulated category
${\mathcal{T}}$ we denote by $\langle S\rangle\subset{\mathcal{T}}$ the
smallest triangulated and by $\mathbf{thick}(S)\subset{\mathcal{T}}$ the
smallest thick (i.e., triangulated and closed under direct summands)
subcategory of ${\mathcal{T}}$ containing $S$.

######  Definition 1.4.

Let $f\colon{\mathcal{X}}\to B$ be a smoothing of $X$ and let $\upiota\colon
X\hookrightarrow{\mathcal{X}}$ be the embedding of the central fiber. We say
that a triangulated subcategory
${\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$ provides a deformation
absorption of singularities or provides a thick deformation absorption,
respectively, (with respect to the smoothing ${\mathcal{X}}$) if
${\mathcal{P}}$ absorbs singularities of $X$ and the triangulated subcategory

$\langle\upiota_{*}{\mathcal{P}}\rangle\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})\qquad\text{or}\qquad\mathbf{thick}(\upiota_{*}{\mathcal{P}})\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$

generated or thickly generated in ${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$
by the pushforwards of objects of ${\mathcal{P}}$ is admissible.

We say that ${\mathcal{P}}$ provides a universal (thick) deformation
absorption of singularities of $X$ if the corresponding property holds for any
smoothing of $X$.

If ${\mathcal{P}}$ provides a deformation absorption, so that
$\langle\upiota_{*}{\mathcal{P}}\rangle\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$
is admissible, then
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})=\langle\upiota_{*}{\mathcal{P}}\rangle$,
so we will uniformly use the notation
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})$ for the resulting admissible
subcategory of ${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$.

To clarify the definition note that the category
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})$ is contained in
${\mathbf{D}}^{\mathrm{b}}_{X}({\mathcal{X}})$, the subcategory of
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ formed by objects with set-
theoretical support on $X$; moreover, we prove in [KS22b, Theorem 1.1],
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})$ is always admissible in
${\mathbf{D}}^{\mathrm{b}}_{X}({\mathcal{X}})$. Thus, the deformation
absorption property amounts to the assumption that
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})$ keeps being admissible in the
larger category ${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$.

Using base change technique [Kuz11] it is easy to see that Definition 1.4
leads to the following result.

######  Theorem 1.5.

Let $X$ be a projective variety and let
${\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$ provide a (thick)
deformation absorption of singularities of $X$ with respect to a smoothing
$f\colon{\mathcal{X}}\to B$. Then the subcategory

${\mathcal{D}}\coloneqq{}^{\perp}(\mathbf{thick}(\upiota_{*}{\mathcal{P}}))\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$

is $B$-linear and there is a $B$-linear semiorthogonal decomposition

${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle\mathbf{thick}(\upiota_{*}{\mathcal{P}}),{\mathcal{D}}\rangle.$

Moreover, the central fiber ${\mathcal{D}}_{o}$ of ${\mathcal{D}}$ can be
described as

(4)
${\mathcal{D}}_{o}\simeq{}^{\perp}{\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X),$

and if $b\neq o$ then
${\mathcal{D}}_{b}\simeq{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{b})$. In
particular, ${\mathcal{D}}$ is smooth and proper over $B$.

######  Remark 1.6.

In the situation of Example 1.2(b), if a smoothing of $X$ is given by a conic
bundle ${\mathcal{X}}/B$ obtained from a ${\mathbb{P}}^{1}$-bundle
${\mathcal{Y}}/B$ as
${\mathcal{X}}=\operatorname{\mathrm{Bl}}_{y_{0}}({\mathcal{Y}})$ with
exceptional divisor $E$, then the object $\upiota_{*}{\mathcal{O}}_{E}(-1)$ is
exceptional on ${\mathcal{X}}$, hence the subcategory it generates in
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is admissible. This means that the
subcategory
${\mathcal{P}}=\langle{\mathcal{O}}_{E}(-1)\rangle\subset{\mathbf{D}}^{\mathrm{b}}(X)$
is an example of a subcategory providing a universal deformation absorption of
singularities. Furthermore, in this case the semiorthogonal decomposition (3)
implies that for the subcategory
${\mathcal{D}}\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ defined in
Theorem 1.5 we have
${\mathcal{D}}\simeq{\mathbf{D}}^{\mathrm{b}}({\mathcal{Y}})$.

Next we observe that the deformation absorption phenomenon of Example 1.2(b)
is not specific to the concrete geometric situation, but rather a consequence
of the equivalence (2). To state this result we introduce the following
definition

######  Definition 1.7.

Let $q\geq 1$. An object ${\mathrm{P}}\in{\mathbf{D}}^{\mathrm{b}}(X)$ is
called a ${\mathbb{P}}^{\infty,{q}}$-object if

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{P}},{\mathrm{P}})\cong{\Bbbk}[\uptheta],$

where $\deg(\uptheta)=q$. When we do not want to specify the parameter $q$, we
just say that ${\mathrm{P}}$ is a ${\mathbb{P}}^{\infty}$-object.

Since $\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{P}},{\mathrm{P}})$ is
infinite-dimensional, ${\mathbb{P}}^{\infty}$-objects can only exist on
singular varieties. In the body of the paper we define
${\mathbb{P}}^{\infty,{q}}$-objects in the general situation of arbitrary
triangulated categories, and then the definition becomes a bit more technical
(see Definition 2.6), but in the geometric situation it is equivalent to the
one stated above (see Remark 2.7). Our main emphasis is on
${\mathbb{P}}^{\infty,{1}}$\- and ${\mathbb{P}}^{\infty,{2}}$-objects,
because, as we will see, they are geometrically meaningful; moreover, we prove
that ${\mathbb{P}}^{\infty,{q}}$-objects with $q\neq 1,2$ do not exist on
smoothable varieties, see Corollary 4.6.

As (2) shows, the situation of Example 1.2(b) corresponds to a
${\mathbb{P}}^{\infty,{2}}$-object. In this case we have the following result,
which we formulate in a slightly more general situation.

######  Theorem 1.8.

Let $X$ be a projective variety. Let
$({\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r})$ be a semiorthogonal collection of
${\mathbb{P}}^{\infty,{2}}$-objects in ${\mathbf{D}}^{\mathrm{b}}(X)$ such
that the subcategory

${\mathcal{P}}\coloneqq\langle{\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r}\rangle$

absorbs singularities of $X$. Then for any smoothing ${\mathcal{X}}\to B$ of
$X$ the collection
$\upiota_{*}{\mathrm{P}}_{1},\dots,\upiota_{*}{\mathrm{P}}_{r}$ in
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is exceptional. In particular,
${\mathcal{P}}$ provides a universal deformation absorption of singularities
of $X$.

If we combine Theorem 1.8 with Theorem 1.5, we obtain a smooth and proper over
$B$ category ${\mathcal{D}}$, and following the analogy of Remark 1.6 we
consider passing from ${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ to
${\mathcal{D}}$ as a categorical incarnation of the contraction
${\mathcal{X}}\to{\mathcal{Y}}$.

We have a similar generalization of the situation of Example 1.2(a), which by
(1) corresponds to a ${\mathbb{P}}^{\infty,{1}}$-object. Again, we state the
result in a slightly more general situation.

######  Theorem 1.9.

Let $X$ be a projective variety. Let
$({\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r})$ be a semiorthogonal collection of
${\mathbb{P}}^{\infty,{1}}$-objects in ${\mathbf{D}}^{\mathrm{b}}(X)$ such
that the subcategory

${\mathcal{P}}\coloneqq\langle{\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r}\rangle$

absorbs singularities of $X$. Then for any smoothing ${\mathcal{X}}\to B$ of
$X$ after an étale base change there are $r$ double coverings
${\mathcal{Z}}_{i}\to B$, $1\leq i\leq r$, étale over $B\setminus\\{o\\}$, and
a $B$-linear semiorthogonal decomposition

${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{1}),\dots,{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{r}),{\mathcal{D}}\rangle,$

where ${\mathcal{D}}$ is smooth and proper over $B$,
${\mathcal{D}}_{o}={}^{\perp}{\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$,
and
${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i})_{o}=\langle{\mathrm{P}}_{i}\rangle$.
Moreover, a semiorthogonal decomposition with these properties is unique
Zariski locally around $o\in B$.

In the paper, besides proving the above results (see §4.2), we discuss
possible approaches to absorbing singularities by
${\mathbb{P}}^{\infty,{q}}$-objects for $q\in\\{1,2\\}$, and obtaining smooth
and proper families of categories out of them. One possibility here is to use
the intrinsic structure of the category $\langle{\mathrm{P}}\rangle$ generated
by a ${\mathbb{P}}^{\infty,{q}}$-object; we call such a category the
categorical ordinary double point and discuss its properties in detail in §2.

Another approach is to start with a resolution of singularities
$\pi\colon{\tilde{X}}\to X$ and find appropriate admissible subcategory
${\tilde{{\mathcal{P}}}}\subset{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$ such
that $\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\tilde{{\mathcal{P}}}}$ and
the Verdier localization

${\mathcal{P}}\coloneqq{\tilde{{\mathcal{P}}}}/\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\mathbf{D}}^{\mathrm{b}}(X)$

is equivalent to a categorical ordinary double point (or has a semiorthogonal
decomposition into several categorical ordinary double points); then
${\mathcal{P}}^{\perp}\simeq{\tilde{{\mathcal{P}}}}^{\perp}$ (see Theorem 4.2)
is equivalent to an admissible subcategory in the derived category of a smooth
and proper variety ${\tilde{X}}$, hence it is smooth and proper, and hence
${\mathcal{P}}$ absorbs singularities of $X$. This approach works well when
$\operatorname{\mathrm{Ker}}(\pi_{*})$ is generated by a _spherical object_
${\mathrm{K}}\in{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$ and in this case one
can choose the category ${\tilde{{\mathcal{P}}}}$ as a subcategory generated
by an exceptional pair $({\mathcal{E}},{\mathcal{E}}^{\prime})$, where
${\mathcal{E}}\in{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$ is an exceptional
object such that

$\dim\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},{\mathrm{K}})=1$

(we say in this case that ${\mathcal{E}}$ is adherent to ${\mathrm{K}}$, see
Definition 3.9; this generalizes the notion introduced in [KKS22]) and
${\mathcal{E}}^{\prime}$ is obtained from ${\mathcal{E}}$ by the spherical
twist functor associated with ${\mathrm{K}}$. We check in Lemma 3.10 that in
this case the pair $({\mathcal{E}},{\mathcal{E}}^{\prime})$ is indeed
exceptional with

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},{\mathcal{E}}^{\prime})\cong{\Bbbk}\oplus{\Bbbk}[-q],\qquad
q\geq 1.$

We call the category generated by such a pair the graded Kronecker quiver
category and discuss its properties in §3. The main result here is Proposition
3.7, where we show that the Verdier localization of the graded Kronecker
quiver category by the spherical object ${\mathrm{K}}$ is equivalent to the
categorical ordinary double point.

However, if $\pi\colon{\tilde{X}}\to X$ is a resolution, the condition that
$\operatorname{\mathrm{Ker}}(\pi_{*})$ is generated by spherical objects is
very restrictive, and for most singularities such resolutions do not exist.
So, to circumvent this problem we replace a geometric resolution by a
_categorical resolution of singularities_ in the sense of [Kuz08, KL15]. In
other words, we consider a smooth and proper triangulated category
${\tilde{{\mathcal{T}}}}$ and a functor

$\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathbf{D}}^{\mathrm{b}}(X)$

that behaves as the derived pushforward for a geometric resolution. There are
several ways to make this assumption precise: one is to assume that $\pi_{*}$
is a Verdier localization, but we prefer a slightly weaker assumption, that
was introduced by Efimov in [Efi20].

######  Definition 1.10.

We say that a functor $\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathcal{T}}$
is a categorical contraction if it is a localization up to direct summands,
i.e., if
$\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to\operatorname{\mathrm{Im}}(\pi_{*})$
is a Verdier localization and
$\operatorname{\mathrm{Im}}(\pi_{*})\subset{\mathcal{T}}$ is a dense
subcategory.

######  Remark 1.11.

In [Efi20, Definition 3.7] the same notion is called a localization. We find
this confusing and to avoid possible misunderstanding change the terminology.

As we mentioned above, it is useful for our goals to have a categorical
contraction
$\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathbf{D}}^{\mathrm{b}}(X)$ such
that the subcategory
$\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\tilde{{\mathcal{T}}}}$ is
generated by spherical objects. Using Serre duality we observe that this
condition implies the following property, discussed in more detail in [KS22b,
§5] (in fact, the definition of crepancy [KS22b, Definition 5.6] is different
from the definition given below, but [KS22b, Lemma 5.7] says that these
definitions are equivalent when ${\tilde{{\mathcal{T}}}}$ is smooth and
proper).

######  Definition 1.12.

We say that a categorical contraction
$\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathcal{T}}$ from a smooth and
proper category ${\tilde{{\mathcal{T}}}}$ is crepant if the orthogonals
$\operatorname{\mathrm{Ker}}(\pi_{*})^{\perp}\subset{\tilde{{\mathcal{T}}}}$
and
${}^{\perp}\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\tilde{{\mathcal{T}}}}$
of the kernel
$\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\tilde{{\mathcal{T}}}}$ coincide.

In the last part of the paper we show that the approach to construction of
absorption of singularities via crepant categorical contractions can be made
completely effective. Actually, in §5 for a variety $X$ of dimension at least
two whose singularities are _ordinary double points_ we construct an
admissible subcategory
${\mathcal{D}}\subset{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$ in the derived
category of the blowup $\pi\colon{\tilde{X}}\to X$ of the singular locus of
$X$ such that the restriction
$\pi_{*}|_{\mathcal{D}}\colon{\mathcal{D}}\to{\mathbf{D}}^{\mathrm{b}}(X)$ of
the pushforward functor for the blowup morphism is a crepant categorical
contraction with $\operatorname{\mathrm{Ker}}(\pi_{*}|_{\mathcal{D}})$
generated by spherical objects, see Theorem 5.8. When this paper was finished
we learned about a work in preparation [CGL+22], where the same crepant
categorical resolutions of nodal varieties have been independently
constructed.

In the last section, §6, we combine the above results by making explicit the
adherence condition in the constructed crepant categorical resolution of a
nodal variety $X$ (see Theorem 6.1). In particular, we show that if the
adherence condition is satisfied then ${\mathbf{D}}^{\mathrm{b}}(X)$ has a
semiorthogonal collection of ${\mathbb{P}}^{\infty,{p+1}}$-objects absorbing
singularities of $X$, where $p\in\\{0,1\\}$ is the _parity_ of $\dim(X)$. This
allows us to apply Theorem 1.9 when $\dim(X)$ is even, or Theorem 1.8 when
$\dim(X)$ is odd. As a baby example we discuss in Proposition 6.4 the case of
nodal quadrics.

Furthermore, in §6.2 we show that the singularity category of a variety that
admits an absorption by categorical ordinary double points is idempotent
complete, and using this observation we deduce explicit necessary conditions
for the existence of such absorption for nodal varieties of dimension at most
$3$, see Proposition 6.12. In the case of threefolds this condition is the
_maximal nonfactoriality_ condition from [KPS21], see Definition 6.10. We also
check in Corollary 6.18 that for a projective threefold $X$ with
${\mathrm{H}}^{>0}(X,{\mathcal{O}}_{X})=0$ and a single ordinary double point
the maximal nonfactoriality condition is also sufficient for the existence of
absorption.

We conclude the paper with a few geometric applications of our results in
§6.3. Namely, we consider nodal curves (§6.3.1), nodal threefolds in general
(§6.3.2), and nodal quintic del Pezzo threefolds in particular (§6.3.3). Even
more interesting examples of prime Fano threefolds will be discussed in a
separate companion paper [KS22a].

When we worked on this paper, one of our principles has been to separate the
geometric aspects from the categorical ones, and to a large extent §§2–3 are
categorical, §4 combines categories with geometry, while §§5–6 are mostly
geometric. On the other hand, we managed to develop the categorical aspects
much further than our geometric applications required, and we feel these
developments are interesting and might be useful for further research; curious
readers can find these in [KS22b].

In fact, [KS22b] includes a systematic treatment of the notion of
homologically finite objects, briefly mentioned in Lemma 2.10, with an
emphasis on the relation between semiorthogonal decompositions of a
triangulated category and the category of its homologically finite-dimensional
objects, that leads to a relation between semiorthogonal decompositions of
${\mathbf{D}}^{\mathrm{b}}(X)$ and ${\mathbf{D}}^{\mathrm{perf}}(X)$, a
development of the concept of categorical contractions and crepancy, and a
generalization of the relation between ${\mathbb{P}}^{\infty,{2}}$-objects on
the special fiber $X$ and exceptional objects on the total space
${\mathcal{X}}$ of a smoothing observed in Theorem 1.8 to any admissible
subcategories of ${\mathbf{D}}^{\mathrm{b}}(X)$.

We would also like to mention that the results of the present paper are
related to the big project aimed at the study of semiorthogonal decompositions
of algebraic varieties in degenerating families. We refer to [Kuz21b] for a
survey of other results in this direction.

#### Notation

Throughout the paper $\Bbbk$ denotes a base field. All schemes are assumed to
be of finite type over $\Bbbk$, and all algebras and categories we work with
are $\Bbbk$-linear.

We write $C^{\mathrm{op}}$ and ${\mathcal{T}}^{\mathrm{op}}$ for the opposite
dg-algebra or category. For a dg-algebra $C$ we denote by

  * •

${\mathbf{D}}(C)$ the unbounded derived category of right dg-modules over $C$;

  * •

${\mathbf{D}}^{\mathrm{perf}}(C)\subset{\mathbf{D}}(C)$ the subcategory of
perfect dg-modules;

  * •

${\mathbf{D}}^{\mathrm{b}}(C)\subset{\mathbf{D}}(C)$ the subcategory of dg-
modules with finite-dimensional total cohomology.

Similarly, for a projective scheme $X$ we write
${\mathbf{D}}_{\mathrm{qc}}(X)$ for the unbounded derived category of
quasicoherent sheaves, while
${\mathbf{D}}^{\mathrm{perf}}(X)\subset{\mathbf{D}}_{\mathrm{qc}}(X)$ and
${\mathbf{D}}^{\mathrm{b}}(X)\subset{\mathbf{D}}_{\mathrm{qc}}(X)$ stand for
the category of perfect complexes and the bounded derived category of coherent
sheaves, respectively.

We write
${\mathcal{T}}=\langle{\mathcal{A}}_{1},\dots,{\mathcal{A}}_{m}\rangle$ for a
semiorthogonal decomposition with components
${\mathcal{A}}_{1},\dots,{\mathcal{A}}_{m}$, and we write
${\mathbf{L}}_{{\mathcal{A}}_{i}}$, ${\mathbf{R}}_{{\mathcal{A}}_{i}}$ for the
left and right mutation functors with respect to the component
${\mathcal{A}}_{i}$, provided it is both left and right admissible.

All pullback, pushforward, and tensor product functors are derived (unless
specified otherwise).

#### Acknowledgements

We would like to thank Sasha Efimov, Haibo Jin, Martin Kalck, Shinnosuke
Okawa, Dima Orlov, Alex Perry, Yura Prokhorov, Michael Wemyss for their help
and interest in this work.

##  2\. Categorical ordinary double points and ${\mathbb{P}}^{\infty}$-objects

In this section we define categorical ordinary double points and
${\mathbb{P}}^{\infty,{q}}$-objects in arbitrary triangulated categories, and
explain how to construct fully faithful functors with admissible image from a
categorical ordinary double point into a given triangulated category. Here we
work over an arbitrary field ${\Bbbk}$.

###  2.1. Categorical ordinary double points

For $p\geq 0$ and $q\geq 1$ consider the following differential graded
algebras

$\displaystyle{{\mathsf{A}}}_{p}$
$\displaystyle\coloneqq\Bbbk[\upepsilon]/(\upepsilon^{2}),$
$\displaystyle\qquad\deg(\upepsilon)$ $\displaystyle=-p,$
$\displaystyle\qquad{\mathrm{d}}(\upepsilon)$ $\displaystyle=0,$
$\displaystyle{{\mathsf{B}}}_{q}$ $\displaystyle\coloneqq\Bbbk[\uptheta],$
$\displaystyle\qquad\deg(\uptheta)$ $\displaystyle=q,$
$\displaystyle\qquad{\mathrm{d}}(\uptheta)$ $\displaystyle=0.$

As the differentials are zero, we will sometimes consider these algebras
simply as graded algebras.

In this subsection we study the categories of dg-modules over
${\mathsf{A}}_{p}$ and ${\mathsf{B}}_{q}$ and their relation. Recall that a
graded algebra $C$ is called intrinsically formal if for every dg-algebra
$C^{\prime}$ such that ${\mathrm{H}}^{\bullet}(C^{\prime})\cong C$ there is a
quasiisomorphism $C\cong C^{\prime}$. The following lemma is well-known (see,
e.g., [KYZ09, §2]).

######  Lemma 2.1.

For $p\geq 0$ and $q\geq 1$ the graded algebras ${\mathsf{A}}_{p}$ and
${\mathsf{B}}_{q}$ are intrinsically formal.

###### Proof.

For ${\mathsf{B}}_{q}$ there is a standard argument. Let $B$ be any dg-algebra
with ${\mathrm{H}}^{\bullet}(B)\cong{\Bbbk}[\uptheta]$. Let $\theta\in B^{q}$
be any lift of $\uptheta\in{\mathrm{H}}^{q}(B)$. Since ${\mathsf{B}}_{q}$ is a
free associative (graded) algebra, there is a unique homomorphism of graded
algebras ${\mathsf{B}}_{q}\to B$, $\uptheta\mapsto\theta$. It follows from
Leibniz rule that it is a homomorphism of dg-algebras, and it is obvious that
it induces an isomorphism on the cohomology; therefore it is a
quasiisomorphism.

For ${\mathsf{A}}_{p}$ we argue as follows. First, any dg-algebra $A$ with
${\mathrm{H}}^{\bullet}(A)\cong{\Bbbk}[\upepsilon]/(\upepsilon^{2})$
corresponds to an $A_{\infty}$-structure on
${\Bbbk}[\upepsilon]/(\upepsilon^{2})$. But for $n\geq 3$ the degree of the
operation ${\mathbf{m}}_{n}$ is negative, hence
${\mathbf{m}}_{n}(\upepsilon,\dots,\upepsilon)=0$, hence any
$A_{\infty}$-structure on $A$ is trivial, so that $A$ is quasiisomorphic to
its cohomology. ∎

Recall that

  * •

a dg-algebra $C$ is proper if $\dim({\mathrm{H}}^{\bullet}(C))<\infty$, where
${\mathrm{H}}^{\bullet}(C)$ is the total cohomology of $C$;

  * •

a dg-algebra $C$ is smooth if the diagonal bimodule $\Delta_{C}$ is _perfect_
, i.e., $\Delta_{C}$ belongs to the idempotent completion of the triangulated
subcategory of the derived category ${\mathbf{D}}(C\otimes C^{\mathrm{op}})$
generated by the free bimodule $C\otimes C^{\mathrm{op}}$.

Here and below tensor products of dg-algebras are taken over $\Bbbk$.

Recall that
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})\subset{\mathbf{D}}({\mathsf{A}}_{p})$
and
${\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})\subset{\mathbf{D}}({\mathsf{B}}_{q})$
are the categories of dg-modules over respective dg-algebras which have total
finite-dimensional cohomology over ${\Bbbk}$. We also denote by
${\Bbbk}_{\mathsf{A}}$ and ${\Bbbk}_{\mathsf{B}}$ the simple (1-dimensional)
dg-modules over ${\mathsf{A}}_{p}$ and ${\mathsf{B}}_{q}$, respectively.

######  Proposition 2.2.

Let $p\geq 0$ and $q\geq 1$.

  1. (i)

The dg-algebra ${\mathsf{B}}_{q}$ is smooth (but not proper) and
$\langle{\Bbbk}_{\mathsf{B}}\rangle={\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})\simeq{\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{q-1})$.

  2. (ii)

The dg-algebra ${\mathsf{A}}_{p}$ is proper (but not smooth) and
$\langle{\Bbbk}_{\mathsf{A}}\rangle={\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})\simeq{\mathbf{D}}^{\mathrm{perf}}({\mathsf{B}}_{p+1})$.

Moreover, the categories ${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})$ and
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{B}}_{q})$ are generated by the free
modules ${\mathsf{A}}_{p}$ and ${\mathsf{B}}_{q}$ as triangulated categories,
i.e., without additional idempotent completion.

###### Proof.

(i) Smoothness of the dg-algebra ${\mathsf{B}}_{q}$ is evident: the diagonal
bimodule can be written as

$\Delta_{{\mathsf{B}}_{q}}\cong\operatorname{\mathrm{Cone}}({\mathsf{B}}_{q}\otimes{\mathsf{B}}_{q}[-q]\xrightarrow{\
\uptheta^{\prime}-\uptheta^{\prime\prime}\
}{\mathsf{B}}_{q}\otimes{\mathsf{B}}_{q}),$

where $\uptheta^{\prime}$ and $\uptheta^{\prime\prime}$ are the generators of
the first and second factor, respectively.

The containment
$\langle{\Bbbk}_{\mathsf{B}}\rangle\subset{\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})$
is evident and to prove the converse containment let
$M\in{\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})$. We prove
$M\in\langle{\Bbbk}_{\mathsf{B}}\rangle$ by induction on
$\dim({\mathrm{H}}^{\bullet}(M))$. Let $i$ be the maximal integer such that
${\mathrm{H}}^{i}(M)\neq 0$ (it exists because ${\mathrm{H}}^{\bullet}(M)$ is
finite-dimensional). Using the isomorphism

(5)
${\Bbbk}_{\mathsf{B}}\cong\operatorname{\mathrm{Cone}}({\mathsf{B}}_{q}[-q]\xrightarrow{\
\uptheta\ }{\mathsf{B}}_{q})$

and
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{B}}_{q}}({\mathsf{B}}_{q},M)={\mathrm{H}}^{\bullet}(M)$,
we obtain an exact sequence

$\dots\to\operatorname{\mathrm{Ext}}^{i}({\Bbbk}_{\mathsf{B}},M)\to{\mathrm{H}}^{i}(M)\to{\mathrm{H}}^{i+q}(M)\to\dots$

Since $q>0$, the group ${\mathrm{H}}^{i+q}(M)$ is zero, hence any class from
${\mathrm{H}}^{i}(M)$ lifts to
$\operatorname{\mathrm{Ext}}^{i}({\Bbbk}_{\mathsf{B}},M)$. Clearly, the
corresponding morphism ${\Bbbk}_{\mathsf{B}}[-i]\to{\mathrm{M}}$ induces an
embedding on ${\mathrm{H}}^{i}$. If $M^{\prime}$ is the cone of this morphism,
we have
$\dim({\mathrm{H}}^{\bullet}(M^{\prime}))=\dim({\mathrm{H}}^{\bullet}(M))-1$,
by induction $M^{\prime}\in\langle{\Bbbk}_{\mathsf{B}}\rangle$, hence also
$M\in\langle{\Bbbk}_{\mathsf{B}}\rangle$. Thus
${\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})=\langle{\Bbbk}_{\mathsf{B}}\rangle$.

Finally, using the above computation for $M={\Bbbk}_{\mathsf{B}}$ we obtain

$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{B}}_{q}}({\Bbbk}_{\mathsf{B}},{\Bbbk}_{\mathsf{B}})\cong{\Bbbk}\oplus{\Bbbk}[q-1],$

and denoting by $\upepsilon$ a generator of the second summand it is easy to
see that $\upepsilon^{2}=0$. Now, by Keller’s Morita Theory [Kel94] we have
${\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})\simeq{\mathbf{D}}^{\mathrm{perf}}(\operatorname{\mathrm{RHom}}({\Bbbk}_{\mathsf{B}},{\Bbbk}_{\mathsf{B}}))$
and the proof is finished since Lemma 2.1 implies that
$\operatorname{\mathrm{RHom}}({\Bbbk}_{\mathsf{B}},{\Bbbk}_{\mathsf{B}})$ is
quasiisomorphic to ${\mathsf{A}}_{q-1}$.

(ii) The dg-algebra ${\mathsf{A}}_{p}$ is proper because it is finite-
dimensional, and the equality
$\langle{\Bbbk}_{\mathsf{A}}\rangle={\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$
can be obtained easily by using the standard t-structure (constructed, e.g.,
in [HKM02, Theorem 1.3]). Thus, it remains to check that
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\cong{\mathsf{B}}_{p+1}$
and apply Lemma 2.1 as in (i). We postpone the computation of
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})$
till Lemma 2.4 below.

The last statement follows from the fact that
${\mathbf{D}}^{\mathrm{b}}({\mathsf{B}}_{q})$ and
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ are generated by the simple
modules ${\Bbbk}_{\mathsf{B}}$ and ${\Bbbk}_{\mathsf{A}}$, without additional
idempotent completions, and from the equivalences of (ii) and (i), since under
these equivalences simple modules go to free modules. ∎

The module ${\Bbbk}_{\mathsf{A}}$ is not contained in
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})$, and thus cannot be directly
expressed in terms of ${\mathsf{A}}_{p}$; to compute
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})$
we use interpretation of ${\Bbbk}_{\mathsf{A}}$ as homotopy colimit. Recall
from [BN93] that the homotopy colimit of an infinite chain of morphisms
$F_{1}\xrightarrow{\ \phi_{1}\ }F_{2}\xrightarrow{\ \phi_{2}\ }\dots$ in a
_cocomplete_ (i.e., admitting arbitrary direct sums) triangulated category is
defined as

$\operatorname{\mathrm{hocolim}}F_{i}\coloneqq\operatorname{\mathrm{Cone}}\left(\bigoplus_{i=0}^{\infty}F_{i}\xrightarrow{\
\operatorname{\mathrm{id}}-\mathop{\oplus}\limits_{i=1}^{\infty}\phi_{i}\
}\bigoplus_{i=0}^{\infty}F_{i}\right).$

Let ${\widehat{{\mathcal{T}}}}$ be a cocomplete triangulated category. Recall
that a set of objects $G_{\alpha}\in{\widehat{{\mathcal{T}}}}$ is a set of
compact generators if the functors $\operatorname{\mathrm{Hom}}(G_{\alpha},-)$
commute with direct sums for each $\alpha$ and the right orthogonal of all
$G_{\alpha}$ in ${\widehat{{\mathcal{T}}}}$ is zero, i.e.,
$\operatorname{\mathrm{Ext}}^{\bullet}(G_{\alpha},F)=0$ for
$F\in{\widehat{{\mathcal{T}}}}$ and all $\alpha$ implies $F=0$.

######  Lemma 2.3.

Let ${\widehat{{\mathcal{T}}}}$ be a cocomplete triangulated category with a
set of compact generators $G_{\alpha}\in{\widehat{{\mathcal{T}}}}$. If for an
object $F\in{\widehat{{\mathcal{T}}}}$ the complex
$\operatorname{\mathrm{Ext}}^{\bullet}(G_{\alpha},F)$ is bounded above for
each $\alpha$ then $\operatorname{\mathrm{hocolim}}F[s_{i}]=0$ for any
increasing sequence $\\{s_{i}\\}$ of integers and any chain of morphisms
$F[s_{1}]\to F[s_{2}]\to\dots$.

###### Proof.

Since $G_{\alpha}$ is compact in ${\widehat{{\mathcal{T}}}}$, the functor
$\operatorname{\mathrm{RHom}}(G_{\alpha},-)$ commutes with homotopy colimits
by [Kuz11, Lemma 2.11], hence [BN93, Remark 2.2] gives for each
$n\in{\mathbb{Z}}$ an isomorphism

$\operatorname{\mathrm{Ext}}^{n}(G_{\alpha},\operatorname{\mathrm{hocolim}}F[s_{i}])\cong\operatorname{\mathrm{colim}}\operatorname{\mathrm{Ext}}^{n}(G_{\alpha},F[s_{i}]).$

The right-hand side is zero because
$\operatorname{\mathrm{Ext}}^{\bullet}(G_{\alpha},F)$ is bounded above. Thus
$\operatorname{\mathrm{hocolim}}F[s_{i}]$ is right orthogonal to all
$G_{\alpha}$, and therefore is zero. ∎

Note that by definition of ${\mathsf{A}}_{p}$ we have a distinguished triangle

(6)
${\mathsf{A}}_{p}\xrightarrow{\quad}{\Bbbk}_{\mathsf{A}}\xrightarrow{\quad}{\Bbbk}_{\mathsf{A}}[p+1].$

The computation of
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})$
is based on using (6) iteratively and passing to the homotopy colimit.

######  Lemma 2.4.

Let $p\geq 0$.

  1. (i)

There is a chain of morphisms
${\mathsf{A}}_{p}={\mathsf{A}}_{p}^{(1)}\to{\mathsf{A}}_{p}^{(2)}\to\dots$ in
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})$ such that

${\mathsf{A}}_{p}^{(i+1)}=\operatorname{\mathrm{Cone}}\Big{(}{\mathsf{A}}_{p}[i(p+1)-1]\to{\mathsf{A}}_{p}^{(i)}\Big{)},$

and a sequence of morphisms ${\mathsf{A}}_{p}^{(i)}\to{\Bbbk}_{\mathsf{A}}$
compatible with the chain maps
${\mathsf{A}}_{p}^{(i-1)}\to{\mathsf{A}}_{p}^{(i)}$ such that

(7)
$\operatorname{\mathrm{Cone}}\Big{(}{\mathsf{A}}_{p}^{(i)}\to{\Bbbk}_{\mathsf{A}}\Big{)}\cong{\Bbbk}_{\mathsf{A}}[i(p+1)].$

  2. (ii)

In ${\mathbf{D}}({\mathsf{A}}_{p})$ one has
$\operatorname{\mathrm{hocolim}}{\Bbbk}_{\mathsf{A}}[i(p+1)]=0$ and
$\operatorname{\mathrm{hocolim}}{\mathsf{A}}_{p}^{(i)}\cong{\Bbbk}_{\mathsf{A}}$.

  3. (iii)

One has
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\cong{\mathsf{B}}_{p+1}$.

###### Proof.

(i) We construct the chain inductively. Consider the composition

${\mathsf{A}}_{p}[i(p+1)-1]\to{\Bbbk}_{\mathsf{A}}[i(p+1)-1]\to{\mathsf{A}}_{p}^{(i)}$

where the first arrow is the (shifted) morphism from (6) and the second arrow
is the connecting morphism in (7), and define ${\mathsf{A}}_{p}^{(i+1)}$ as
the cone of the composition
${\mathsf{A}}_{p}[i(p+1)-1]\to{\mathsf{A}}_{p}^{(i)}$. Then (7) for $i+1$
follows easily from the octahedral axiom.

(ii) As $p+1>0$, we have
$\operatorname{\mathrm{hocolim}}{\Bbbk}_{\mathsf{A}}[i(p+1)]=0$ by Lemma 2.3,
because
$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\mathsf{A}}_{p},{\Bbbk}_{\mathsf{A}})\cong{\Bbbk}$
is bounded above and ${\mathsf{A}}_{p}$ is a compact generator for
${\mathbf{D}}({\mathsf{A}}_{p})$. To compute the second homotopy colimit we
apply the functor $\operatorname{\mathrm{hocolim}}$ to the chain of morphisms
${\mathsf{A}}_{p}^{(i)}\to{\Bbbk}_{\mathsf{A}}$ constructed in (i). From (7)
and the exactness of homotopy colimit we deduce a distinguished triangle

$\operatorname{\mathrm{hocolim}}{\mathsf{A}}_{p}^{(i)}\to\operatorname{\mathrm{hocolim}}{}{\Bbbk}_{\mathsf{A}}\to\operatorname{\mathrm{hocolim}}{\Bbbk}_{\mathsf{A}}[i(p+1)],$

and since the last term was shown to be zero, we have
$\operatorname{\mathrm{hocolim}}{\mathsf{A}}_{p}^{(i)}\cong\operatorname{\mathrm{hocolim}}{\Bbbk}_{\mathsf{A}}\cong{}{\Bbbk}_{\mathsf{A}}$.

(iii) Applying the functor
$\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}(-,{\Bbbk}_{\mathsf{A}})$
to the triangle
${\mathsf{A}}_{p}^{(i)}\to{\mathsf{A}}_{p}^{(i+1)}\to{\mathsf{A}}_{p}[i(p+1)]$
we deduce that

$\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}({\mathsf{A}}_{p}^{(i)},{\Bbbk}_{\mathsf{A}})\cong{\Bbbk}[\uptheta]/\uptheta^{i},\qquad\text{where}\qquad\deg(\uptheta)=p+1.$

Now passing to the homotopy colimit and using the isomorphism of (ii), we
conclude that

$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\cong\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}(\operatorname{\mathrm{hocolim}}{\mathsf{A}}_{p}^{(i)},{\Bbbk}_{\mathsf{A}})\cong\operatorname{\mathrm{holim}}\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{p}}({\mathsf{A}}_{p}^{(i)},{\Bbbk}_{\mathsf{A}})\cong\operatorname{\mathrm{holim}}{\Bbbk}[\uptheta]/\uptheta^{i},$

where $\operatorname{\mathrm{holim}}$ is the homotopy limit defined as
$\operatorname{\mathrm{holim}}F_{i}\coloneqq\operatorname{\mathrm{Cone}}(\prod
F_{i}\to\prod F_{i})[-1]$, analogously to homotopy colimit.

It follows that
$\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\cong{\Bbbk}[\uptheta]$
as graded vector spaces, and it remains to identify the algebra structure. For
this, using (6), we obtain a distinguished triangle

$\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})[-p-1]\xrightarrow{\
\uptheta\
}\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\xrightarrow{\quad}{\Bbbk},$

which implies that the multiplication by $\uptheta$ is injective, hence
$\operatorname{\mathrm{Ext}}_{{\mathsf{A}}_{p}}^{\bullet}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})\cong{\Bbbk}[\uptheta]$
as algebras. ∎

Later we will use the following simple observation.

######  Corollary 2.5.

The category ${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})$ is thickly
generated by ${\mathsf{A}}_{p}^{(2)}$, i.e.,
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})=\mathbf{thick}({\mathsf{A}}_{p}^{(2)})$.

###### Proof.

Recall the defining triangle ${\mathsf{A}}_{p}[p]\xrightarrow{\ \upepsilon\
}{\mathsf{A}}_{p}\xrightarrow{\quad}{\mathsf{A}}_{p}^{(2)}$ and consider the
composition

${\mathsf{A}}_{p}^{(2)}[-1-p]\to{\mathsf{A}}_{p}\to{\mathsf{A}}_{p}^{(2)},$

where the first arrow is a shift of the connecting morphism in the above
triangle. Using the octahedral axiom it is easy to see that its cone is
isomorphic to the cone of the morphism ${\mathsf{A}}_{p}[p]\xrightarrow{\
\upepsilon^{2}\ }{\mathsf{A}}_{p}[-p]$, and as $\upepsilon^{2}=0$, it is
isomorphic to ${\mathsf{A}}_{p}[-p]\oplus{\mathsf{A}}_{p}[p+1]$. Thus,
${\mathsf{A}}_{p}$ belongs to the thick subcategory generated by
${\mathsf{A}}_{p}^{(2)}$, and hence the latter thickly generates
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})$. ∎

###  2.2. ${\mathbb{P}}^{\infty}$-objects

As by Proposition 2.2 the category
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ is generated by the simple
module ${\Bbbk}_{\mathsf{A}}$, to embed
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})\simeq{\mathbf{D}}^{\mathrm{perf}}({\mathsf{B}}_{p+1})$
into another triangulated category ${\mathcal{T}}$ we need to find in
${\mathcal{T}}$ an object that could serve as ${\Bbbk}_{\mathsf{A}}$.
Comparing with the properties of ${\Bbbk}_{\mathsf{A}}$ proved in Lemma 2.4,
we arrive at the following

######  Definition 2.6.

Let $q\geq 1$. An object ${\mathrm{P}}$ in a triangulated category
${\mathcal{T}}$ is called a ${\mathbb{P}}^{\infty,{q}}$-object if

  1. (a)

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{P}},{\mathrm{P}})\cong{\Bbbk}[\uptheta]$,
where $\deg(\uptheta)=q$, and

  2. (b)

$\operatorname{\mathrm{hocolim}}({\mathrm{P}}\xrightarrow{\ \uptheta\
}{\mathrm{P}}[q]\xrightarrow{\ \uptheta\ }{\mathrm{P}}[2q]\xrightarrow{\
\uptheta\ }\dots)=0$,

where the homotopy colimit in (b) is taken in any cocomplete category
${\widehat{{\mathcal{T}}}}$ containing ${\mathcal{T}}$, e.g., in the derived
category ${\mathbf{D}}({\mathcal{T}})$ of ${\mathcal{T}}$. When we do not want
to specify the parameter $q$, we just say that ${\mathrm{P}}$ is a
${\mathbb{P}}^{\infty}$-object.

######  Remark 2.7.

When ${\mathcal{T}}={\mathbf{D}}^{\mathrm{b}}(X)$, the property (b) is
satisfied automatically by Lemma 2.3, because if
${\mathrm{P}}\in{\mathbf{D}}^{\mathrm{b}}(X)$ then
$\operatorname{\mathrm{Ext}}^{\bullet}(G,{\mathrm{P}})$ is bounded above for
any compact (i.e., perfect) object $G\in{\mathbf{D}}_{\mathrm{qc}}(X)$. On the
other hand, there are examples of objects for which (a) is true but (b) fails.
The simplest such example (maybe somewhat unexpectedly) is the free module
${\mathsf{B}}_{q}$ in
${\mathcal{T}}={\mathbf{D}}^{\mathrm{perf}}({\mathsf{B}}_{q})$. Indeed, in
this case

$\widehat{\mathsf{B}}_{q}\coloneqq\operatorname{\mathrm{hocolim}}({\mathsf{B}}_{q}\xrightarrow{\
\uptheta\ }{\mathsf{B}}_{q}[q]\xrightarrow{\ \uptheta\
}{\mathsf{B}}_{q}[2q]\xrightarrow{\ \uptheta\
}\dots)\cong{\Bbbk}[\uptheta,\uptheta^{-1}].$

On the other hand, one can check that the object
$\operatorname{\mathrm{Cone}}({\mathsf{B}}_{q}\to\widehat{\mathsf{B}}_{q})\cong\uptheta^{-1}{\Bbbk}[\uptheta^{-1}]$
is a ${\mathbb{P}}^{\infty,{q}}$-object.

######  Definition 2.8.

For any ${\mathbb{P}}^{\infty,{q}}$-object ${\mathrm{P}}\in{\mathcal{T}}$ we
define another object ${\mathrm{M}}$ from the distinguished triangle

(8) ${\mathrm{M}}\xrightarrow{\quad}{\mathrm{P}}\xrightarrow{\ \uptheta\
}{\mathrm{P}}[q].$

We call ${\mathrm{M}}\in{\mathcal{T}}$ the canonical self-extension of
${\mathrm{P}}$.

It follows easily from (8) and Definition 2.6(a) that

(9)
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}},{\mathrm{P}})\cong{\Bbbk}.$

Furthermore, the following computation is obvious.

######  Lemma 2.9.

If a pair of objects $({\mathrm{M}},{\mathrm{P}})$ in a triangulated category
${\mathcal{T}}$ satisfies (8) and (9) then

(10)
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}},{\mathrm{M}})\cong{\mathsf{A}}_{q-1}.$

In these terms we can state a criterion for a full embedding of a categorical
double point into a ${\Bbbk}$-linear dg-enhanced triangulated category
${\mathcal{T}}$. We will say that an object $F\in{\mathcal{T}}$ is left or
right homologically finite if $\operatorname{\mathrm{Ext}}^{\bullet}(F,T)$ or
$\operatorname{\mathrm{Ext}}^{\bullet}(T,F)$ is finite-dimensional over
${\Bbbk}$ for any $T\in{\mathcal{T}}$, respectively (cf. [KS22b, §4.1]).

######  Lemma 2.10.

Let ${\mathcal{T}}$ be a dg-enhanced triangulated category and let
$({\mathrm{M}},{\mathrm{P}})$ be a pair of objects in ${\mathcal{T}}$
satisfying (9) and such that
$\operatorname{\mathrm{Cone}}({\mathrm{M}}\to{\mathrm{P}})\cong{\mathrm{P}}[q]$
and $\operatorname{\mathrm{hocolim}}{\mathrm{P}}[iq]=0$ for the chain from
Definition 2.6(b).

  1. (i)

We have
$\operatorname{\mathrm{RHom}}_{\mathcal{T}}({\mathrm{M}},{\mathrm{M}})\cong{\mathsf{A}}_{q-1}$
and the functor

$\Phi_{\mathrm{M}}\colon{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})\to{\mathcal{T}},\qquad
F\mapsto F\otimes_{{\mathsf{A}}_{q-1}}{\mathrm{M}}$

is fully faithful, with
$\Phi_{\mathrm{M}}({\mathsf{A}}_{q-1})\cong{\mathrm{M}}$ and
$\Phi_{\mathrm{M}}({\Bbbk}_{\mathsf{A}})\cong{\mathrm{P}}$. Furthermore,

$\Phi_{\mathrm{M}}({\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1}))=\langle{\mathrm{P}}\rangle\subset{\mathcal{T}}\qquad\text{and}\qquad\Phi_{\mathrm{M}}({\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{q-1}))=\langle{\mathrm{M}}\rangle\subset{\mathcal{T}}.$

In particular, ${\mathrm{P}}$ is a ${\mathbb{P}}^{\infty,{q}}$-object and
${\mathrm{M}}$ is its canonical self-extension.

  2. (ii)

The functor
$\Phi_{\mathrm{M}}\colon{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})\to{\mathcal{T}}$
has right adjoint if and only if the subcategory
$\langle{\mathrm{P}}\rangle\subset{\mathcal{T}}$ is right admissible if and
only if ${\mathrm{M}}$ is left homologically finite.

  3. (iii)

The functor
$\Phi_{\mathrm{M}}\colon{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})\to{\mathcal{T}}$
has left adjoint if and only if the subcategory
$\langle{\mathrm{P}}\rangle\subset{\mathcal{T}}$ is left admissible, and if
${\mathcal{T}}\simeq{\mathcal{T}}^{\mathrm{op}}$ this is equivalent to
${\mathrm{M}}$ being right homologically finite.

In particular, if ${\mathcal{T}}={\mathbf{D}}^{\mathrm{b}}(X)$, $X$ is
Gorenstein, and ${\mathrm{M}}\in{\mathbf{D}}^{\mathrm{perf}}(X)$, then
$\langle{\mathrm{P}}\rangle\subset{\mathbf{D}}^{\mathrm{b}}(X)$ is admissible.

###### Proof.

(i) By Lemma 2.9 a combination of (9) with the assumption
$\operatorname{\mathrm{Cone}}({\mathrm{M}}\to{\mathrm{P}})\cong{\mathrm{P}}[q]$
implies (10). Using Lemma 2.1 we deduce
$\operatorname{\mathrm{RHom}}_{\mathcal{T}}({\mathrm{M}},{\mathrm{M}})\cong{\mathsf{A}}_{q-1}$
hence ${\mathrm{M}}$ is a
${\mathsf{A}}_{q-1}\operatorname{-}{\mathcal{T}}$-bimodule, and it gives a
continuous (i.e., commuting with arbitrary direct sums) fully faithful derived
tensor product functor

$\Phi_{\mathrm{M}}\colon{\mathbf{D}}({\mathsf{A}}_{q-1})\to{\mathbf{D}}({\mathcal{T}}),\qquad
F\mapsto F\otimes_{{\mathsf{A}}_{q-1}}{\mathrm{M}},$

see [Kel06, §3.8]. Obviously,
$\Phi_{\mathrm{M}}({\mathsf{A}}_{q-1})\cong{\mathrm{M}}$. Furthermore, it
follows from continuity of $\Phi_{\mathrm{M}}$ and Lemma 2.4 that

$\Phi_{\mathrm{M}}({\Bbbk}_{\mathsf{A}})\cong\Phi_{\mathrm{M}}(\operatorname{\mathrm{hocolim}}{\mathsf{A}}_{q-1}^{(i)})\cong\operatorname{\mathrm{hocolim}}{\mathrm{M}}^{(i)},$

where
${\mathrm{M}}^{(i)}\coloneqq\Phi_{\mathrm{M}}({\mathsf{A}}_{q-1}^{(i)})$.
Using the inductive construction of the objects ${\mathsf{A}}_{q-1}^{(i)}$ in
Lemma 2.4 and the octahedral axiom, it is easy to find distinguished triangles

(11) ${\mathrm{M}}^{(i)}\to{\mathrm{P}}\to{\mathrm{P}}[iq]$

for all $i\geq 1$ such that the maps ${\mathrm{M}}^{(i)}\to{\mathrm{P}}$ are
compatible with the chain maps
${\mathrm{M}}^{(1)}\to{\mathrm{M}}^{(2)}\to\dots$. Therefore, taking the
homotopy colimit of the above triangles we obtain the triangle

$\operatorname{\mathrm{hocolim}}{\mathrm{M}}^{(i)}\to\operatorname{\mathrm{hocolim}}{}{\mathrm{P}}\to\operatorname{\mathrm{hocolim}}{\mathrm{P}}[iq],$

and since the last term is zero by assumption, we obtain an isomorphism
$\Phi_{\mathrm{M}}({\Bbbk}_{\mathsf{A}})\cong\operatorname{\mathrm{hocolim}}{\mathrm{P}}\cong{\mathrm{P}}$.
As the functor $\Phi_{\mathrm{M}}$ is fully faithful, we conclude from Lemma
2.4(iii) that ${\mathrm{P}}$ is a ${\mathbb{P}}^{\infty,{q}}$-object.

Finally, since by Proposition 2.2 the categories
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})$ and
${\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{q-1})$ are generated by the
objects ${\Bbbk}_{\mathsf{A}}$ and ${\mathsf{A}}_{q-1}$, respectively, we
conclude that their images are $\langle{\mathrm{P}}\rangle$ and
$\langle{\mathrm{M}}\rangle$.

(ii) Since the functor $\Phi_{\mathrm{M}}$ is fully faithful, right
admissibility of $\langle{\mathrm{P}}\rangle$ is equivalent to the existence
of a right adjoint for $\Phi_{\mathrm{M}}$ on
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})$; this proves the first
equivalence.

Now note that the functor $\Phi_{\mathrm{M}}$ on
${\mathbf{D}}({\mathsf{A}}_{q-1})$ always has the right adjoint given by

$\operatorname{\mathrm{RHom}}_{\mathcal{T}}({\mathrm{M}},-)\colon{\mathbf{D}}({\mathcal{T}})\to{\mathbf{D}}({\mathsf{A}}_{q-1}),$

see [Kel06, §3.8]. If ${\mathrm{M}}$ is left homologically finite, this
functor takes ${\mathcal{T}}$ to
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})$, hence it gives a right
adjoint to $\Phi_{\mathrm{M}}$ on ${\mathcal{T}}$. Conversely, if a right
adjoint functor
$\Phi_{\mathrm{M}}^{!}\colon{\mathcal{T}}\to{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{q-1})$
exists then for any $T\in{\mathcal{T}}$

$\operatorname{\mathrm{Ext}}^{\bullet}_{\mathcal{T}}({\mathrm{M}},T)\cong\operatorname{\mathrm{Ext}}^{\bullet}_{\mathcal{T}}(\Phi_{\mathrm{M}}({\mathsf{A}}_{q-1}),T)\cong\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathsf{A}}_{q-1}}({\mathsf{A}}_{q-1},\Phi_{\mathrm{M}}^{!}(T))\cong{\mathrm{H}}^{\bullet}(\Phi_{\mathrm{M}}^{!}(T))$

is finite-dimensional, hence ${\mathrm{M}}$ is left homologically finite. This
proves the second equivalence.

(iii) We apply (ii) to the opposite categories, using an equivalence
${\mathsf{A}}_{q-1}^{\mathrm{op}}\simeq{\mathsf{A}}_{q-1}$ (which holds
because ${\mathsf{A}}_{q-1}$ is commutative) and the equivalence
${\mathcal{T}}\simeq{\mathcal{T}}^{\mathrm{op}}$. Passing to opposite
categories swaps left and right adjoints and left and right homological
finiteness and the result follows.

The final statement is a combination of parts (ii) or (iii) because left
homological finiteness for an object
${\mathrm{M}}\in{\mathbf{D}}^{\mathrm{b}}(X)$ is equivalent to the condition
${\mathrm{M}}\in{\mathbf{D}}^{\mathrm{perf}}(X)$ by [Orl06, Proposition 1.11],
and when $X$ is Gorenstein the Grothendieck duality implies that the same is
true for right homological finiteness. ∎

We conclude this section with an obvious consequence of Corollary 2.5.

######  Corollary 2.11.

Under the assumptions of Lemma 2.10 we have
${\mathrm{M}}\in\mathbf{thick}({\mathrm{M}}^{(2)})\subset{\mathcal{T}}$.

Despite the apparent symmetry between the algebras ${\mathsf{A}}_{p}$ and
${\mathsf{B}}_{q}$ and their derived categories, the category we will use
mostly is the category ${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$. In
particular, in Section §6 we will show that varieties with ordinary double
points often admit subcategories equivalent to
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ in their bounded derived
categories. For this reason we call
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ categorical ordinary double
point of degree $p$.

##  3\. Kronecker quiver and categorical ordinary double points

In this section we introduce the graded Kronecker quiver category, study its
properties, and show that the categorical ordinary double point can be
obtained as its Verdier localization with respect to a spherical object. Also
we explain how to construct fully faithful functors with admissible image from
the graded Kronecker quiver category. We keep working over an arbitrary field
${\Bbbk}$.

###  3.1. The graded Kronecker quiver

Let $q\geq 1$ be a positive integer. We define the graded Kronecker quiver of
degree $q$ as the dg-quiver

(12) $\mathsf{Kr}_{q}\coloneqq\left\\{\ \lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
17.99097pt\raise 11.32639pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.6875pt\hbox{$\scriptstyle{\upalpha_{q}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
43.49815pt\raise 2.1806pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
17.95554pt\raise-11.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\upalpha_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
43.49815pt\raise-2.1806pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
43.50006pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet,}$}}}}}}}\ignorespaces}}}}\ignorespaces\qquad\deg(\upalpha_{0})=0,\quad\deg(\upalpha_{q})=q,\qquad{\mathrm{d}}(\upalpha_{0})={\mathrm{d}}(\upalpha_{q})=0\
\right\\}.$

We will also use the same notation $\mathsf{Kr}_{q}$ for the path dg-algebra
of this quiver.

######  Remark 3.1.

The same definition makes sense for any $q$, not necessarily positive.
However, for $q<0$ the resulting dg-algebra $\mathsf{Kr}_{q}$ is Morita
equivalent to $\mathsf{Kr}_{-q}$, while for $q=0$ the derived category of
$\mathsf{Kr}_{0}$ is equivalent to the derived category of the projective line
${\mathbb{P}}^{1}$ (cf. the proof of Lemma 3.5).

######  Lemma 3.2.

The dg-algebra $\mathsf{Kr}_{q}$ is intrinsically formal, smooth and proper.

###### Proof.

Intrinsic formality is proved by the argument of Lemma 2.1, see also [Kuz21a,
proof of Proposition A.3]. Smoothness is clear, since the quiver (12) is
directed, and properness is evident. ∎

A combination of smoothness and properness of the dg-algebra $\mathsf{Kr}_{q}$
implies that the category ${\mathbf{D}}^{\mathrm{perf}}(\mathsf{Kr}_{q})$ of
perfect dg-modules over $\mathsf{Kr}_{q}$ coincides with the category
${\mathbf{D}}^{\mathrm{b}}(\mathsf{Kr}_{q})$ of dg-modules with bounded total
cohomology. We denote this category by

$\mathcal{K}\\!\mathit{r}_{q}\coloneqq{\mathbf{D}}^{\mathrm{b}}(\mathsf{Kr}_{q})={\mathbf{D}}^{\mathrm{b}}(\mathsf{Kr}_{q}),$

and call it the graded Kronecker quiver category of degree $q$.

The vertices of the quiver provide two idempotents in $\mathsf{Kr}_{q}$, and
since the quiver is directed, the corresponding direct summands of the free
module form an exceptional pair $({\mathcal{E}},{\mathcal{E}}^{\prime})$ in
$\mathcal{K}\\!\mathit{r}_{q}$ generating the category. In other words, we
have a semiorthogonal decomposition

$\mathcal{K}\\!\mathit{r}_{q}\coloneqq\langle{\mathcal{E}},{\mathcal{E}}^{\prime}\rangle,$

with exceptional objects ${\mathcal{E}}$ and ${\mathcal{E}}^{\prime}$ and

(13)
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},{\mathcal{E}}^{\prime})={\Bbbk}\oplus{\Bbbk}[-q].$

Intrinsic formality (Lemma 3.2) of the dg-algebra $\mathsf{Kr}_{q}$ has the
following useful consequence.

######  Corollary 3.3.

Every dg-enhanced triangulated category ${\mathcal{T}}$ generated by an
exceptional pair $({\mathcal{E}},{\mathcal{E}}^{\prime})$ satisfying (13) is
equivalent to $\mathcal{K}\\!\mathit{r}_{q}$.

We define the objects
${\mathrm{K}}_{+},{\mathrm{K}}_{-}\in\mathcal{K}\\!\mathit{r}_{q}$ from the
distinguished triangles

(14) $\displaystyle{\mathrm{K}}_{+}$
$\displaystyle\xrightarrow{\quad}{\mathcal{E}}\xrightarrow{\ \upalpha_{0}\
}{\mathcal{E}}^{\prime},$ (15) $\displaystyle{\mathrm{K}}_{-}$
$\displaystyle\xrightarrow{\quad}{\mathcal{E}}\xrightarrow{\ \upalpha_{q}\
}{\mathcal{E}}^{\prime}[q].$

######  Lemma 3.4.

The $\operatorname{\mathrm{Ext}}$-groups between the objects
${\mathrm{K}}_{-},{\mathcal{E}},{\mathcal{E}}^{\prime},{\mathrm{K}}_{+}\in\mathcal{K}\\!\mathit{r}_{q}$
are given in the table below

$\begin{array}[]{|c||c|c|c|c|}\hline\cr&\hphantom{1234.}{\mathrm{K}}_{-}\hphantom{1234.}&\hphantom{12345}{\mathcal{E}}\hphantom{12345}&\hphantom{12345}{\mathcal{E}}^{\prime}\hphantom{12345}&{\mathrm{K}}_{+}\\\
\hline\cr\hline\cr\hphantom{12}{\mathrm{K}}_{-}\hphantom{12}&{\Bbbk}\oplus{\Bbbk}[q-1]&{\Bbbk}&{\Bbbk}&0\\\
\hline\cr{\mathcal{E}}&{\Bbbk}[q-1]&{\Bbbk}&{\Bbbk}\oplus{\Bbbk}[-q]&{\Bbbk}[-q-1]\\\
\hline\cr{\mathcal{E}}^{\prime}&{\Bbbk}[q-1]&0&{\Bbbk}&{\Bbbk}[-1]\\\
\hline\cr{\mathrm{K}}_{+}&0&{\Bbbk}&{\Bbbk}[-q]&{\Bbbk}\oplus{\Bbbk}[-q-1]\\\
\hline\cr\end{array}$

where each entry shows the $\operatorname{\mathrm{Ext}}$-groups from the
object in the given row to the object in the given column.

###### Proof.

These are easily computed from the defining triangles (14) and (15), using
exceptionality of the pair $({\mathcal{E}},{\mathcal{E}}^{\prime})$ and (13).
∎

Recall that an object ${\mathrm{K}}\in{\mathcal{T}}$ in a triangulated
category with a Serre functor ${\mathbf{S}}_{\mathcal{T}}$ is $r$-spherical
[ST01] if

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}},{\mathrm{K}})={\Bbbk}\oplus{\Bbbk}[-r]\qquad\text{and}\qquad{\mathbf{S}}_{\mathcal{T}}({\mathrm{K}})\cong{\mathrm{K}}[r].$

Note that the Kronecker quiver category $\mathcal{K}\\!\mathit{r}_{q}$ has a
Serre functor (e.g., by [BK89, Proposition 3.8]).

######  Lemma 3.5.

The objects ${\mathrm{K}}_{+}$ and ${\mathrm{K}}_{-}$ in
$\mathcal{K}\\!\mathit{r}_{q}$ are $(1+q)$-spherical and $(1-q)$-spherical,
respectively.

###### Proof.

The spaces
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}}_{+},{\mathrm{K}}_{+})$ and
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}}_{-},{\mathrm{K}}_{-})$
have been already computed in Lemma 3.4, so it remains to compute
${\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathrm{K}}_{\pm})$. We start
with a few remarks about the structure of the category
$\mathcal{K}\\!\mathit{r}_{q}$, which is very similar to the structure of
${\mathbf{D}}^{\mathrm{b}}({\mathbb{P}}^{1})\simeq{}\mathcal{K}\\!\mathit{r}_{0}$
(e.g., the objects ${\mathcal{E}}_{i}$ constructed below are analogous to the
line bundles ${\mathcal{O}}(i)$, while ${\mathrm{K}}_{+}$ and
${\mathrm{K}}_{-}$ are analogous to the structure sheaves of the points
$0,\infty\in{\mathbb{P}}^{1}$).

Consider the infinite sequence of exceptional objects
${\mathcal{E}}_{i}\in\mathcal{K}\\!\mathit{r}_{q}$, $i\in{\mathbb{Z}}$,
defined by

${\mathcal{E}}_{0}={\mathcal{E}},\qquad{\mathcal{E}}_{1}={\mathcal{E}}^{\prime},\qquad{\mathcal{E}}_{i+1}={\mathbf{R}}_{{\mathcal{E}}_{i}}({\mathcal{E}}_{i-1})[1-q],\qquad{\mathcal{E}}_{i-1}={\mathbf{L}}_{{\mathcal{E}}_{i}}({\mathcal{E}}_{i+1})[q-1],$

where ${\mathbf{R}}$ and ${\mathbf{L}}$ stand for the right and left mutation
functors. Spelling out the definition of ${\mathcal{E}}_{2}$, we have a
distinguished triangle (which is analogous to the Euler sequence on
${\mathbb{P}}^{1}$)

(16) ${\mathcal{E}}_{0}\xrightarrow{\ (\upalpha_{0},\upalpha_{q})\
}{\mathcal{E}}_{1}\oplus{\mathcal{E}}_{1}[q]\xrightarrow{\
\hphantom{(\upalpha_{0},\upalpha_{q})}\ }{\mathcal{E}}_{2}[q].$

It follows from this that
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{1},{\mathcal{E}}_{2})\cong{\Bbbk}\oplus{\Bbbk}[-q]\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{0},{\mathcal{E}}_{1})$.
Since both $({\mathcal{E}}_{0},{\mathcal{E}}_{1})$ and
$({\mathcal{E}}_{1},{\mathcal{E}}_{2})$ are exceptional pairs generating
$\mathcal{K}\\!\mathit{r}_{q}$, Corollary 3.3 implies that there is an
autoequivalence

$\tau\colon\mathcal{K}\\!\mathit{r}_{q}\to\mathcal{K}\\!\mathit{r}_{q},\qquad\text{such
that $\tau({\mathcal{E}}_{0})={\mathcal{E}}_{1}$ and
$\tau({\mathcal{E}}_{1})={\mathcal{E}}_{2}$},$

(analogous to the ${\mathcal{O}}(1)$-twist in
${\mathbf{D}}^{\mathrm{b}}({\mathbb{P}}^{1})$) and upon replacing (16) by an
isomorphic triangle we may assume that the second arrow in (16) takes the form
$(-\tau(\upalpha_{q}),\tau(\upalpha_{0}))$. Then it follows from the
definition of the sequence ${\mathcal{E}}_{i}$ that

(17) $\tau({\mathcal{E}}_{i})\cong{\mathcal{E}}_{i+1}\qquad\text{for
all~{}$i\in{\mathbb{Z}}$.}$

Note also that the octahedral axiom applied to (16) implies that

(18)
$\displaystyle\operatorname{\mathrm{Cone}}({\mathcal{E}}_{1}[q]\xrightarrow{\
\tau(\upalpha_{0})\ }{\mathcal{E}}_{2}[q])$
$\displaystyle{}\cong\operatorname{\mathrm{Cone}}({\mathcal{E}}_{0}\xrightarrow{\
\upalpha_{0}\ }{\mathcal{E}}_{1})$ $\displaystyle{}\cong{\mathrm{K}}_{+}[1],$
(19)
$\displaystyle\operatorname{\mathrm{Cone}}({\mathcal{E}}_{1}\xrightarrow{\
\tau(\upalpha_{q})\ }{\mathcal{E}}_{2}[q])$
$\displaystyle{}\cong\operatorname{\mathrm{Cone}}({\mathcal{E}}_{0}\xrightarrow{\
\upalpha_{q}\ }{\mathcal{E}}_{1}[q])$
$\displaystyle{}\cong{\mathrm{K}}_{-}[1],$

hence $\tau({\mathrm{K}}_{+})\cong{\mathrm{K}}_{+}[-q]$,
$\tau({\mathrm{K}}_{-})\cong{\mathrm{K}}_{-}$, and therefore, we have
distinguished triangles

(20) ${\mathrm{K}}_{+}[-iq]\xrightarrow{\qquad}{\mathcal{E}}_{i}\xrightarrow{\
\tau^{i}(\upalpha_{0})\
}{\mathcal{E}}_{i+1},\qquad\text{and}\qquad{\mathrm{K}}_{-}\xrightarrow{\qquad}{\mathcal{E}}_{i}\xrightarrow{\
\tau^{i}(\upalpha_{q})\ }{\mathcal{E}}_{i+1}[q],$

for any $i\in{\mathbb{Z}}$. On the other hand, it follows from (16) that
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{2},{\mathcal{E}}_{0})\cong{\Bbbk}[q-1]$,
hence

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{0},{\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{2}))\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{2},{\mathcal{E}}_{0})^{\vee}\cong{\Bbbk}[1-q],\qquad\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{1},{\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{2}))\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{2},{\mathcal{E}}_{1})^{\vee}=0.$

Since the category $\mathcal{K}\\!\mathit{r}_{q}$ is generated by the
exceptional pair $({\mathcal{E}}_{0},{\mathcal{E}}_{1})$, the above
isomorphisms imply that
${\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{2})\cong{\mathcal{E}}_{0}[1-q]$.
Since the Serre functor commutes with any autoequivalence, we conclude from
this and (17) that for all $i\in{\mathbb{Z}}$ we have

${\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{i})\cong{\mathcal{E}}_{i-2}[1-q].$

Applying this to triangle (14) and using the first triangle in (20) for $i=-2$
we conclude that

${\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathrm{K}}_{+})\cong\operatorname{\mathrm{Cone}}({\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{0})\to{\mathbf{S}}_{\mathcal{K}\\!\mathit{r}_{q}}({\mathcal{E}}_{1}))[-1]\cong\operatorname{\mathrm{Cone}}({\mathcal{E}}_{-2}[-q]\to{\mathcal{E}}_{-1}[-q])\cong{\mathrm{K}}_{+}[1+q].$

Thus, ${\mathrm{K}}_{+}$ is $(1+q)$-spherical, and a similar computation shows
${\mathrm{K}}_{-}$ is $(1-q)$-spherical. ∎

######  Remark 3.6.

The category $\mathcal{K}\\!\mathit{r}_{q}$ also has a topological
interpretation (see [BBD82] and [dCM09] for the required material). Let
${\mathrm{S}}^{q+1}$ be a $(q+1)$-dimensional real sphere with a point
$\\{P\\}\hookrightarrow{\mathrm{S}}^{q+1}$ and its open complement
$U\cong{\mathbb{R}}^{q+1}\hookrightarrow{\mathrm{S}}^{q+1}$. Then
$\mathcal{K}\\!\mathit{r}_{q}$ is equivalent to the category of complexes of
sheaves of $\Bbbk$-vector spaces on ${\mathrm{S}}^{q+1}$ constructible with
respect to the stratification $\\{P,U\\}$. Under this equivalence the object
${\mathrm{K}}_{+}$ corresponds to the constant sheaf
$\Bbbk_{{\mathrm{S}}^{q+1}}$.

Note that this interpretation gives an alternative explanation for the fact
that ${\mathrm{K}}_{+}=\Bbbk_{{\mathrm{S}}^{q+1}}$ is spherical. Indeed, the
self Ext-algebra of the constant sheaf $\Bbbk_{{\mathrm{S}}^{q+1}}$ is
isomorphic to the cohomology of the sphere, and using Verdier duality one can
show that the Serre functor of $\mathcal{K}\\!\mathit{r}_{q+1}$ acts on
$\Bbbk_{{\mathrm{S}}^{q+1}}$ as the shift $[q+1]$.

In fact, in [KY18] a more general “orbifold version” of the graded Kronecker
quiver has been studied. Its special case, the graded quiver $\Gamma(1,1,r)$
defined in [KY18, Theorem 1.1] is isomorphic to the quiver $\mathsf{Kr}_{r}$,
and Lemma 3.5 can be deduced from [KY18, Theorem 1.2(a)].

###  3.2. Localization and adherence

The next proposition relates the graded Kronecker quiver category
$\mathcal{K}\\!\mathit{r}_{p+1}$ to the categorical double point
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$.

######  Proposition 3.7.

There is an equivalence

$\mathcal{K}\\!\mathit{r}_{p+1}/\langle{\mathrm{K}}_{+}\rangle\simeq{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p}),$

of the Verdier localization of the Kronecker quiver category and the
categorical double point, such that the corresponding localization functor
$\uprho_{*}\colon\mathcal{K}\\!\mathit{r}_{p+1}\to{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$
satisfies

  1. (i)

$\uprho_{*}({\mathcal{E}})\cong\uprho_{*}({\mathcal{E}}^{\prime})={\Bbbk}_{\mathsf{A}}$,
$\uprho_{*}(\upalpha_{0})=\operatorname{\mathrm{id}}_{{\Bbbk}_{\mathsf{A}}}\in\operatorname{\mathrm{Hom}}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})$,
$\uprho_{*}(\upalpha_{q})=\uptheta\in\operatorname{\mathrm{Ext}}^{q}_{{\mathsf{A}}_{p}}({\Bbbk}_{\mathsf{A}},{\Bbbk}_{\mathsf{A}})$;

  2. (ii)

$\uprho_{*}({\mathrm{K}}_{-})={\mathsf{A}}_{p}$.

Since $\operatorname{\mathrm{Ker}}(\uprho_{*})=\langle{\mathrm{K}}_{+}\rangle$
is generated by a spherical object, the functor $\uprho_{*}$ is an example of
a crepant categorical contraction (see Definitions 1.10 and 1.12). One can
also show that $\uprho_{*}$ has a left adjoint functor
$\uprho^{*}\colon{\mathbf{D}}^{\mathrm{perf}}({\mathsf{A}}_{p})\to\mathcal{K}\\!\mathit{r}_{p+1}$
such that $\uprho^{*}({\mathsf{A}}_{p})\cong{\mathrm{K}}_{-}$ and $\uprho^{*}$
is a weakly crepant categorical resolution in the sense of [Kuz08].

###### Proof.

Denote
${\mathcal{T}}\coloneqq\mathcal{K}\\!\mathit{r}_{p+1}/\langle{\mathrm{K}}_{+}\rangle$
and let $\uprho_{*}\colon\mathcal{K}\\!\mathit{r}_{p+1}\to{\mathcal{T}}$ be
the localization functor. We will use Lemma 2.10 to construct the equivalence
${\mathcal{T}}\simeq{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$. So, let

${\mathrm{P}}\coloneqq\uprho_{*}({\mathcal{E}})\cong\uprho_{*}({\mathcal{E}}^{\prime})\in{\mathcal{T}},\qquad{\mathrm{M}}\coloneqq\uprho_{*}({\mathrm{K}}_{-})\in{\mathcal{T}}$

(the isomorphism
$\uprho_{*}({\mathcal{E}})\cong\uprho_{*}({\mathcal{E}}^{\prime})$ follows
from (14); it is induced by $\uprho_{*}(\upalpha_{0})$). Since
$\mathcal{K}\\!\mathit{r}_{p+1}$ is generated by ${\mathcal{E}}$ and
${\mathcal{E}}^{\prime}$, it follows that ${\mathcal{T}}$ is generated by
${\mathrm{P}}$. By Lemma 3.4 the object ${\mathrm{K}}_{-}$ is orthogonal to
${\mathrm{K}}_{+}$, hence by definition of Verdier localization we have

$\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathcal{T}}}({\mathrm{M}},{\mathrm{P}})\cong\operatorname{\mathrm{Ext}}^{\bullet}_{{\mathcal{K}\\!\mathit{r}_{p+1}}}({\mathrm{K}}_{-},{\mathcal{E}})\cong\Bbbk,$

where the second isomorphism is again by Lemma 3.4. Furthermore, applying
$\uprho_{*}$ to (15) we conclude that
$\operatorname{\mathrm{Cone}}({\mathrm{M}}\to{\mathrm{P}})\cong{\mathrm{P}}[p+1]$,
with the morphism ${\mathrm{P}}\to{\mathrm{P}}[p+1]$ given by
$\uprho_{*}(\upalpha_{q})$. Finally, note that
$\operatorname{\mathrm{hocolim}}{\mathcal{E}}[i(p+1)]=0$ in
${\mathbf{D}}(\mathsf{Kr}_{p+1})$ by Lemma 2.3, because ${\mathcal{E}}$ and
${\mathcal{E}}^{\prime}$ compactly generate ${\mathbf{D}}(\mathsf{Kr}_{p+1})$,
and since $\uprho_{*}$ induces a continuous functor
${\mathbf{D}}(\mathsf{Kr}_{p+1})\to{{\mathbf{D}}({\mathcal{T}})}$, it follows
that $\operatorname{\mathrm{hocolim}}{\mathrm{P}}[i(p+1)]=0$.

Now applying Lemma 2.10(i) we obtain a fully faithful functor
$\Phi_{\mathrm{M}}\colon{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})\to{\mathcal{T}}$
that takes ${\Bbbk}_{\mathsf{A}}$ to ${\mathrm{P}}$. Since ${\mathcal{T}}$ is
generated by ${\mathrm{P}}$, this functor is essentially surjective, hence it
is an equivalence. This completes the proof of the first statement of the
proposition.

Since $\uptheta\in\operatorname{\mathrm{Ext}}^{q}({\Bbbk},{\Bbbk})$ is taken
by the functor $\Phi_{\mathrm{M}}$ to a non-trivial element in
$\operatorname{\mathrm{Hom}}({\mathrm{P}},{\mathrm{P}}[p+1])$, we conclude
that rescaling $\uptheta$ if necessary, we obtain
$\uprho_{*}(\upalpha_{q})=\uptheta$. This proves (i). Similarly, since
$\Phi_{\mathrm{M}}$ takes ${\mathsf{A}}_{p}$ to ${\mathrm{M}}$, comparing with
the definition of ${\mathrm{M}}$ we deduce (ii). ∎

######  Remark 3.8.

Conversely, it is easy to check that the dg-algebra $\mathsf{Kr}_{p+1}$ is
Morita equivalent to the _Auslander resolution_ of the dg-algebra
${\mathsf{A}}_{p}$, see the definition in [KL15, §5] for the case of algebras,
[Orl20, §2.3] for the case of dg-algebras, and [KL15, Example 5.3] for a
computation in the case $p=0$. Therefore, the graded Kronecker quiver category
$\mathcal{K}\\!\mathit{r}_{p+1}$ provides a categorical resolution in the
sense of [KL15] for the categorical ordinary double point
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$.

We conclude this section by explaining how an admissible subcategory
equivalent to $\mathcal{K}\\!\mathit{r}_{q}$ can be embedded into a given
triangulated category. For this the following definition, generalizing [KKS22,
Definition 3.6], is convenient.

######  Definition 3.9.

Let ${\tilde{{\mathcal{T}}}}$ be a proper triangulated category with a Serre
functor, and let ${\mathrm{K}}\in{\tilde{{\mathcal{T}}}}$ be a spherical
object. We say that an exceptional object
${\mathcal{E}}\in{\tilde{{\mathcal{T}}}}$ is adherent to ${\mathrm{K}}$ if

(21)
$\dim\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}},{\mathcal{E}})=\dim\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},{\mathrm{K}})=1.$

Note that the first equality follows from Serre duality and the definition of
a spherical object.

Recall that a spherical object ${\mathrm{K}}$ in a dg-enhanced proper
triangulated category ${\tilde{{\mathcal{T}}}}$ gives rise to an
autoequivalence

(22)
${\mathbf{T}}_{\mathrm{K}}(-)\coloneqq{}\operatorname{\mathrm{Cone}}(\operatorname{\mathrm{RHom}}^{\bullet}({\mathrm{K}},-)\otimes{\mathrm{K}}\xrightarrow{\
\mathrm{ev}\ }-),$

which is called a spherical twist [ST01].

######  Lemma 3.10.

Let ${\tilde{{\mathcal{T}}}}$ be a dg-enhanced proper triangulated category
with a Serre functor ${\mathbf{S}}_{\tilde{{\mathcal{T}}}}$, and let
${\mathrm{K}}\in{\tilde{{\mathcal{T}}}}$ be a $(p+2)$-spherical object with
$p\geq 0$. If an exceptional object ${\mathcal{E}}\in{\tilde{{\mathcal{T}}}}$
is adherent to ${\mathrm{K}}$ then

(23) $({\mathcal{E}},{\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}}))$

is an exceptional pair and the subcategory
$\langle{\mathcal{E}},{\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}})\rangle\subset{\tilde{{\mathcal{T}}}}$
generated by (23) is equivalent to $\mathcal{K}\\!\mathit{r}_{p+1}$ and
admissible in ${\tilde{{\mathcal{T}}}}$.

###### Proof.

Shifting ${\mathrm{K}}$ appropriately (note that this does not affect the
corresponding spherical twist), we may assume that
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}},{\mathcal{E}})=\Bbbk$.
Since ${\mathbf{T}}_{{\mathrm{K}}}$ is an autoequivalence,
${\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}})$ is also an exceptional object.
Furthermore, by definition (22) of the spherical twist, we have a
distinguished triangle

${\mathrm{K}}\to{\mathcal{E}}\to{\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}}).$

Applying the functor $\operatorname{\mathrm{Ext}}^{\bullet}(-,{\mathcal{E}})$
to this triangle we see that the pair
$({\mathcal{E}},{\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}}))$ is exceptional.
On the other hand, using Serre duality in ${\tilde{{\mathcal{T}}}}$ we obtain

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},{\mathrm{K}})\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathbf{S}}_{\tilde{{\mathcal{T}}}}^{-1}({\mathrm{K}}),{\mathcal{E}})^{\vee}\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}}[-p-2],{\mathcal{E}})^{\vee}=\Bbbk[-p-2]$

and so, applying $\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}},-)$ to
the above triangle we see that the $\operatorname{\mathrm{Ext}}$-groups
between ${\mathcal{E}}$ and ${\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}})$ are
those of the graded Kronecker quiver of degree $p+1$. Thus, by Corollary 3.3
the subcategory in ${\tilde{{\mathcal{T}}}}$ generated by the pair
$({\mathcal{E}},{\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}}))$ is equivalent to
$\mathcal{K}\\!\mathit{r}_{p+1}$, and since this subcategory is generated by
an exceptional pair in a proper triangulated category, it is admissible. ∎

In Section 6 we will present several geometric applications of Lemma 3.10 (or
rather of Theorem 4.2 that relies on it), where the category
${\tilde{{\mathcal{T}}}}$ will be taken to be a crepant categorical resolution
of ${\mathbf{D}}^{\mathrm{b}}(X)$. Here we mention the following simpler
example, see [KKS22, Example 2.14] for details.

######  Example 3.11.

Let ${\tilde{{\mathcal{T}}}}={\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$ where
${\tilde{X}}$ is a smooth projective surface such that
${\mathrm{H}}^{>0}({\tilde{X}},{\mathcal{O}}_{{\tilde{X}}})=0$. If
$E\subset{\tilde{X}}$ is a $(-2)$-curve then
${\mathrm{K}}={\mathcal{O}}_{E}(-1)$ is a $2$-spherical object. Furthermore,
if $D\in\operatorname{\mathrm{Pic}}({\tilde{X}})$ is a divisor class
satisfying $D\cdot E=1$ then the exceptional object
${\mathcal{E}}={\mathcal{O}}_{\tilde{X}}(D)$ is adherent to ${\mathrm{K}}$ and
${\mathbf{T}}_{{\mathrm{K}}}({\mathcal{E}})={\mathcal{O}}(E+D)$. In this
situation Lemma 3.10 says that the subcategory

$\langle{\mathcal{O}}_{\tilde{X}}(D),{\mathcal{O}}_{\tilde{X}}(E+D)\rangle$

is admissible in ${\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$, and is equivalent
to $\mathcal{K}\\!\mathit{r}_{1}$.

##  4\. Adherence and categorical absorption of singularities

In this section we relate the notion of adherence introduced in Definition 3.9
to the notion of categorical absorption of singularities from Definition 1.1.
After that we prove Theorems 1.5, 1.8, and 1.9 from the Introduction. We keep
working over an arbitrary field ${\Bbbk}$.

###  4.1. From adherence to categorical absorption

The following result allows us to descend semiorthogonal decompositions along
Verdier localizations.

######  Proposition 4.1.

Let $\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathcal{T}}$ be a Verdier
localization with the kernel category
${\mathcal{K}}\coloneqq\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\tilde{{\mathcal{T}}}}$.
Let

${\tilde{{\mathcal{T}}}}=\langle{\tilde{{\mathcal{T}}}}_{1},\dots,{\tilde{{\mathcal{T}}}}_{n}\rangle$

be a semiorthogonal decomposition compatible with ${\mathcal{K}}$, i.e., such
that the subcategories
${\mathcal{K}}_{i}\coloneqq{\mathcal{K}}\cap{\tilde{{\mathcal{T}}}}_{i}$ form
a semiorthogonal decomposition
${\mathcal{K}}=\langle{\mathcal{K}}_{1},\dots,{\mathcal{K}}_{n}\rangle$. Then
there is a semiorthogonal decomposition

${\mathcal{T}}=\langle{\mathcal{T}}_{1},\dots,{\mathcal{T}}_{n}\rangle$

with
${\mathcal{T}}_{i}\coloneqq\pi_{*}({\tilde{{\mathcal{T}}}}_{i})\subset{\mathcal{T}}$,
and the restriction
$\pi_{*}|_{{\tilde{{\mathcal{T}}}}_{i}}\colon{\tilde{{\mathcal{T}}}}_{i}\to{\mathcal{T}}_{i}$
is a Verdier localization with kernel ${\mathcal{K}}_{i}$.

###### Proof.

We assume $n=2$, as the general case follows by induction. The subcategories

${\mathcal{T}}_{i}\simeq{\tilde{{\mathcal{T}}}}_{i}/{\mathcal{K}}_{i}\subset{\tilde{{\mathcal{T}}}}/{\mathcal{K}}={\mathcal{T}}$

are semiorthogonal by [Orl06, Lemma 1.1], and they generate ${\mathcal{T}}$
because $\pi_{*}$ is essentially surjective and ${\tilde{{\mathcal{T}}}}_{i}$
generate ${\tilde{{\mathcal{T}}}}$; therefore,
${\mathcal{T}}=\langle{\mathcal{T}}_{1},{\mathcal{T}}_{2}\rangle$ is a
semiorthogonal decomposition. ∎

We now explain how to construct subcategories absorbing singularities. Let
$\delta_{ij}$ be the Kronecker delta. Recall the definition of a crepant
categorical contraction (Definitions 1.10 and 1.12).

######  Theorem 4.2.

Let ${\tilde{{\mathcal{T}}}}$ be a dg-enhanced smooth and proper triangulated
category, and let
${\mathrm{K}}_{1},\dots,{\mathrm{K}}_{r}\in{\tilde{{\mathcal{T}}}}$ be a
collection of $r$ completely orthogonal $(p+2)$-spherical objects with $p\geq
0$. Assume that there is a (non-full) exceptional collection
${\mathcal{E}}_{1},\dots,{\mathcal{E}}_{r}$ in ${\tilde{{\mathcal{T}}}}$ such
that

(24)
$\dim\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{i},{\mathrm{K}}_{j})=\delta_{ij},$

in particular ${\mathcal{E}}_{i}$ is adherent to ${\mathrm{K}}_{i}$ for each
$i$. Set
${\mathcal{K}}\coloneqq\langle{\mathrm{K}}_{1},\dots,{\mathrm{K}}_{r}\rangle\subset{\tilde{{\mathcal{T}}}}$
and consider the localization

$\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathcal{T}}\coloneqq{\tilde{{\mathcal{T}}}}/{\mathcal{K}}.$

Then the following holds:

  1. (i)

For each $1\leq i\leq r$ the triangulated subcategory

${\tilde{{\mathcal{P}}}}_{i}\coloneqq\langle{\mathcal{E}}_{i},{\mathbf{T}}_{{\mathrm{K}}_{i}}{\mathcal{E}}_{i}\rangle$

generated in ${\tilde{{\mathcal{T}}}}$ by ${\mathcal{E}}_{i}$ and
${\mathbf{T}}_{{\mathrm{K}}_{i}}({\mathcal{E}}_{i})$ (or, equivalently, by
${\mathcal{E}}_{i}$ and ${\mathrm{K}}_{i}$) is equivalent to the graded
Kronecker quiver category $\mathcal{K}\\!\mathit{r}_{p+1}$ and is admissible
in ${\tilde{{\mathcal{T}}}}$. Moreover, the collection of subcategories
${\tilde{{\mathcal{P}}}}_{1},\dots,{\tilde{{\mathcal{P}}}}_{r}\subset{\mathcal{T}}$
is semiorthogonal in ${\tilde{{\mathcal{T}}}}$ and the subcategory

${\tilde{{\mathcal{P}}}}\coloneqq\langle{\tilde{{\mathcal{P}}}}_{1},\dots,{\tilde{{\mathcal{P}}}}_{r}\rangle$

is admissible in ${\tilde{{\mathcal{T}}}}$.

  2. (ii)

For each $i$ the object ${\mathrm{P}}_{i}\coloneqq\pi_{*}({\mathcal{E}}_{i})$
is a ${\mathbb{P}}^{\infty,{p+1}}$-object, the category
${\mathcal{P}}_{i}\coloneqq\pi_{*}({\tilde{{\mathcal{P}}}}_{i})=\langle{\mathrm{P}}_{i}\rangle$
is equivalent to the categorical ordinary double point
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ and is admissible in
${\mathcal{T}}$. Moreover, the collection
${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r}$ is semiorthogonal in
${\mathcal{T}}$, the subcategory

${\mathcal{P}}\coloneqq\langle{\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r}\rangle=\langle{\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r}\rangle$

is equal to $\pi_{*}({\tilde{{\mathcal{P}}}})$ and absorbs singularities of
${\mathcal{T}}$, and the functor $\pi_{*}$ induces equivalences

(25)
${}^{\perp}{\tilde{{\mathcal{P}}}}\simeq{}^{\perp}{\mathcal{P}},\qquad{\tilde{{\mathcal{P}}}}^{\perp}\simeq{\mathcal{P}}^{\perp}.$

  3. (iii)

The functor $\pi_{*}$ is a crepant categorical contraction.

###### Proof.

(i) By Lemma 3.10 the subcategory
${\tilde{{\mathcal{P}}}}_{i}\subset{\tilde{{\mathcal{T}}}}$ is equivalent to
$\mathcal{K}\\!\mathit{r}_{p+1}$ and admissible. To check that the
subcategories
${\tilde{{\mathcal{P}}}}_{1},\dots,{\tilde{{\mathcal{P}}}}_{r}\subset{\tilde{{\mathcal{T}}}}$
are semiorthogonal it is enough to note that the generating sets of these
subcategories are semiorthogonal:

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{i},{\mathcal{E}}_{j})=0,\qquad\operatorname{\mathrm{Ext}}^{\bullet}({\mathcal{E}}_{i},{\mathrm{K}}_{j})=0,\qquad\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}}_{i},{\mathcal{E}}_{j})=0,\qquad\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{K}}_{i},{\mathrm{K}}_{j})=0.$

Indeed, the first holds for $i>j$ since the collection
${\mathcal{E}}_{1},\dots,{\mathcal{E}}_{r}$ is exceptional, the second holds
for $i\neq j$ by (24), the third follows from the second and Serre duality
since ${\mathrm{K}}_{i}$ is spherical, and the last holds for all $i\neq j$
since the spherical objects are assumed to be orthogonal. Since each
${\tilde{{\mathcal{P}}}}_{i}$ is admissible in ${\tilde{{\mathcal{T}}}}$, the
subcategory ${\tilde{{\mathcal{P}}}}$ is admissible as well.

(ii) Consider the semiorthogonal decompositions

${\tilde{{\mathcal{T}}}}=\langle{\tilde{{\mathcal{P}}}}_{1},\dots,{\tilde{{\mathcal{P}}}}_{r},{}^{\perp}{\tilde{{\mathcal{P}}}}\rangle,\qquad{\tilde{{\mathcal{T}}}}=\langle{\tilde{{\mathcal{P}}}}^{\perp},{\tilde{{\mathcal{P}}}}_{1},\dots,{\tilde{{\mathcal{P}}}}_{r}\rangle.$

Applying Proposition 4.1 to the subcategories
$\langle{\mathrm{K}}_{1}\rangle\subset{\tilde{{\mathcal{P}}}}_{1}$, …,
$\langle{\mathrm{K}}_{r}\rangle\subset{\tilde{{\mathcal{P}}}}_{r}$, and
$0\subset{}^{\perp}{\tilde{{\mathcal{P}}}}$ or
$0\subset{\tilde{{\mathcal{P}}}}^{\perp}$ (and defining
${\mathcal{K}}\subset{}{\tilde{{\mathcal{T}}}}$ as
${\mathcal{K}}\coloneqq\langle{\mathrm{K}}_{1},\dots,{\mathrm{K}}_{r}\rangle$),
we obtain semiorthogonal decompositions

${\tilde{{\mathcal{T}}}}/{\mathcal{K}}=\langle{\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r},{}^{\perp}{\mathcal{P}}\rangle,\qquad{\tilde{{\mathcal{T}}}}/{\mathcal{K}}=\langle{\mathcal{P}}^{\perp},{\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r}\rangle,$

where
${\mathcal{P}}_{i}=\pi_{*}({\tilde{{\mathcal{P}}}}_{i})\simeq{\tilde{{\mathcal{P}}}}_{i}/\langle{\mathrm{K}}_{i}\rangle$,
${\mathcal{P}}=\pi_{*}({\tilde{{\mathcal{P}}}})\subset{\mathcal{T}}/{\mathcal{K}}$
is the subcategory generated by ${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r}$,
and the functor $\pi_{*}$ induces equivalences (25). Furthermore, we deduce
equivalence
${\mathcal{P}}_{i}\simeq{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{p})$ from
Proposition 3.7, and since $\pi_{*}|_{{\tilde{{\mathcal{P}}}}_{i}}=\uprho_{*}$
we also conclude that ${\mathrm{P}}_{i}\coloneqq\pi_{*}({\mathcal{E}}_{i})$ is
the ${\mathbb{P}}^{\infty,{p+1}}$-generator of ${\mathcal{P}}_{i}$.

To check that ${\mathcal{P}}_{i}$ is admissible we note that using the Serre
functor of ${\tilde{{\mathcal{T}}}}$ and the spherical property of
${\mathrm{K}}_{i}$ we can rearrange its semiorthogonal decomposition in such a
way that ${\tilde{{\mathcal{P}}}}_{i}$ stands on the left or on the right;
applying Proposition 4.1 to this we obtain semiorthogonal decompositions of
${\mathcal{T}}$ where ${\mathcal{P}}_{i}$ stands on the left and on the right,
respectively, hence ${\mathcal{P}}_{i}$ is left and right admissible.

Finally, ${\mathcal{P}}$ absorbs singularities of ${\mathcal{T}}$ because
${}^{\perp}{\mathcal{P}}$ and ${\mathcal{P}}^{\perp}$ are equivalent to
admissible subcategories ${}^{\perp}{\tilde{{\mathcal{P}}}}$ and
${\tilde{{\mathcal{P}}}}^{\perp}$ in the smooth and proper category
${\tilde{{\mathcal{T}}}}$, hence they are smooth and proper.

(iii) This is obvious because $\operatorname{\mathrm{Ker}}(\pi_{*})$ is
generated by spherical objects. ∎

See Section 6, in particular Theorem 6.1 for geometric applications of Theorem
4.2 to nodal varieties. Meanwhile just note that in the situation of Example
3.11 if $\pi\colon{\tilde{X}}\to X$ is the contraction of $E$, we obtain a
fully faithful embedding
${\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{0})\hookrightarrow{\mathbf{D}}^{\mathrm{b}}(X)$,
cf. [KKS22, Example 3.17].

###  4.2. Absorptions and deformation absorptions

Recall Definition 1.1 of absorption. We will need the following observation.
Let $\omega_{X}^{\bullet}$ be the dualizing complex of $X$.

######  Lemma 4.3.

Assume ${\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$ absorbs
singularities of a projective scheme $X$. Then

${}^{\perp}{\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{perf}}(X),\qquad\text{and}\qquad{\mathcal{P}}^{\perp}\subset{\mathbf{D}}^{\mathrm{perf}}(X)\otimes\omega_{X}^{\bullet},$

and these embeddings are admissible.

###### Proof.

Let $F\in{}^{\perp}{\mathcal{P}}$. Then
$\operatorname{\mathrm{Ext}}^{\bullet}(F,F^{\prime})$ is finite-dimensional
for any $F^{\prime}\in{}^{\perp}{\mathcal{P}}$, because
${}^{\perp}{\mathcal{P}}$ is proper, and
$\operatorname{\mathrm{Ext}}^{\bullet}(F,P)=0$ for any $P\in{\mathcal{P}}$ by
definition of the orthogonal. Therefore,
$\operatorname{\mathrm{Ext}}^{\bullet}(F,G)$ is finite-dimensional for any
$G\in{\mathbf{D}}^{\mathrm{b}}(X)$, and hence $F$ is perfect by [Orl06,
Proposition 1.11], and the first inclusion is proved. The second inclusion
follows from the first and Grothendieck duality.

Finally, since the category
${}^{\perp}{\mathcal{P}}\simeq{\mathcal{P}}^{\perp}$ is smooth, it has a
strong generator by [Lun10, Lemma 3.5 and Lemma 3.6(a)], and since it is also
proper and idempotent complete, it is saturated by [BVdB03, Theorem 1.3].
Therefore, its image in the proper categories
${\mathbf{D}}^{\mathrm{perf}}(X)$ and
${\mathbf{D}}^{\mathrm{perf}}(X)\otimes\omega_{X}^{\bullet}$ is admissible. ∎

######  Example 4.4.

Assume ${\mathcal{P}}$ absorbs singularities of a projective Gorenstein
variety $X$. Then $\omega_{X}^{\bullet}$ is a shift of a line bundle and by
Lemma 4.3 we have
${}^{\perp}{\mathcal{P}},{\mathcal{P}}^{\perp}\subset{\mathbf{D}}^{\mathrm{perf}}(X)$.
If in addition ${\mathcal{P}}$ has an admissible semiorthogonal decomposition

(26)
${\mathcal{P}}=\langle{\mathbf{D}}^{\mathrm{b}}(R_{1}),\dots,{\mathbf{D}}^{\mathrm{b}}(R_{m})\rangle,$

where $R_{i}$ are finite-dimensional associative algebras, then
$\langle{\mathcal{P}}^{\perp},{\mathcal{P}}\rangle$ is a “Kawamata
semiorthogonal decomposition” in the sense of [KPS21]. Conversely, a Kawamata
semiorthogonal decomposition provides absorption of singularities. For
examples of Kawamata decompositions see [KPS21, PS21a, Xie21].

Recall Definition 1.3 of smoothing and Definition 1.4 of (thick) deformation
absorption. We denote by

(27) $\upiota\colon X\hookrightarrow{\mathcal{X}}$

the embedding of the central fiber. Note that since $X$ is a hypersurface in a
smooth variety ${\mathcal{X}}$, it follows that $X$ has (at worst) Gorenstein
singularities. Furthermore, since $X\subset{\mathcal{X}}$ is a Cartier divisor
with trivial normal bundle, for each $F\in{\mathbf{D}}^{\mathrm{b}}(X)$ we
have the standard distinguished triangle

(28) $F[1]\to\upiota^{*}\upiota_{*}(F)\to F\to F[2].$

Recall the definition of a ${\mathbb{P}}^{\infty,{q}}$-object (Definition 2.6)
and its canonical self-extension (Definition 2.8).

######  Lemma 4.5.

Let $X$ be a Gorenstein projective variety with a smoothing (27). If
${\mathrm{P}}\in{\mathbf{D}}^{\mathrm{b}}(X)$ is a
${\mathbb{P}}^{\infty,{q}}$-object then $q\in\\{1,2\\}$. Moreover,

  * •

if $q=2$ the morphism ${\mathrm{P}}\to{\mathrm{P}}[2]$ in (28) for
$F={\mathrm{P}}$ is given by $\uptheta$ and
$\upiota^{*}\upiota_{*}{\mathrm{P}}\cong{\mathrm{M}}$,

  * •

if $q=1$ the morphism ${\mathrm{P}}\to{\mathrm{P}}[2]$ in (28) for
$F={\mathrm{P}}$ is given by $\uptheta^{2}$ and
$\upiota^{*}\upiota_{*}{\mathrm{P}}\cong{\mathrm{M}}^{(2)}$,

where ${\mathrm{M}}$ is the canonical self-extension of ${\mathrm{P}}$ and the
object ${\mathrm{M}}^{(2)}$ is defined in (11). In both cases ${\mathrm{M}}$
is a perfect complex on $X$.

###### Proof.

First, we note that the smoothness of ${\mathcal{X}}$ implies that
$\upiota_{*}{\mathrm{P}}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is a
perfect complex, hence

$\upiota^{*}\upiota_{*}{\mathrm{P}}\in{\mathbf{D}}^{\mathrm{perf}}(X).$

On the other hand, consider the triangle (28) for $F={\mathrm{P}}$. Since
${\mathrm{P}}$ itself is not perfect (because
$\operatorname{\mathrm{Ext}}^{*}({\mathrm{P}},{\mathrm{P}})$ is infinite-
dimensional), it follows that the connecting morphism
${\mathrm{P}}\to{\mathrm{P}}[2]$ is nonzero. By definition of a
${\mathbb{P}}^{\infty,{q}}$-object it follows that $q\leq 2$, and moreover, if
$q=2$ the connecting morphism is isomorphic to $\uptheta$, and if $q=1$ it is
isomorphic to $\uptheta^{2}$. Finally, if $q=2$ it follows that
$\upiota^{*}\upiota_{*}{\mathrm{P}}\cong{\mathrm{M}}$ is perfect, and if $q=1$
it follows that $\upiota^{*}\upiota_{*}{\mathrm{P}}\cong{\mathrm{M}}^{(2)}$ is
perfect, hence ${\mathrm{M}}$ is also perfect by Corollary 2.11. ∎

We would like to emphasize the following consequence:

######  Corollary 4.6.

If $X$ is a smoothable projective variety and
${\mathrm{P}}\in{\mathbf{D}}^{\mathrm{b}}(X)$ is a
${\mathbb{P}}^{\infty,{q}}$-object then $q\in\\{1,2\\}$.

We are ready to prove Theorem 1.5. Recall that for a variety ${\mathcal{X}}/B$
a subcategory ${\mathcal{D}}\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$
is called $B$-linear if it is closed under tensor products with pullbacks of
perfect complexes on $B$. Given a $B$-linear triangulated category
${\mathcal{D}}\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ and a morphism
$\varphi\colon B^{\prime}\to B$, base change category
${\mathcal{D}}_{B^{\prime}}\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}\times_{B}B^{\prime})$
is defined in [Kuz11, Theorem 5.6] under appropriate assumptions
(admissibility of ${\mathcal{D}}$ and of either of its orthogonals and
finiteness of cohomological amplitude of the projection functor to
${\mathcal{D}}$) as

${\mathcal{D}}_{B^{\prime}}\coloneqq\operatorname{\mathrm{hocolim}}\Big{(}{\tilde{\varphi}}^{*}({\mathcal{D}}\cap{\mathbf{D}}^{\mathrm{perf}}({\mathcal{X}}))\Big{)},$

where
${\tilde{\varphi}}\colon{\mathcal{X}}\times_{B}B^{\prime}\to{\mathcal{X}}$ is
the morphism induced by $\varphi$. In other words, first one considers “the
perfect part” ${\mathcal{D}}\cap{\mathbf{D}}^{\mathrm{perf}}({\mathcal{X}})$
of ${\mathcal{D}}$, then one considers its pullback
${\tilde{\varphi}}^{*}({\mathcal{D}}\cap{\mathbf{D}}^{\mathrm{perf}}({\mathcal{X}}))$
along ${\tilde{\varphi}}$ (this gives “the perfect part” of the base change
category ${\mathcal{D}}_{B^{\prime}}$), and finally one considers all homotopy
colimits contained in
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}\times_{B}B^{\prime})$ of chains of
morphisms of objects in
${\tilde{\varphi}}^{*}({\mathcal{D}}\cap{\mathbf{D}}^{\mathrm{perf}}({\mathcal{X}}))$.
In the special case where $B^{\prime}$ is a point $b\in B$, we obtain the
_fiber_ ${\mathcal{D}}_{b}$ of ${\mathcal{D}}$.

###### Proof of Theorem 1.5.

By definition of deformation absorption the subcategory
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$
is admissible, hence we have a semiorthogonal decomposition

(29)
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle\mathbf{thick}(\upiota_{*}{\mathcal{P}}),{\mathcal{D}}\rangle.$

Its first component is supported set-theoretically on the central fiber of
${\mathcal{X}}$ over $B$, hence it is $B$-linear. By [Kuz06, Lemma 2.36] the
second component
${\mathcal{D}}\coloneqq{}^{\perp}(\mathbf{thick}(\upiota_{*}{\mathcal{P}}))$
is $B$-linear as well. The components of (29) are both admissible because
${\mathcal{X}}$ is smooth over ${\Bbbk}$ and proper over $B$, and their
projection functors have finite cohomology amplitude by [Kuz08, Proposition
2.5] since ${\mathcal{X}}$ is smooth and quasiprojective. Therefore, by
[Kuz11, Theorem 5.6] we can talk about base change of the $B$-linear category
${\mathcal{D}}$; in particular, the fibers ${\mathcal{D}}_{b}$ and
${\mathcal{D}}_{o}$ are defined.

Now the equality
${\mathcal{D}}_{b}={\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{b})$ for $b\neq o$
follows from base change along the point embedding $\\{b\\}\hookrightarrow B$
applied to (29), since the base change of the first component of (29) is zero
(because the support of any object in it does not intersect the fiber
${\mathcal{X}}_{b}$). Similarly, by base change we have a semiorthogonal
decomposition

${\mathbf{D}}^{\mathrm{b}}(X)=\langle\mathbf{thick}(\upiota_{*}{\mathcal{P}})_{o},{\mathcal{D}}_{o}\rangle.$

Thus, we need to identify its first component with ${\mathcal{P}}$.

On the one hand, by [Kuz11, Corollary 5.7] we have
${\mathcal{P}}\subset\mathbf{thick}(\upiota_{*}{\mathcal{P}})_{o}$. On the
other hand, triangle (28) applied to an object $F\in{\mathcal{P}}$ shows that
$\upiota^{*}\upiota_{*}F\in{\mathcal{P}}$. Since the functor $\upiota^{*}$ is
triangulated, we have $\upiota^{*}F\in{\mathcal{P}}$ for any $F$ in
$\langle\upiota_{*}{\mathcal{P}}\rangle$ and since ${\mathcal{P}}$ is
idempotent complete, $\upiota^{*}F\in{\mathcal{P}}$ for any
$F\in\mathbf{thick}(\upiota_{*}{\mathcal{P}})$. This proves that the perfect
part of $\mathbf{thick}(\upiota_{*}{\mathcal{P}})_{o}$ is contained in
${\mathcal{P}}$, and since ${\mathcal{P}}$ is closed under homotopy colimits
(because by Lemma 4.3 the category ${\mathcal{P}}$ is an orthogonal to a
subcategory consisting of perfect objects), we have the inclusion
$\mathbf{thick}(\upiota_{*}{\mathcal{P}})_{o}\subset{\mathcal{P}}$. Combining
the two inclusions proved above we obtain the equality
${\mathcal{P}}=\mathbf{thick}(\upiota_{*}{\mathcal{P}})_{o}$.

Finally, smoothness and properness of ${\mathcal{D}}$ over $B$ follow from
[Kuz22, Theorem 2.10]. ∎

###### Proof of Theorem 1.8.

Let $f\colon{\mathcal{X}}\to B$ be a smoothing of $X$ and recall that
$\upiota\colon X\to{\mathcal{X}}$ denotes the embedding of the central fiber.
Consider the distinguished triangle (28) for $F={\mathrm{P}}_{i}$. Applying
Lemma 4.5 we conclude that

$\upiota^{*}\upiota_{*}({\mathrm{P}}_{i})\cong{\mathrm{M}}_{i}$

is the canonical self-extension of ${\mathrm{P}}_{i}$. As a consequence, using
adjunction and (10) we obtain

$\operatorname{\mathrm{Ext}}^{\bullet}(\upiota_{*}({\mathrm{P}}_{i}),\upiota_{*}({\mathrm{P}}_{i}))\cong\operatorname{\mathrm{Ext}}^{\bullet}(\upiota^{*}\upiota_{*}({\mathrm{P}}_{i}),{\mathrm{P}}_{i})\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}}_{i},{\mathrm{P}}_{i})\cong{\Bbbk},$

which means that
$\upiota_{*}({\mathrm{P}}_{i})\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is
an exceptional object. Similarly, if $i>j$ we have

$\operatorname{\mathrm{Ext}}^{\bullet}(\upiota_{*}({\mathrm{P}}_{i}),\upiota_{*}({\mathrm{P}}_{j}))\cong\operatorname{\mathrm{Ext}}^{\bullet}(\upiota^{*}\upiota_{*}({\mathrm{P}}_{i}),{\mathrm{P}}_{j})\cong\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}}_{i},{\mathrm{P}}_{j}),$

and using the triangle (8) for ${\mathrm{M}}_{i}$ and ${\mathrm{P}}_{i}$, we
see that
$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}}_{i},{\mathrm{P}}_{j})=0$.
Thus, $\upiota_{*}({\mathrm{P}}_{1}),\dots,\upiota_{*}({\mathrm{P}}_{r})$ is
an exceptional collection of compactly supported objects in
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$, hence the subcategory
${\mathcal{P}}\coloneqq\langle{\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r}\rangle$
provides deformation absorption with

$\langle\upiota_{*}{\mathcal{P}}\rangle=\langle\upiota_{*}({\mathrm{P}}_{1}),\dots,\upiota_{*}({\mathrm{P}}_{r})\rangle\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}).$

Since this is true for any smoothing of $X$, the subcategory
${\mathcal{P}}\subset{\mathbf{D}}^{\mathrm{b}}(X)$ provides a universal
deformation absorption. ∎

######  Remark 4.7.

The fact that $\upiota_{*}{\mathrm{P}}$ is an exceptional object on the total
space of a smoothing is an analog and the limiting case of the fact that the
direct image of a ${\mathbb{P}}^{n}$-object to the total space of a
deformation is a $(2n+1)$-spherical object, see [HT06].

Combining Theorem 1.8 with Theorem 4.2 we obtain the following

######  Corollary 4.8.

Under the conditions of Theorem 4.2, if
${\mathcal{T}}\simeq{\mathbf{D}}^{\mathrm{b}}(X)$ for a projective variety $X$
and $p=1$ then the category ${\mathcal{P}}$ constructed therein provides a
universal deformation absorption of singularities of $X$.

Finally, we prove Theorem 1.9.

###### Proof of Theorem 1.9.

Let $f\colon{\mathcal{X}}\to B$ be a smoothing of $X$ and recall that
$\upiota\colon X\to{\mathcal{X}}$ denotes the embedding of the central fiber.
Consider the stack
${\mathcal{D}}^{\mathbf{b}}_{\mathrm{pug}}({\mathcal{X}}/B)$ of universally
gluable $B$-perfect complexes on ${\mathcal{X}}$ as defined in [Lie06, §2.1].
Note that under the assumption of smoothness of the base $B$ an object
${\mathcal{F}}\in{\mathbf{D}}_{\mathrm{qc}}({\mathcal{X}})$ is $B$-perfect if
and only if ${\mathcal{F}}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$. Recall
also that the “gluability condition” for an object
${\mathcal{F}}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is just

$f_{*}\operatorname{\mathrm{R}\mathcal{H}\mathit{om}}({\mathcal{F}},{\mathcal{F}})\in{\mathbf{D}}^{\mathrm{b}}(B)^{\geq
0},$

where the right-hand side stands for the subcategory of objects with zero
sheaf cohomology in negative degrees, and “universal gluability” is gluability
after arbitrary base change.

For each $1\leq i\leq r$ consider the canonical self-extension
${\mathrm{M}}_{i}\in{\mathbf{D}}^{\mathrm{b}}(X)$ of the
${\mathbb{P}}^{\infty,{1}}$-object
${\mathrm{P}}_{i}\in{\mathbf{D}}^{\mathrm{b}}(X)$. By Lemma 4.5 we have
${\mathrm{M}}_{i}\in{\mathbf{D}}^{\mathrm{perf}}(X)$ and by Lemma 2.9 we have

$\operatorname{\mathrm{Ext}}^{\bullet}({\mathrm{M}}_{i},{\mathrm{M}}_{i})\cong{\mathsf{A}}_{0}\cong{\Bbbk}[\upepsilon]/\upepsilon^{2},\qquad\deg(\upepsilon)=0,$

hence ${\mathrm{M}}_{i}$ gives a ${\Bbbk}$-point of the stack
${\mathcal{D}}^{\mathbf{b}}_{\mathrm{pug}}({\mathcal{X}}/B)$ over the point
$o\in B$. Moreover, the stack
${\mathcal{D}}^{\mathbf{b}}_{\mathrm{pug}}({\mathcal{X}}/B)$ is locally of
finite presentation by [Lie06, Theorem 4.2.1] and it follows from [Lie06,
Theorem 3.1.1] that the natural morphism of stacks

${\mathcal{D}}^{\mathbf{b}}_{\mathrm{pug}}({\mathcal{X}}/B)\to B$

is smooth at the point $[{\mathrm{M}}_{i}]$. Therefore, étale locally this
morphism admits a section passing through $[{\mathrm{M}}_{i}]$. In other
words, after an étale base change there is an object
${\mathcal{M}}_{i}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ such that

$\upiota^{*}({\mathcal{M}}_{i})\cong{\mathrm{M}}_{i}.$

Now consider the object

${\mathcal{R}}_{i}\coloneqq
f_{*}\operatorname{\mathrm{R}\mathcal{H}\mathit{om}}({\mathcal{M}}_{i},{\mathcal{M}}_{i})\cong
f_{*}({\mathcal{M}}_{i}^{\vee}\otimes{\mathcal{M}}_{i})\in{\mathbf{D}}^{\mathrm{b}}(B).$

By base change the (derived) restriction of ${\mathcal{R}}_{i}$ to the point
$o\in B$ is isomorphic to the algebra ${\mathsf{A}}_{0}$, in particular, it is
a 2-dimensional vector space sitting in degree zero. Therefore, shrinking $B$
further we may assume that ${\mathcal{R}}_{i}$ is a locally free sheaf of rank
$2$. Note also that by construction ${\mathcal{R}}_{i}$ is a sheaf of
${\mathcal{O}}_{B}$-algebras, and since it has rank $2$, it is commutative.
Consider now the scheme

${\mathcal{Z}}_{i}\coloneqq\operatorname{\mathrm{Spec}}_{B}({\mathcal{R}}_{i}).$

Since ${\mathcal{R}}_{i}$ is locally free of rank $2$, the scheme
${\mathcal{Z}}_{i}$ is flat over $B$ of degree $2$. Furthermore, the sheaf
${\mathcal{M}}_{i}$ has a structure of a ${\mathcal{R}}_{i}$-module, hence it
is isomorphic to the pushforward of a sheaf ${\tilde{{\mathcal{M}}}}_{i}$ on
${\mathcal{X}}\times_{B}{\mathcal{Z}}_{i}$ under the natural projection
${\mathcal{X}}\times_{B}{\mathcal{Z}}_{i}\to{\mathcal{X}}$. Consider the
$B$-linear Fourier–Mukai functor

$\Phi_{{\tilde{{\mathcal{M}}}}_{i}}\colon{\mathbf{D}}^{\mathrm{b}}(B,{\mathcal{R}}_{i})={\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i})\to{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}).$

Its fiber over the point $o\in B$ is the functor
$\Phi_{{\mathrm{M}}_{i}}\colon{\mathbf{D}}^{\mathrm{b}}({\mathsf{A}}_{0})={\mathbf{D}}^{\mathrm{b}}(({\mathcal{Z}}_{i})_{o})\to{\mathbf{D}}^{\mathrm{b}}(X)$
which is fully faithful by Lemma 2.10. By [FK18, Proposition 2.16] the functor
$\Phi_{{\tilde{{\mathcal{M}}}}_{i}}$ is fully faithful over an open
neighborhood of $o\in B$, so shrinking $B$ further, we may assume it is fully
faithful everywhere. A similar argument using semiorthogonality of the images
$\langle{\mathrm{P}}_{i}\rangle$ of the functors $\Phi_{{\mathrm{M}}_{i}}$
shows that after shrinking $B$ further we may assume that the images of
functors $\Phi_{{\tilde{{\mathcal{M}}}}_{i}}$ are semiorthogonal. As
Fourier–Mukai functors always have adjoints, we conclude that there is a
$B$-linear semiorthogonal decomposition

(30)
${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{1}),\dots,{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{r}),{\mathcal{D}}\rangle,$

where the first $r$ components are the images of
$\Phi_{{\tilde{{\mathcal{M}}}}_{i}}$, and the last component ${\mathcal{D}}$
is their orthogonal complement.

Now, consider the base change of this decomposition to any geometric point
$b\neq o$. We obtain

${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{b})=\langle{\mathbf{D}}^{\mathrm{b}}(({\mathcal{Z}}_{1})_{b}),\dots,{\mathbf{D}}^{\mathrm{b}}(({\mathcal{Z}}_{r})_{b}),{\mathcal{D}}_{b}\rangle.$

On the one hand,
${\mathbf{D}}^{\mathrm{b}}(({\mathcal{Z}}_{i})_{b})\simeq{\mathbf{D}}^{\mathrm{b}}(({\mathcal{R}}_{i})_{b})$
and $({\mathcal{R}}_{i})_{b}$, which is a 2-dimensional algebra. On the other
hand, since this is a semiorthogonal component in the derived category of a
smooth projective variety ${\mathcal{X}}_{b}$, the algebra should have finite
homological dimension, hence it should be étale over the residue field of the
point $b$. Therefore, ${\mathcal{Z}}_{i}\to B$ is étale over
$B\setminus\\{o\\}$. Moreover, ${\mathcal{D}}_{b}$ is also a smooth and proper
category.

Furthermore, by construction the base change of
${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i})$ to the point $o$ coincides with
the subcategory
$\langle{\mathrm{P}}_{i}\rangle\subset{\mathbf{D}}^{\mathrm{b}}(X)$, hence
${\mathcal{D}}_{o}\simeq{}^{\perp}{\mathcal{P}}$; in particular, it is smooth
and proper. Finally, we use [Kuz22, Theorem 2.10] to conclude that
${\mathcal{D}}$ is smooth and proper over $B$.

It remains to prove the uniqueness of (30). For this we note that if
${\mathcal{F}}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})$ is an object such
that
$\upiota^{*}{\mathcal{F}}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i})_{o}$
for some $1\leq i\leq r$ then there is a Zariski neighborhood $U\subset B$ of
$o$ such that the pullback ${\mathcal{F}}_{U}$ of ${\mathcal{F}}$ to
${\mathcal{X}}_{U}={\mathcal{X}}\times_{B}U$ is contained in
${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i}\times_{B}U)\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{X}}_{U})$.
Indeed, if ${\mathcal{F}}_{j}\in{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{j})$
and ${\mathcal{F}}_{\mathcal{D}}\subset{\mathcal{D}}$ are the components of
${\mathcal{F}}$ with respect to (30) then the assumption means that

$\upiota^{*}{\mathcal{F}}_{j}=0\quad\text{for $j\neq
i$}\qquad\text{and}\qquad\upiota^{*}{\mathcal{F}}_{\mathcal{D}}=0.$

But then the objects ${\mathcal{F}}_{j}$ for $j\neq i$ and
${\mathcal{F}}_{\mathcal{D}}$ vanish on a Zariski neighborhood of
$X\subset{\mathcal{X}}$, and since $f\colon{\mathcal{X}}\to B$ is proper, they
vanish on the preimage of a Zariski neighborhood $U\subset B$ of $o$, hence
their pullbacks to ${\mathcal{X}}_{U}$ vanish, and hence the pullback of
${\mathcal{F}}$ is equal to the pullback of ${\mathcal{F}}_{i}$ and therefore
it is contained in ${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i}\times_{B}U)$.

Now, if

${\mathbf{D}}^{\mathrm{b}}({\mathcal{X}})=\langle{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{1}),\dots,{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{r}),{\mathcal{D}}^{\prime}\rangle$

is another $B$-linear semiorthogonal decomposition such that its base change
to $o$ gives the decomposition
${\mathbf{D}}^{\mathrm{b}}(X)=\langle{\mathrm{P}}_{1},\dots,{\mathrm{P}}_{r},{}^{\perp}{\mathcal{P}}\rangle$,
then applying the above observation to generators
${\mathcal{G}}_{1},\dots,{\mathcal{G}}_{r}$ of the components
${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{1}),\dots,{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{r})$
we find Zariski neighbourhoods $U_{1},\dots,U_{r}\subset B$ of $o$ such that
the pullback of ${\mathcal{G}}_{i}$ to ${\mathcal{X}}\times_{B}U_{i}$ is
contained in ${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i}\times_{B}U_{i})$.
Taking $U=U_{1}\cap\dots\cap U_{r}$, we deduce that

${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{i}\times_{B}U)\subset{\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i}\times_{B}U)$

for all $i$. The same argument shows the opposite inclusion (possibly after
shrinking the base further), and therefore we finally obtain an equality
${\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}^{\prime}_{i}\times_{B}U)={\mathbf{D}}^{\mathrm{b}}({\mathcal{Z}}_{i}\times_{B}U)$.
∎

##  5\. Crepant categorical resolutions for nodal varieties

In this section we construct crepant categorical resolutions for varieties
with ordinary double points or nodal varieties. In §5.1 we consider the blowup
morphism $\pi\colon\operatorname{\mathrm{Bl}}_{x}(X)\to X$ for an ordinary
double point $x\in X$ and check that the pushforward functor
$\pi_{*}\colon{\mathbf{D}}^{\mathrm{b}}(\operatorname{\mathrm{Bl}}_{x}(X))\to{\mathbf{D}}^{\mathrm{b}}(X)$
is a Verdier localization. In §5.2, we consider a nodal variety with singular
points $x_{1},\dots,x_{r}\in X$ and find an admissible subcategory
${\mathcal{D}}\subset{\mathbf{D}}^{\mathrm{b}}(\operatorname{\mathrm{Bl}}_{x_{1},\dots,x_{r}}(X))$
such that
$\pi_{*}|_{\mathcal{D}}\colon{\mathcal{D}}\to{\mathbf{D}}^{\mathrm{b}}(X)$ is
a crepant categorical resolution and
$\operatorname{\mathrm{Ker}}(\pi_{*}|_{\mathcal{D}})$ is generated by a
completely orthogonal collection of spherical objects.

Starting from §5.2 we work over an algebraically closed field ${\Bbbk}$ of
characteristic not equal to $2$.

###  5.1. Bondal–Orlov localization

It is a conjecture of Bondal and Orlov (see [BO02, §5] and [Efi20, Conjecture
1.9]) that if $X$ is a variety with rational singularities, and
$\pi\colon{\tilde{X}}\to X$ is a resolution then the pushforward functor
$\pi_{*}\colon{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})\to{\mathbf{D}}^{\mathrm{b}}(X)$
is a Verdier localization. The main advance in this direction was obtained in
[Efi20]; we prove a slight extension of his results in a convenient form.
Recall from Definition 1.10 the notion of a categorical contraction.

######  Lemma 5.1.

A functor $\pi_{*}\colon{\tilde{{\mathcal{T}}}}\to{\mathcal{T}}$ is a Verdier
localization if and only if it is a categorical contraction and the induced
map on the Grothendieck groups
${\mathrm{K}}_{0}({\tilde{{\mathcal{T}}}})\to{\mathrm{K}}_{0}({\mathcal{T}})$
is surjective.

###### Proof.

If $\pi_{*}$ is a categorical contraction and
${\mathrm{K}}_{0}({\tilde{{\mathcal{T}}}})\to{\mathrm{K}}_{0}({\mathcal{T}})$
is surjective, then
${\mathrm{K}}_{0}(\operatorname{\mathrm{Im}}(\pi_{*}))={\mathrm{K}}_{0}({\mathcal{T}})$
and by Thomason’s theorem on classification of dense subcategories [Tho97,
Theorem 2.1], we have the equivalence
${\tilde{{\mathcal{T}}}}/\operatorname{\mathrm{Ker}}(\pi_{*})\simeq\operatorname{\mathrm{Im}}(\pi_{*})={\mathcal{T}}$.
The other implication is obvious. ∎

We denote

${\mathrm{G}}_{0}(X)\coloneqq{\mathrm{K}}_{0}({\mathbf{D}}^{\mathrm{b}}(X))$

the Grothendieck group of the bounded derived category of $X$.

######  Theorem 5.2 ([Efi20]).

Let $\pi\colon{\tilde{X}}\to X$ be a proper birational morphism. Let
$\upzeta\colon Z\hookrightarrow X$ be a closed subscheme such that
$E\coloneqq\pi^{-1}(Z)$ is a Cartier divisor, the restriction
$\pi\colon{\tilde{X}}\setminus E\to X\setminus Z$ is an isomorphism. Assume
that

(31)
$\pi_{*}{\mathcal{O}}_{{\tilde{X}}}(-mE)\cong{\mathcal{J}}^{m}_{Z}\qquad\text{for
all~{}$m\geq 0$.}$

Let $p\colon E\to Z$ and $\upeta\colon E\hookrightarrow{\tilde{X}}$ be the
natural morphisms so that we have a Cartesian square

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\upeta}$$\textstyle{{\tilde{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upzeta}$$\textstyle{X}$

If $p_{*}\colon{\mathbf{D}}^{\mathrm{b}}(E)\to{\mathbf{D}}^{\mathrm{b}}(Z)$ is
a Verdier localization or a categorical contraction, then
$\pi_{*}\colon{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})\to{\mathbf{D}}^{\mathrm{b}}(X)$
is also a Verdier localization or a categorical contraction, respectively. In
both cases the category $\operatorname{\mathrm{Ker}}(\pi_{*})$ is generated by
$\upeta_{*}(\operatorname{\mathrm{Ker}}(p_{*}))$.

###### Proof.

Assume that $p_{*}$ is a categorical contraction. Then conditions of [Efi20,
Theorem 8.22] (recall the difference in terminology mentioned in Remark 1.11)
are satisfied, so that $\pi_{*}$ is also a categorical contraction and
$\operatorname{\mathrm{Ker}}(\pi_{*})$ is thickly generated by
$\upeta_{*}(\operatorname{\mathrm{Ker}}(p_{*}))$.

To show that $\operatorname{\mathrm{Ker}}(\pi_{*})$ is generated by
$\upeta_{*}(\operatorname{\mathrm{Ker}}(p_{*}))$ as a triangulated category,
by Thomason’s theorem (see proof of Lemma 5.1) it suffices to check that the
induced map on the Grothendieck group

(32) ${\mathrm{K}}_{0}(\operatorname{\mathrm{Ker}}(p_{*}))\xrightarrow{\
\upeta_{*}\ }{\mathrm{K}}_{0}(\operatorname{\mathrm{Ker}}(\pi_{*}))$

is surjective. For this we consider the diagram of categories

$\textstyle{\operatorname{\mathrm{Ker}}(p_{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upeta_{*}}$$\textstyle{{\mathbf{D}}^{\mathrm{b}}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upeta_{*}}$$\scriptstyle{p_{*}}$$\textstyle{{{\mathbf{D}}^{\mathrm{b}}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upzeta_{*}}$$\textstyle{\operatorname{\mathrm{Ker}}(\pi_{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{D}}^{\mathrm{b}}_{E}({\tilde{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{{{\mathbf{D}}^{\mathrm{b}}_{Z}(X)}.}$

The horizontal rows are Verdier localization up to direct summands “exact
sequences” by [Efi20, Theorem 8.13]. Now we apply Schlichting’s K-theory
machinery as explained in [PS21b, §1.2] to this diagram; in this way we obtain
a commutative diagram of ${\mathrm{K}}$-groups

$\textstyle{{\mathrm{G}}_{1}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upeta_{*}}$$\scriptstyle{p_{*}}$$\textstyle{{\mathrm{G}}_{1}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upzeta_{*}}$$\textstyle{{\mathrm{K}}_{0}(\operatorname{\mathrm{Ker}}(p_{*}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upeta_{*}}$$\textstyle{{\mathrm{G}}_{0}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upeta_{*}}$$\scriptstyle{p_{*}}$$\textstyle{{\mathrm{G}}_{0}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\upzeta_{*}}$$\textstyle{{\mathrm{K}}_{1}({\mathbf{D}}^{\mathrm{b}}_{E}({\tilde{X}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{{\mathrm{K}}_{1}({\mathbf{D}}^{\mathrm{b}}_{Z}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{K}}_{0}(\operatorname{\mathrm{Ker}}(\pi_{*}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{K}}_{0}({\mathbf{D}}^{\mathrm{b}}_{E}({\tilde{X}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{{\mathrm{K}}_{0}({\mathbf{D}}^{\mathrm{b}}_{Z}(X))}$

Applying Quillen’s devissage (see [Qui73, §5, Theorem 4] and [Orl11, Lemma
2.1]) we conclude that the first two and the last two vertical arrows in the
diagram are isomorphisms, hence the middle arrow (32) is also an isomorphism.

Assume $p_{*}$ is a Verdier localization. By Lemma 5.1 we only need to show
that the map

(33) $\pi_{*}\colon{\mathrm{G}}_{0}({\tilde{X}})\to{\mathrm{G}}_{0}(X),$

is surjective. For any coherent sheaf ${\mathcal{F}}$ on $X$ the natural
morphism
${\mathcal{F}}\to{\mathrm{R}}^{0}\pi_{*}({\mathrm{L}}_{0}\pi^{*}{\mathcal{F}})\to\pi_{*}({\mathrm{L}}_{0}\pi^{*}{\mathcal{F}})$
is an isomorphism on the complement of $Z$, so it is enough to show that the
classes of sheaves set-theoretically supported at $Z$ are in the image of
(33). Since any such sheaf is filtered by sheaves scheme-theoretically
supported on $Z$, and $p_{*}\colon{\mathrm{G}}_{0}(E)\to{\mathrm{G}}_{0}(Z)$
is surjective (again by Lemma 5.1), we conclude that (33) is surjective as
well. ∎

Let $x\in X$ be a normal isolated singularity and let
$\pi\colon{\tilde{X}}\coloneqq\operatorname{\mathrm{Bl}}_{x}(X)\to X$ be the
blowup of $x\in X$ with exceptional divisor $E\subset X$; we denote by
${\mathcal{O}}_{E}(1)\coloneqq{\mathcal{O}}_{E}(-E)$ its conormal bundle. We
will use the following definition to verify the assumption (31).

######  Definition 5.3.

We say that a normal isolated singularity $x\in X$ with the maximal ideal
${\mathfrak{m}}_{X,x}\subset{\mathcal{O}}_{X,x}$ is acyclic projectively
normal if the following conditions hold for all $m\geq 0$:

  1. (a)

the canonical map
${\mathfrak{m}}_{X,x}^{m}/{\mathfrak{m}}_{X,x}^{m+1}\to{\mathrm{H}}^{0}(E,{\mathcal{O}}_{E}(m))$
is an isomorphism;

  2. (b)

${\mathrm{H}}^{i}(E,{\mathcal{O}}_{E}(m))=0$ for all $i>0$.

######  Example 5.4.

If $X$ is a cone over a smooth Fano complete intersection of positive
dimension, or more generally $x\in X$ is an isolated singular point such that
the tangent cone ${\mathrm{C}}_{X,x}$ is a Fano complete intersection, then
$x\in X$ is acyclic projectively normal. Indeed, complete intersections are
projectively normal which using [Har77, Ex. II.5.14] implies (a). The
vanishing in (b) follows easily from the standard computation of cohomology of
line bundles on a complete intersection, cf. the proof of Lemma 5.7.

######  Lemma 5.5.

Let $x\in X$ be an acyclic projectively normal singularity. Then
$\pi_{*}({\mathcal{O}}_{\tilde{X}})\cong{\mathcal{O}}_{X}$; in particular, if
${\tilde{X}}$ is smooth, then $(X,x)$ is a rational singularity. Moreover, the
condition (31) is satisfied for $Z=\\{x\\}$.

###### Proof.

Let $m\geq 1$. The structure sheaf
${\mathcal{O}}_{mE}\cong{\mathcal{O}}_{\tilde{X}}/{\mathcal{I}}_{E}^{m}$ of
the $m$-th infinitesimal neighborhood of the exceptional divisor $E$ of the
blowup has a filtration with factors
${\mathcal{I}}_{E}^{k}/{\mathcal{I}}_{E}^{k+1}\cong{\mathcal{O}}_{E}(k)$,
$0\leq k\leq m-1$; all these sheaves have no higher cohomology by Definition
5.3(b), hence

${\mathrm{H}}^{>0}({\tilde{X}},{\mathcal{O}}_{mE})=0$

for all $m\geq 1$. It follows from the formal functions theorem [Har77,
Theorem III.11.1] that

${\mathrm{R}}^{>0}\pi_{*}({\mathcal{O}}_{\tilde{X}})=0.$

Since $X$ is normal and $\pi$ birational, we have
${\mathrm{R}}^{0}\pi_{*}({\mathcal{O}}_{\tilde{X}})\cong{\mathcal{O}}_{X}$, so
that $\pi_{*}({\mathcal{O}}_{\tilde{X}})\cong{\mathcal{O}}_{X}$.

We now prove that
$\pi_{*}({\mathcal{O}}_{{{\tilde{X}}}}(-mE))\cong{\mathcal{J}}_{X,x}^{m}$ by
induction on $m\geq 0$. The base of induction, $m=0$, is proved above. Now
assume the claim is proved for some $m\geq 0$ and consider the exact sequence

$0\to{\mathcal{O}}_{{{\tilde{X}}}}(-(m+1)E)\to{\mathcal{O}}_{{{\tilde{X}}}}(-mE)\to{\mathcal{O}}_{E}(m)\to
0.$

Applying $\pi_{*}$ and using the induction hypothesis and Definition 5.3(b) we
obtain a distinguished triangle

$\pi_{*}({\mathcal{O}}_{{{\tilde{X}}}}(-(m+1)E))\to{\mathcal{J}}_{X,x}^{m}\to{\mathrm{H}}^{0}(E,{\mathcal{O}}_{E}(m))\otimes{\mathcal{O}}_{x}.$

The second arrow in it factors as the composition

${\mathcal{J}}_{X,x}^{m}\twoheadrightarrow{\mathcal{J}}_{X,x}^{m}/{\mathcal{J}}_{X,x}^{m+1}={\mathfrak{m}}_{X,x}^{m}/{\mathfrak{m}}_{X,x}^{m+1}\to{\mathrm{H}}^{0}(E,{\mathcal{O}}_{E}(m)).$

By Definition 5.3(a) the last arrow is an isomorphism, hence
$\pi_{*}({\mathcal{O}}_{{{\tilde{X}}}}(-(m+1)E))\cong{\mathcal{J}}_{X,x}^{m+1}$.
∎

Now we deduce the Bondal–Orlov localization conjecture for acyclic
projectively normal singularities. Note that Definition 5.3 for $m=0$ ensures
that ${\mathcal{O}}_{E}$ is an exceptional object in
${\mathbf{D}}^{\mathrm{b}}(E)$, hence we have a semiorthogonal decomposition
${\mathbf{D}}^{\mathrm{b}}(E)=\langle{\mathcal{O}}_{E}^{\perp},{\mathcal{O}}_{E}\rangle$;
its first component ${\mathcal{O}}_{E}^{\perp}$ plays the crucial role.

######  Corollary 5.6.

Let $\pi\colon{\tilde{X}}=\operatorname{\mathrm{Bl}}_{x}(X)\to X$ be the
blowup of an acyclic projectively normal point. Then the functor
$\pi_{*}\colon{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})\to{\mathbf{D}}^{\mathrm{b}}(X)$
is a Verdier localization and the subcategory
$\operatorname{\mathrm{Ker}}(\pi_{*})\subset{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$
is generated by
$\upeta_{*}({\mathcal{O}}_{E}^{\perp})\subset{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})$
where $\upeta\colon E\to{\tilde{X}}$ is the embedding of the exceptional
divisor.

Moreover, the map
$\pi_{*}\colon{\mathrm{G}}_{0}({\tilde{X}})\to{\mathrm{G}}_{0}(X)$ is
surjective with kernel generated by
$\upeta_{*}([{\mathcal{O}}_{E}^{\perp}])\subset{\mathrm{G}}_{0}({\tilde{X}})$.

###### Proof.

Since ${\mathcal{O}}_{E}\in{\mathbf{D}}^{\mathrm{b}}(E)$ is an exceptional
object the pushforward
$p_{*}\colon{\mathbf{D}}^{\mathrm{b}}(E)\to{\mathbf{D}}^{\mathrm{b}}(\operatorname{\mathrm{Spec}}({\Bbbk}))$
is a Verdier localization with
$\operatorname{\mathrm{Ker}}(p_{*})={\mathcal{O}}_{E}^{\perp}$. By Lemma 5.5
the conditions of Theorem 5.2 are satisfied for $\pi$, hence $\pi_{*}$ is a
Verdier localization with kernel generated by
$\upeta_{*}({\mathcal{O}}_{E}^{\perp})$. Finally, the statement about
${\mathrm{G}}_{0}$ follows from the localization exact sequence

${\mathrm{K}}_{0}(\operatorname{\mathrm{Ker}}(\pi_{*}))\to{\mathrm{K}}_{0}({\mathbf{D}}^{\mathrm{b}}({\tilde{X}}))\to{\mathrm{K}}_{0}({\mathbf{D}}^{\mathrm{b}}(X))\to
0,$

see [SGA77, Ch. VIII, Proposition 3.1]. ∎

###  5.2. Crepant localization for ordinary double points

Recall that an isolated singularity $x\in X$ is an ordinary double point or a
node if it is a hypersurface singularity, the blowup
$\operatorname{\mathrm{Bl}}_{x}(X)$ is smooth, its exceptional divisor
$E\subset\operatorname{\mathrm{Bl}}_{x}(X)$ is a smooth quadric, and its
conormal bundle ${\mathcal{O}}_{E}(-E)$ is isomorphic to the hyperplane line
bundle ${\mathcal{O}}_{E}(1)$ on the quadric.

######  Lemma 5.7.

An ordinary double point is an acyclic projectively normal singularity.

###### Proof.

Let $(X,x)$ be an ordinary double point. Since the question is étale local at
$x$ and $(X,x)$ is a hypersurface singularity, we can assume that $X$ is a
hypersurface in the affine space ${\mathbb{A}}^{n+1}$ with equation $f=0$ and
$x\in X$ is the origin. Then it is easy to see that the exceptional divisor of
$\operatorname{\mathrm{Bl}}_{x}(X)$ is given in ${\mathbb{P}}^{n}$ by the
lowest degree homogeneous component of $f$. Since this is a smooth quadric by
assumption, we have $f=f_{2}+f_{\geq 3}$, where $f_{2}$ is a non-degenerate
quadratic form and $f_{\geq 3}$ is the sum of components of higher degree.
Then denoting by $z_{i}$ coordinates on ${\mathbb{A}}^{n+1}$ it is easy to see
that

${\mathfrak{m}}_{X,x}^{m}/{\mathfrak{m}}_{X,x}^{m+1}={\Bbbk}[z_{0},z_{1},\dots,z_{n}]_{m}/f_{2}\cdot{\Bbbk}[z_{0},z_{1},\dots,z_{n}]_{m-2}\cong{\mathrm{H}}^{0}(E,{\mathcal{O}}_{E}(m)),$

hence (a) is satisfied. The condition (b) is evident since $E$ is a quadric. ∎

Let $X$ be a variety of dimension $n\geq 2$ with ordinary double points and
let $\pi\colon{\tilde{X}}\to X$ be the blowup of its singular locus. A
combination of Lemma 5.7 with Corollary 5.6 shows that the functor
$\pi_{*}\colon{\mathbf{D}}^{\mathrm{b}}({\tilde{X}})\to{\mathbf{D}}^{\mathrm{b}}(X)$
is a Verdier localization. However, unless $n=2$, the functor $\pi_{*}$ is not
crepant (see Definition 1.12). The goal of this subsection is to construct a
crepant categorical resolution of $X$ in all dimensions $n\geq 2$.