vector bundles with flat connections on $X$, as well as
$\mathcal{F}_{\widehat{D}}$ \- the stack of vector bundles $W$ on the formal
completion endowed with flat connections with poles of order at most $n$ along
$D\hookrightarrow\widehat{D}$. Finally, we have
$\mathcal{F}_{\widehat{\partial}X}:=\mathsf{Vect}^{\nabla}(\widehat{\partial}X)_{0}$.
The formal gluing of [He-Po-Ve] again implies that there exists a cartesian
square of underived stacks

$\textstyle{\mathcal{F}_{\mathfrak{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{\widehat{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{\widehat{\partial}X}.}$

The stack $\mathcal{F}_{\widehat{D}}$ is a limit of Artin stacks locally of
finite type, and thus satisfies the condition $(5)$ of Theorem B.3. The stack
$\mathcal{F}_{\mathfrak{X}}$ satisfies the conditions $(5)$ and $(6)$ of
theorem B.3. This implies that the fiber of the top horizontal map, taken at
$(\overline{V},\overline{\nabla})$, will satisfy condition $(5)$. But by
construction this fiber is the truncated stack
$\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)_{0}$. This implies that
$\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)_{0}$ satisfies the condition $(5)$ of
the theorem B.3, as desired.

Finally we need to show that $\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)$
satisfies condition $(6)$ of Theorem B.3. For this, let
$B=\operatorname{colim}_{i}B_{i}$ as in $(6)$, and assume that each $B_{i}$ as
well as $B$ are noetherian rings. We consider

$\operatorname{colim}_{i}\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)(B_{i})\longrightarrow\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)(B).$

By Lemma 5.3 this map is injective and so we need to show it is surjective as
well. Let us fix an object in $\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)(B)$,
represented by a pair $(E,\alpha)$, of $E\in\mathsf{Vect}^{\nabla}(X)(B)$ and
$\alpha:R(E)\simeq V_{\infty}\times_{k}B$ in
$\mathsf{Vect}^{\nabla}(\widehat{\partial}X)(B)$. Since the stack
$\mathsf{Vect}^{\nabla}(X)$ of flat bundles on $X$ is locally of finite
presentation, there is an $i$ and $E_{i}\in\mathsf{Vect}^{\nabla}(X)(B_{i})$
such that $E_{i}\otimes_{B_{i}}B\simeq E$.

We now consider the sheaf $\mathcal{I}$ of isomorphisms between $R(E_{i})$ and
$V_{\infty}\otimes_{k}B_{i}$, which is a sheaf on the big étale site of affine
schemes over $S_{i}=Spec\,B_{i}$. This sheaf is a subsheaf in $\mathcal{J}$
the sheaf of all morphisms from $R(E_{i})$ to $V_{\infty}\otimes_{k}B_{i}$.

###### Lemma 5.5

There exists a non-empty Zariski open $U_{i}\subset S_{i}=Spec\,B_{i}$ such
that the restriction of the sheaf $\mathcal{J}$ is representable by a scheme
of finite type over $U_{i}$.

Proof of the lemma: This is similar to the argument we used in Lemma 5.3. What
we have to prove is that if we set

$E_{i}^{\prime}:=R(E_{i})^{\vee}\otimes
V_{\infty}\otimes_{k}B_{i}\in\mathsf{Vect}^{\nabla}(\widehat{\partial}X)(B_{i}),$

then the Tate object
$\widetilde{\mathbb{H}}_{DR}(\widehat{\partial}X,E^{\prime})[f^{-1}]$ is a
perfect $B_{i}[f^{-1}]$-module, for some non-zero localization
$B_{i}[f^{-1}]$. For this we use the criterion from Corollary 4.8 and
Proposition A.4. We let $j:X\hookrightarrow\mathfrak{X}$ be the embedding in
the good compactification. We first notice that $j_{*}(E^{\vee})$ is a perfect
$\mathcal{D}_{\mathfrak{X},B}$-module on $\mathfrak{X}\times Spec\,B$. This is
a local statement on $\mathfrak{X}$ which reduces to the following algebraic
fact.

Let $A$ be a smooth $k$-algebra of finite type and $f\in A$. We consider
$\widehat{A\otimes_{k}B}$, the formal completion of $A\otimes_{k}B$ at
$f\otimes 1$. We denote by $\widehat{\mathcal{D}}_{\mathfrak{X},B}$ the ring
of completed relative differential operators. As a module it is
$\widehat{A\otimes_{k}B}_{A\otimes_{k}B}(\mathcal{D}_{\mathfrak{X}}\otimes_{k}B)$,
where the ring structure is defined naturally by making
$\mathcal{D}_{\mathfrak{X}}\otimes_{k}B$ act on the completion
$\widehat{A\otimes_{k}B}$ by extending derivations to the completion. In the
same manner we let $\widehat{\mathcal{D}}_{X,B}$ be
$\widehat{\mathcal{D}}_{\mathfrak{X},B}[f^{-1}]$. Using the formal gluing of
[Bh] we have a cartesian square of $\infty$-categories

$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{\mathfrak{X},B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\widehat{\mathcal{D}}_{\mathfrak{X},B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\widehat{\mathcal{D}}_{X,B}).}$

Therefore, for
$j_{*}(E^{\vee})\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{\mathfrak{X},B})$ to
be perfect it is enough that its restrictions as $\mathcal{D}_{X,B}$ and
$\widehat{\mathcal{D}}_{\mathfrak{X},B}$ modules are both perfect. But, the
first of these restrictions is $E$ which is perfect over $\mathcal{D}_{X,B}$,
and the second of these restrictions corresponds to
$j_{*}(V_{s}^{\vee}\otimes_{k}B)$. This is perfect because it is the
restriction to $\widehat{\partial}X$ of
$j_{*}(V_{0}^{\vee})\otimes_{k}B\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$,
which is perfect because of Bertsein’s theorem asserting that
$j_{*}(V_{0}^{\vee})$ is a coherent and holonomic complex of
$\mathcal{D}_{X}$-modules.

We thus have that $j_{*}(E^{\vee})$ is a perfect
$\mathcal{D}_{\mathfrak{X},B}$-module. Moreover it is also holonomic (see
Proposition A.4). Indeed, because $R(E)$ is isomorphic to
$j_{*}(V_{s})\otimes_{k}B$, its characteristic cycle is contained in
$\Lambda\times Spec\,B\subset T^{*}X\times Spec\,B$, where
$\Lambda=Char(j_{*}(V_{\infty}))$. Since the $\infty$-functor sending $B$ to
perfect $\mathcal{D}_{\mathfrak{X},B}$-modules is locally of finite
presentation, we can chose $i$ and
$F_{i}\in\mathcal{D}_{\mathsf{perf}}(\mathcal{D}_{\mathfrak{X},B})$ so that
$j_{*}(E^{\vee})\simeq F_{i}\otimes_{B_{i}}B$. By enlarging $i$ if necessary,
we can also assume that the characteristic variety of $F_{i}$ is contained in
$\Lambda\times Spec\,B_{i}$, and thus that $F_{i}$ is moreover holonomic.
Also, we can assume that $F_{i}$ and $j^{*}(E_{i}^{\vee})$ are isomorphic as
objects in $\mathsf{Vect}^{\nabla}(X)(B_{i})$.

As now both $F_{i}$ and $j_{*}(V_{\infty})\otimes_{k}B_{i}$ are perfect and
holonomic, Proposition A.4 implies that
$F_{i}\otimes_{\mathcal{O}}(j_{*}(V_{\infty})\otimes_{k}B_{i})\simeq
j_{*}(E_{i}^{\prime})$ remains perfect over
$\mathcal{D}_{\mathfrak{X},B_{i}[f^{-1}]}$ for some non-zero localization of
$B_{i}$. Working with $V_{\infty}^{\vee}$ and $E^{\vee}$ from the start we
prove the same manner that $j_{*}((E_{i}^{\prime})^{\vee})$ is also perfect.
By Corollary 4.8 this implies that
$\widetilde{\mathbb{H}}_{DR}(\widehat{\partial}X,E_{i}^{\prime})$ is perfect
and thus the lemma. $\Box$

By the above lemma $\mathcal{J}$ is representable by a scheme of finite type.
The sheaf $\mathcal{I}$ clearly is an open subsheaf of $\mathcal{J}$ and thus
is also representable by a scheme of finite type over an non-empty open on
$Spec\,B_{i}$. The canonical isomorphism $\alpha:R(E)\simeq
V_{\infty}\otimes_{k}B$, which an element in $\mathcal{I}(B)$ is then
definable over some $B_{i}[f^{-1}]$ for some $i$ and non-zero localization,
say $\alpha_{i}:R(E_{0})[f^{-1}]\simeq V_{\infty}\otimes_{k}B_{i}[f^{-1}]$.
The pair $(E_{0},\alpha_{i})$ defines an object in
$\mathsf{Vect}^{\nabla}(X)_{V_{\infty}}(B_{i}[f^{-1}])$ whose image in
$\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)(B[f^{-1}])$ is the restriction of our
original object $E$.

This finishes the proof of condition $(6)$ of theorem B.3, and thus of theorem
5.1. $\Box$

Unfortunately, we do not know if Theorem 5.1 can be strengthened to the
statement that $\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)$ is representable by an
algebraic space locally of finite type over $k$. The only missing condition
would be that $\mathsf{Vect}^{\nabla}_{V_{\infty}}(X)$ is also locally of
finite presentation, a condition that we havent been able to prove or
disprove.

Appendix A: Perfect relative $\mathcal{D}$-modules

In this section we have gathered some basic results about
$\mathcal{D}$-modules in the relative setting. Most of these results are
already contained in [Ga-Ro], and this part does not claim originality. We
include it here since we were unable to find a reference treating the
algebraic situation allowing for $k$ non-algebraically closed, and also
relative to bases $Spec\,B$ with $B$ an arbitrary connective cdga.

First we discuss the compact generation, and characterization of compact
objects inside quasi-coherent relative $\mathcal{D}$-modules. Fix $X$ a smooth
variety over $k$, and $S=Spec\,B$ an affine derived scheme. We have
$\mathcal{D}_{X}\otimes_{k}B$, which is a sheaf of dg-algebras over $X$. We
can therefore consider the dg-category of all sheaves of
$\mathcal{D}_{X}\otimes_{k}B$-modules, whose underlying
$\mathcal{O}_{X}\otimes_{k}B$-module is quasi-coherent on $X\times S$. We
denote this category by $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$, and
call it the dg-category of relative $\mathcal{D}$-modules on $X\times S$ over
$S$. An object $E\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$ will be
called _perfect_ if locally on $X$ it is given by a perfect dg-module over the
dg-algebra $\mathcal{D}_{X}\otimes_{k}B$. In the special case when $B$ is a
regular discrete $k$-algebra, $\mathcal{D}_{X}\otimes_{k}B$ is locally a
finitely generated algebra of finite homological dimension (and thus is of
finite type in the sense of [To-Va]) which implies that the perfect objects
are precisely the bounded coherent $\mathcal{D}_{X}\otimes_{k}B$-modules. In
general the two notions do not coincide since being perfect implies in
particular being of finite tor dimension over $B$. Nevertheless we have the
following

###### Proposition A.1

The dg-category $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$ is compactly
generated and its compact objects are the perfect
$\mathcal{D}_{X}\otimes_{k}B$-modules.

Proof: There is a forgetful functor

$\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})\longrightarrow\mathsf{D}_{\text{qcoh}}(X\times
S)$

to the dg-category of quasi-coherent complexes on $X\times S$. This dg-functor
is conservative and continuous. Moreover, it has a left adjoint

$\mathsf{ind}:\mathsf{D}_{\text{qcoh}}(X\times
S)\longrightarrow\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$

which sends a quasi-coherent complex $E$ on $X\times S$ to
$\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}E$, with its natural
$\mathcal{D}_{X}\otimes_{k}B$-module structure. It is well known that perfect
complexes in $\mathsf{D}_{\text{qcoh}}(X\times S)$ are the compact generators,
and it is a formal consequence from this that $\mathsf{ind}$-images of perfect
complexes will be compact generators of
$\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$. These are obviously perfect
$\mathcal{D}_{X}\otimes_{k}B$-modules. Finally, any perfect
$\mathcal{D}_{X}\otimes_{k}B$-module is locally compact, and thus compact by
quasi-compactness of $X$. $\Box$

Let now $f:X\longrightarrow Y$ be a morphism between smooth varieties over
$k$. The usual definition gives a direct image dg-functor

$f_{*,B}:\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})\longrightarrow\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{Y,B}).$

We will often drop the $B$ in the notation and simply write $f_{*}$. On the
level of compact generators, $f_{*}$ acts as follows. Let $E$ be a perfect
complex on $X\times S$, and
$\mathsf{ind}(E)=\mathcal{D}_{X}\otimes_{\mathcal{O}_{X}}E$. Then we have a
canonical isomorphism $f_{*}(\mathsf{ind}(E))\simeq\mathsf{ind}(f_{*}(E))$,
where $f_{*}(E)$ is the direct image of $E$ as a quasi-coherent complex on
$X\times S$. In particular, when $f$ is proper the dg-functor $f_{*}$
preserves perfect objects. It is easy to check that the formation of $f_{*}$
commutes with base changes: for any morphism $B\rightarrow B^{\prime}$ of
connective cdga, the square

$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes_{B}B^{\prime}}$$\scriptstyle{f_{*,B}}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*,B^{\prime}}}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{Y,B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes_{B}B^{\prime}}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{Y,B^{\prime}})}$

canonically commutes. We have thus proved the following proposition, which is
well known when $B$ is itself a smooth algebra but for which we could not find
any general reference.

###### Proposition A.2

If $f$ is proper, then $f_{*}$ preserves perfect objects, and its formation
commutes with change of bases $B$.

We recall also the following notion of holonomicity. For this, we recall that
any coherent $\mathcal{D}_{X}\otimes_{k}B$-module admits a good filtration,
and that the support of the associated graded sheaf is a well defined
algebraic subset inside $T^{*}X\times S$.

###### Definition A.3

Let $E\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$ be a quasi-coherent
$\mathcal{D}_{X}\otimes_{k}B$-module. We say that $E$ is _holonomic_ if it
satisfies the following two conditions.

  1. (1)

$E$ is perfect.

  2. (2)

There exists a conic Lagrangian algebraic subset $\Lambda\subset T^{*}X$ such
that the characteristic variety of $E$ is contained in $\Lambda\times S$.

In contrast with the case of a base field, it is not true that holonomic
$\mathcal{D}$-modules are stable by all operations. However, this holds on a
dense open subset in $S$, and for us the following proposition will be useful.

###### Proposition A.4

Suppose that $B$ is a discrete noetherian $k$-algebra. Let $E$ and $F$ be two
holonomic objects in $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$. There
exists a non-empty open derived sub-scheme $Spec\,B[f^{-1}]\subset Spec\,B$
such that the tensor product $E\otimes_{\mathcal{O}}F$ is a perfect
$\mathcal{D}_{X,B[f^{-1}]}$-module on $X\times Spec\,B[f^{-1}]$.

Proof: Write $B_{0}=B_{red}$ for the reduced algebra of $B$. Note that a given
object $E\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$ is perfect if and
only if its restriction to $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B_{0}})$
is perfect. Indeed, we can use induction on the power annihilating the nil-
radical of $B$ to restrict to the case where $B$ is a square zero extension of
$B_{0}$ by an ideal $I$. It is easy to see that the functor sending a cdga $B$
to the space of all quasi-coherent $\mathcal{D}_{X,B}$-module is 1-proximate
in the sense of formal deformation theory (see [Lu3]). More precisely, given a
discrete noetherian $k$-algebra $B_{0}$, an ideal $I_{0}\subset B_{0}$ and a
derivation $d:B_{0}\longrightarrow I[1]$, consider the square-zero extension
$B=B_{0}\oplus_{d}I$ classified by $d$. The square of $\infty$-categories

$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B_{0}\oplus
I})}$

induces a full-embedding from $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$ to
the fiber product of the three other terms. This implies that for a given
$E\in\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B})$, $E$ is a compact object if
its restriction in $\mathsf{D}_{\text{qcoh}}(\mathcal{D}_{X,B_{0}})$ is
compact.

Thus it is sufficient to tackle the case where $B$ is a reduced noetherian
$k$-algebra. By picking a dense open in an irreducible component we can even
assume that $B$ is a smooth domain. Let $K=Frac(B)$ be its fraction field and
we consider $X\times_{k}K$ as a smooth variety over $K$. It is known that for
any smooth $K$-variety $Z$, any smooth sub-variety $j:Y\subset Z$ given by a
single equation $f=0$ on $X$, and any holonomic coherent
$\mathcal{D}_{Z/K}$-module $M$, there exists a Berstein polynomial $b(M)$ for
$M$ with respect to the equation $f$. The polynomial exists as a monic
polynomial over a localization $B[f^{-1}]$ of $B$. Replacing $B$ by
$B[f^{-1}]$ we can assume that $b(M)$ exists as a monic polynomial over $B$.
Now the standard Bernstein-KAshiwara argument gives that the existence of
$b(M)$ implies that the pull-back $j^{*}(E)$ is a bounded coherent complex of
$\mathcal{D}_{Y,B}$-modules with coherent cohomology, and thus is perfect as
$B$ is smooth.

Since the statement of the proposition is local on $X$, we can apply the above
reasonng to the diagonal $X\subset X\times X$, by writting it as a complete
intersection, and to the holonomic $\mathcal{D}_{X\times X,B}$-module
$E\boxtimes F$ \- the exterior tensor product of $E$ and $F$. The proposition
follows. $\Box$

Appendix B: Derived quasi-algebraic spaces and Artin’s representability

In this section we have gathered some definitions and results on _derived
quasi-algebraic spaces_ and the corresponding representability criterion.
Derived quasi-algebraic spaces are slight gneralizations of derived algebraic
spaces for which atlases only exist generically. These derived stacks are not
algebraic in general, but are algebraic as soon as the functors they represent
are locally of finite presentation.

To make sense of such spaces, we will need the following notion of a dominant
morphism to a not necessarily algebraic derived stack $F$. Assume that $F$ is
a derived stack which has a perfect global cotangent complex, and is nil-
complete and infinitesimally cartesian. We will also assume that $F$ is
integrable, that is for any local complete noetherian discrete $k$-algebra
$A=\lim_{i}A/\mathfrak{m}^{i}$, the natural morphism

$F(A)\longrightarrow\lim_{i}F(A/\mathfrak{m}^{i})$

is bijective.

For any such $F$, any field $K$ which is finitely generated over $k$, and any
point $x:Spec\,K\longrightarrow F$, there exists by [Lu2, Theorem 18.2.5.1] a
complete local noetherian cdga $A$ with residue field $K$, and a formally
smooth morphism

$Spf(A)\longrightarrow F$

extending the point $x$. We get this way a morphism from its truncation

$Spf(\pi_{0}(A))\longrightarrow F,$

and by integrability a well defined morphism

$\widehat{x}:Spec(\pi_{0}(A))\longrightarrow F.$

A morphism $\widehat{x}$ obtained this way will be called a _formally smooth
lift of $x$_.

###### Definition B.1

For a derived stack $F$ as above and a derived scheme $X$ locally of finite
presentation over $k$, with a morphism $f:X\longrightarrow F$. We say that $f$
is _dominant_ if for any finitely generated $k$-field $K$, any point
$x:Spec\,K\longrightarrow F$ and any formally smooth lift
$\widehat{x}:Spec(\pi_{0}(A))\longrightarrow F$, the derived scheme
$X\times_{F}Spec(\pi_{0}(A))$ is non-empty.

Note that if $F$ is itself representable by a derived algebraic space locally
of finite presentation, then $f:X\longrightarrow F$ is dominant in the sense
above if and only if for any étale morphism $Spec\,B\longrightarrow F$ we have
$X\times_{F}Spec\,B\neq\varnothing$. Indeed, assume that there is an étale map
$Spec\,B\longrightarrow F$ whose pull-back to $X$ is empty. We pick a point
$x$ of $Spec\,B$ and consider its formal completion $\widehat{B}_{x}$. Since
$Spec\,B\longrightarrow F$ is étale the composition

$Spec\,\pi_{0}(\widehat{B}_{x})\longrightarrow Spec\,B\longrightarrow F$

is a formally smooth lift of $x$. By construction the pull-back
$Spec\,\pi_{0}(\widehat{B}_{x})\times_{F}X$ is empty. This shows that the
above notion of dominant map is a generalization of the notion of a morphism
with Zariski dense image.

We can now give the definition of a derived quasi-algebraic spaces as derived
stacks with dominant smooth atlases as follows.

###### Definition B.2

A derived stack $F$ is a _derived quasi-algebraic space_ (locally of
presentation with schematic diagonal of finite presentation) if it satisfies
the following conditions.

  1. (i)

The diagonal of the stack $F\longrightarrow F\times F$ is representable by a
derived scheme of finite presentation.

  2. (ii)

The derived stack $F$ has a perfect global cotangent complex, and is nil
complete and infinitesimally cartesian.

  3. (iii)

The derived stack $F$ is integrable: for any local complete noetherian
discrete $k$-algebra $A=\lim_{i}A/\mathfrak{m}^{i}$, the natural morphism

$F(A)\longrightarrow\lim_{i}F(A/\mathfrak{m}^{i})$

is bijective.

  4. (iv)

There exists a family of cdga $A_{i}$ of finite presentation over $k$ and a
morphism $p:\sqcup Spec\,A_{i}\longrightarrow F$ such that

    1. (a)

For each $i$ the morphism $Spec\,A_{i}\longrightarrow F$ is smooth.

    2. (b)

The morphism $p$ is dominant in the sense of definition B.1 above.

A derived quasi-algebraic space is algebraic if and only if the functor $F$ is
futhermore locally of finite presentation. This follows from Artin-Lurie’s
representability theorem [Lu2, Theorem 18.3.0.1]. Similarly derived quasi-
algebraic spaces can be characterized by the following version of Artin’s
representability.

###### Theorem B.3

A derived stack $F$ is a derived quasi-algebraic space if it satisfies the
following conditions.

  1. (1)

For any discrete cdga $B$ the simplicial set $F(B)$ is $0$-truncated.

  2. (2)

The diagonal morphism of its truncation is representable by a scheme of finite
presentation.

  3. (3)

The derived stack $F$ has a perfect global cotangent complex.

  4. (4)

The derived stack $F$ is nil-complete and infinitesimally cartesian.

  5. (5)

For any discrete local $k$-algebra $(A,\mathfrak{m})$ essentially of finite
type, with completion $\widehat{A}=\lim_{i}A/\mathfrak{m}^{i}$, the morphism
$F(\widehat{A})\longrightarrow\lim_{i}F(A/\mathfrak{m}^{i})$ is an
equivalence.

  6. (6)

For any filtered system of noetherian discrete commutative $k$-algebras
$B=\operatorname{colim}_{i}B_{i}$ and any $x\in F(B)$, there exists an index
$i$ and a non-empty Zariski open $U_{i}\subset Spec\,B_{i}$ with
$U=U_{i}\times_{Spec\,B_{i}}Spec\,B$ non-empty, and such that the restriction
of $x$ lies in the image of $F(U_{i})\longrightarrow F(U)$.

Sketch of a proof: The proof is essentially the same as the usual
representability theorem in [Lu2].

Consider fields $K$ which are finitely generated over $k$. For any morphism

$x:Spec\,K\longrightarrow F$

we can use [Lu2, Theorem 18.2.5.1] to find a local complete and noetherian
cdga $(A,\mathfrak{m})$ with residue field $K=A/\mathfrak{m}$ and a
factorisation

$Spec\,K\hookrightarrow Spf(\widehat{A})\longrightarrow F,$

where the second map is formally smooth (i.e. its relative contangent complex
is a vector bundle). We write $B=\pi_{0}(\widehat{A})$, which is a complete
local discrete $k$-algebra with residue field $K$, and consider the induced
morphism on the truncation $\widehat{x}:Spf(B)\longrightarrow F$. We can use
condition (4) to lift this to a factorization

$Spec\,K\hookrightarrow Spec(B)\longrightarrow F.$

As explained in the proof of [Lu2, Theorem 18.2.5.1], there exists a
$k$-algebra of finite type $B^{\prime}\subset B$, such that if
$p=\mathfrak{m}\cap B^{\prime}$, then the induced morphism on formal
completions

$\widehat{B}^{\prime}_{p}\longrightarrow B$

is surjective (take $A^{\prime}$ big enough so that it contains generators for
$K$ over $k$ as well as generators of the $k$-vector space
$\mathfrak{m}/\mathfrak{m}^{2}$). We can now apply Popescu’s theorem to the
regular morphism $B^{\prime}\longrightarrow\widehat{B}^{\prime}_{p}$ and thus
write $\widehat{B}^{\prime}_{p}=\operatorname{colim}_{i}B^{\prime}_{i}$ as a
filetered colimit of smooth $B^{\prime}$-algebras. Since $B$ is finitely
presented as a $\widehat{B}^{\prime}_{p}$-algebra, we can find an index $i$
and a $B^{\prime}_{i}$-algebra $C^{\prime}_{i}$ of finite presentation such
that

$C\simeq\operatorname{colim}_{i}(\widehat{B}^{\prime}_{p}B\otimes_{B^{\prime}_{i}}C^{\prime}_{i}).$

We let
$C_{i}:=\widehat{B}^{\prime}_{p}B\otimes_{B^{\prime}_{i}}C^{\prime}_{i}$,
which is a $B^{\prime}_{i}$-algebra of finite presentation, and thus is itself
of finite presentation over $k$.

We now apply condition $(6)$ to the morphism $Spec\,B\longrightarrow F$, and
get that there exists an integer $i$ and a Zariski open
$U_{i}=Spec\,C_{i}[f^{-1}]\subset Spec\,C_{i}$, with $U=SpecB[f^{-1}]$ non-
empty, and which fits in a commutative diagram

$\textstyle{U_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F}$$\textstyle{Spec\,B.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###### Lemma B.4

With the notation above, and enlarging $i$ is necessary, the morphism
$p:U_{i}\longrightarrow F$ constructed above is formally smooth is the
underived sense: $\tau_{\leq-1}(\mathbb{L}_{U_{i}/F})$ is a vector bundle in
degree $0$.

Proof of the lemma: First of all $U_{i}$ being of finite type together with
the fact that the diagonal of $F$ is representable of locally of finite
presentation implies that $p$ is representable and locally of finite type in
the underived sense. It thus only remains to show that $p$ is also formally
smooth in the underived sense, i.e. that its relative $1$-truncated cotangent
complex $\tau_{\leq 1}(\mathbb{L}_{U_{i}/F})$ is a vector bundle.

For this we first notice that $\mathbb{L}_{U_{i}/F}$ is almost perfect (i.e.
quasi-isomorphic to a complex of free modules of finite rank over
$C_{i}[f^{-1}]$ concentrated in degree $(-\infty,0]$). Since we are only
interested in its truncation $\tau_{\leq 1}(\mathbb{L}_{U_{i}/F})$ we will be
able to act as if $\mathbb{L}_{U_{i}/F}$ is in fact perfect (simply replace it
by a perfect complex having the same cohomology in degree $[-n,0]$ for n big
enough). We start by computing the pull-back of $\mathbb{L}_{U_{i}}$ to
$U=Spec\,B[f^{-1}]$.

Consider the exact triangle of complexes of $B$-modules (where
$\mathbb{L}_{A}$ stands for $\mathbb{L}_{A/k}$ for any $k$-algebra $A$).

$\textstyle{\mathbb{L}_{C_{i}}\otimes_{C_{i}}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{B/C_{i}}.}$

Since $B$ is complete with respect to its maximal ideal $\mathfrak{m}$, for
any connective dg-module $E$ over $B$, we have its completion
$\widehat{E}:=\lim_{i}E\otimes_{B}B/\mathfrak{m}^{i}$, which is another
connective$B$-dg-module together with a natural morphism
$E\longrightarrow\widehat{E}$. Moreover, when $E$ is almost perfect this
morphism is a quasi-isomorphism. We can then complete the terms in the above
triangle to get a new triangle

$\textstyle{\widehat{\mathbb{L}_{C_{i}}\otimes_{C_{i}}B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{L}_{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{L}_{B/C_{i}}}.}$

As $\mathbb{L}_{C_{i}}$ is almost perfect the first term is simply
$\mathbb{L}_{C_{i}}\otimes_{C_{i}}B$. Moreover, by base change
$\mathbb{L}_{B/C_{i}}$ is naturally equivalent to
$\mathbb{L}_{\widehat{B}^{\prime}/C_{i}^{\prime}}\otimes_{B^{\prime}}B$. Since
$B^{\prime}\longrightarrow B$ is a surjective local morphism we see that the
base change of $\widehat{\mathbb{L}_{\widehat{B}^{\prime}/C_{i}^{\prime}}}$,
considered as a pro-object in connective $B^{\prime}$-dg-modules, by the map
$\widehat{B}^{\prime}\longrightarrow B$, is the pro-object
$\widehat{\mathbb{L}_{B/C_{i}}}$.

We now use that $\widehat{B}^{\prime}$ is the completion of $B^{\prime}$ along
the maximal ideal $\mathfrak{m}\subset B^{\prime}$, and so for all $i$ we have
$\mathbb{L}_{\widehat{B}^{\prime}/B^{\prime}}\otimes_{B^{\prime}}B^{\prime}/\mathfrak{m}^{i}\simeq
0$. We thus have an equivalence of pro-objects

$\widehat{\mathbb{L}_{\widehat{B}^{\prime}/C_{i}}}\simeq\widehat{\mathbb{L}_{C_{i}/B^{\prime}}\otimes_{B}^{\prime}\widehat{B}^{\prime}}[1].$

Since $C_{i}$ is smooth over $B^{\prime}$ we therefore conclude that the pro-
object $\widehat{\mathbb{L}_{B/C_{i}}}$ is a vector bundle in degree $1$, and
so its realization as a connective $B$-dg-module is
$\mathbb{L}_{C_{i}/B^{\prime}}\otimes_{C_{i}}B[1]$. Our original triangle can
therefore be written as

$\textstyle{\mathbb{L}_{C_{i}}\otimes_{C_{i}}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{L}_{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V[1]}$

with $V$ a vector bundle on $Spec\,B$. We can now localize this triangle to
the open $U=Spec\,B[f^{-1}]$ in order to get new triangle on $U$

$\textstyle{\mathbb{L}_{C_{i}[f^{-1}]}\otimes_{C_{i}[f^{-1}]}B[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{L}_{B}}[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V[f^{-1}][1].}$

The morphisms $q:U\rightarrow U_{i}\longrightarrow F$ induces a morphism

$\textstyle{q^{*}(\mathbb{L}_{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{C_{i}[f^{-1}]}\otimes_{C_{i}[f^{-1}]}B[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{B}[f^{-1}],}$

which factors throught completions since $q^{*}(\mathbb{L}_{F})$ is a perfect
complex by our condition $(3)$. We get

$\textstyle{q^{*}(\mathbb{L}_{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}_{C_{i}[f^{-1}]}\otimes_{C_{i}[f^{-1}]}B[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{L}_{B}}[f^{-1}],}$

and the induced morphism on the cones sits in an exact triangle

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.8261pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{L}_{U_{i}/F}\otimes_{C_{i}[f^{-1}]}B[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.8261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.8261pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\widehat{\mathbb{L}_{Spec\
B/F}}[f^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
111.57616pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
111.57616pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V[1]}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Because $Spf\longrightarrow F$ was chosen to be formally smooth in the
underived sense we have that $\tau_{\leq-1}(\widehat{\mathbb{L}_{Spec\
B/F}}[f^{-1}])$ is a vector bundle in degree $0$. The conclusion is that
$\mathbb{L}_{U_{i}/F}$ is an almost perfect complex over $C_{i}[f^{-1}]$ such
that its base change to $B[f^{-1}]=colim(C_{i}[f^{-1}])$ has vanishing
$H^{-}1$ and a vector bundle as $H^{0}$. This implies that the same is true
for $\mathbb{L}_{U_{i}/F}\otimes_{C_{i}[f^{-1}]}C_{j}[f^{-1}]$ for some big
enough $j$. $\Box$

Going back to the proof of the theorem, we use again [Lu2, Theorem 18.2.5.1]
but this time for $U_{i}\longrightarrow F$, which by the lemma can be chosen
to be formally smooth in the underived sense. We can thus produce a smooth
morphism

$W_{i}\longrightarrow F$

where $W_{i}$ is a derived affine scheme whose truncations coincides with the
given map $U_{i}\longrightarrow F$. The derived scheme $W_{i}$ is itself of
finite presentation over $k$ as its truncation is of finite type and its
cotangent complex is perfect (because its maps smoothly to $F$).

Taking the union of all morphisms $W_{i}\longrightarrow F$ constructed above
provides the required generic atlas for $F$ as in definition B.2. $\Box$

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Tony Pantev, University of Pennsylvania<EMAIL_ADDRESS>
Bertrand Toën, Université de Toulouse & CNRS<EMAIL_ADDRESS>toulouse.fr