# Contraherent cosheaves

Leonid Positselski  Faculty of Mathematics and Laboratory of Algebraic
Geometry, National Research University Higher School of Economics, Moscow
117312; and
Sector of Algebra and Number Theory, Institute for Information Transmission
Problems, Moscow 127994, Russia<EMAIL_ADDRESS>
###### Abstract.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules
over commutative rings obtained by gluing together over a scheme. The category
of contraherent cosheaves over a scheme is a Quillen exact category with exact
functors of infinite product. Over a quasi-compact semi-separated scheme or a
Noetherian scheme of finite Krull dimension (in a different version—over any
locally Noetherian scheme), it also has enough projectives. We construct the
derived co-contra correspondence, meaning an equivalence between appropriate
derived categories of quasi-coherent sheaves and contraherent cosheaves, over
a quasi-compact semi-separated scheme and, in a different form, over a
Noetherian scheme with a dualizing complex. The former point of view allows us
to obtain an explicit construction of Neeman’s extraordinary inverse image
functor $f^{!}$ for a morphism of quasi-compact semi-separated schemes
$f\colon X\longrightarrow Y$. The latter approach provides an expanded version
of the covariant Serre–Grothendieck duality theory and leads to Deligne’s
extraordinary inverse image functor $f^{!}$ (which we denote by $f^{+}$) for a
morphism of finite type $f$ between Noetherian schemes. Semi-separated
Noetherian stacks, affine Noetherian formal schemes, and ind-affine ind-
schemes (together with the noncommutative analogues) are briefly discussed in
the appendices.

###### Contents

  1. 1 Contraadjusted and Cotorsion Modules
    1. 1.1 Contraadjusted and very flat modules
    2. 1.2 Affine geometry of contraadjusted and very flat modules
    3. 1.3 Cotorsion modules
    4. 1.4 Exact categories of contraadjusted and cotorsion modules
    5. 1.5 Very flat and cotorsion dimensions
    6. 1.6 Coherent rings, finite morphisms, and coadjusted modules
    7. 1.7 Very flat morphisms of schemes
  2. 2 Contraherent Cosheaves over a Scheme
    1. 2.1 Cosheaves of modules over a sheaf of rings
    2. 2.2 Exact category of contraherent cosheaves
    3. 2.3 Direct and inverse images of contraherent cosheaves
    4. 2.4 $\operatorname{\mathfrak{Cohom}}$ from a quasi-coherent sheaf to a contraherent cosheaf
    5. 2.5 Contraherent cosheaves of $\operatorname{\mathfrak{Hom}}$ between quasi-coherent sheaves
    6. 2.6 Contratensor product of sheaves and cosheaves
  3. 3 Locally Contraherent Cosheaves
    1. 3.1 Exact category of locally contraherent cosheaves
    2. 3.2 Contraherent and locally contraherent cosheaves
    3. 3.3 Direct and inverse images of locally contraherent cosheaves
    4. 3.4 Coflasque contraherent cosheaves
    5. 3.5 Contrahereable cosheaves and the contraherator
    6. 3.6 $\operatorname{\mathfrak{Cohom}}$ into a locally derived contrahereable cosheaf
    7. 3.7 Contraherent tensor product
    8. 3.8 Compatibility of direct and inverse images with the tensor operations
  4. 4 Quasi-compact Semi-separated Schemes
    1. 4.1 Contraadjusted and cotorsion quasi-coherent sheaves
    2. 4.2 Colocally projective contraherent cosheaves
    3. 4.3 Colocally flat contraherent cosheaves
    4. 4.4 Projective contraherent cosheaves
    5. 4.5 Homology of locally contraherent cosheaves
    6. 4.6 The “naïve” co-contra correspondence
    7. 4.7 Homotopy locally injective complexes
    8. 4.8 Derived functors of direct and inverse image
    9. 4.9 Finite flat and locally injective dimension
    10. 4.10 Morphisms of finite flat dimension
    11. 4.11 Finite injective and projective dimension
    12. 4.12 Derived tensor operations
  5. 5 Noetherian Schemes
    1. 5.1 Projective locally cotorsion contraherent cosheaves
    2. 5.2 Flat contraherent cosheaves
    3. 5.3 Homology of locally cotorsion locally contraherent cosheaves
    4. 5.4 Background equivalences of triangulated categories
    5. 5.5 Co-contra correspondence over a regular scheme
    6. 5.6 Co-contra correspondence over a Gorenstein scheme
    7. 5.7 Co-contra correspondence over a scheme with a dualizing complex
    8. 5.8 Co-contra correspondence over a non-semi-separated scheme
    9. 5.9 Compact generators
    10. 5.10 Homotopy projective complexes
    11. 5.11 Special inverse image of contraherent cosheaves
    12. 5.12 Derived functors of direct and special inverse image
    13. 5.13 Adjoint functors and bounded complexes
    14. 5.14 Compatibilities for a smooth morphism
    15. 5.15 Compatibilities for finite and proper morphisms
    16. 5.16 The extraordinary inverse image
  6. A Derived Categories of Exact Categories and Resolutions
    1. A.1 Derived categories of the second kind
    2. A.2 Fully faithful functors
    3. A.3 Infinite left resolutions
    4. A.4 Homotopy adjusted complexes
    5. A.5 Finite left resolutions
    6. A.6 Finite homological dimension
  7. B Co-Contra Correspondence over a Flat Coring
    1. B.1 Contramodules over a flat coring
    2. B.2 Base rings of finite weak dimension
    3. B.3 Gorenstein base rings
    4. B.4 Corings with dualizing complexes
    5. B.5 Base ring change
  8. C Affine Noetherian Formal Schemes
    1. C.1 Torsion modules and contramodules
    2. C.2 Contraadjusted and cotorsion contramodules
    3. C.3 Very flat contramodules
    4. C.4 Affine geometry of $(R,I)$-contramodules
    5. C.5 Noncommutative Noetherian rings
  9. D Ind-Affine Ind-Schemes
    1. D.1 Flat and projective contramodules
    2. D.2 Co-contra correspondence
    3. D.3 Very flat and contraadjusted contramodules
    4. D.4 Cotorsion contramodules

## Introduction

Quasi-coherent sheaves resemble comodules. Both form abelian categories with
exact functors of infinite direct sum (and in fact, even of filtered inductive
limit) and with enough injectives. When one restricts to quasi-coherent
sheaves over Noetherian schemes and comodules over (flat) corings over
Noetherian rings, both abelian categories are locally Noetherian. Neither has
projective objects or exact functors of infinite product, in general.

In fact, quasi-coherent sheaves _are_ comodules. Let $X$ be a quasi-compact
semi-separated scheme and $\\{U_{\alpha}\\}$ be its finite affine open
covering. Denote by $T$ the disconnected union of the schemes $U_{\alpha}$; so
$T$ is also an affine scheme and the natural morphism $T\longrightarrow X$ is
affine. Then quasi-coherent sheaves over $X$ can be described as quasi-
coherent sheaves ${\mathcal{F}}$ over $T$ endowed with an isomorphism
$\phi\colon p_{1}^{*}({\mathcal{F}})\simeq p_{2}^{*}({\mathcal{F}})$ between
the two inverse images under the natural maps $p_{1}$, $p_{2}\colon
T\times_{X}T\birarrow T$. The isomorphism $\phi$ has to satisfy a natural
associativity constraint.

In other words, this means that the ring of functions
${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ has a natural structure of a
coring over the ring $A={\mathcal{O}}(T)$. The quasi-coherent sheaves over $X$
are the same thing as (left or right) comodules over this coring. The quasi-
coherent sheaves over a (good enough) stack can be also described in such way
[36].

There are two kinds of module categories over a coalgebra or coring: in
addition to the more familiar comodules, there are also _contramodules_.
Introduced originally by Eilenberg and Moore [16] in 1965 (see also the
notable paper [4]), they were all but forgotten for four decades, until the
author’s preprint and then monograph [50] attracted some new attention to them
towards the end of 2000’s.

Assuming a coring ${\mathcal{C}}$ over a ring $A$ is a projective left
$A$-module, the category of left ${\mathcal{C}}$-contramodules is abelian with
exact functors of infinite products and enough projectives. Generally,
contramodules are “dual-analogous” to comodules in most respects, i. e., they
behave as though they formed two opposite categories—which in fact they don’t
(or otherwise it wouldn’t be interesting).

On the other hand, there is an important homological phenomenon of _comodule-
contramodule correspondence_ , or a _covariant_ equivalence between
appropriately defined (“exotic”) derived categories of left comodules and left
contramodules over the same coring. This equivalence is typically obtained by
deriving certain adjoint functors which act between the abelian categories of
comodules and contramodules and induce a covariant equivalence between their
appropriately picked exact or additive subcategories (e. g., the equivalence
of _Kleisli categories_ , which was empasized in the application to comodules
and contramodules in the paper [7]).

_Contraherent cosheaves_ are geometric module objects over a scheme that are
similar to (and, sometimes, particular cases of) contramodules in the same way
as quasi-coherent sheaves are similar to (or particular cases of) comodules.
Thus the simplest way to define contramodules would be to assume one’s scheme
$X$ to be quasi-compact and semi-separated, pick its finite affine covering
$\\{U_{\alpha}\\}$, and consider contramodules over the related coring
${\mathcal{C}}$ over the ring $A$ as constructed above.

This indeed largely agrees with our approach in this paper, but there are
several problems to be dealt with. First of all, ${\mathcal{C}}$ is not a
projective $A$-module, but only a flat one. The most immediate consequence is
that one cannot hope for an abelian category of ${\mathcal{C}}$-contramodules,
but at best for an exact category. This is where the _cotorsion modules_ [63,
19] (or their generalizations which we call the _contraadjusted_ modules) come
into play. Secondly, it turns out that the exact category of
${\mathcal{C}}$-contramodules, however defined, depends on the choice of an
affine covering $\\{U_{\alpha}\\}$.

In the exposition below, we strive to make our theory as similar (or rather,
dual-analogous) to the classical theory of quasi-coherent sheaves as possible,
while refraining from the choice of a covering to the extent that it remains
practicable. We start with defining cosheaves of modules over a sheaf of rings
on a topological space, and proceed to introducing the exact subcategory of
contraherent cosheaves in the exact category of cosheaves of
${\mathcal{O}}_{X}$-modules on an arbitrary scheme $X$.

Several attempts to develop a theory of cosheaves have been made in the
literature over the years (see, e. g., [9, 59, 11]). The main difficulty
arising in this connection is that the conventional theory of sheaves depends
on the exactness property of filtered inductive limits in the categories of
abelian groups or sets in an essential way. E. g., the most popular approach
is based on the wide use of the construction of stalks, which are defined as
filtered inductive limits. It is important that the functors of stalks are
exact on both the categories of presheaves and sheaves.

The problem is that the costalks of a co(pre)sheaf would be constructed as
filtered projective limits, and filtered projective limits of abelian groups
are not exact. One possible way around this difficulty is to restrict oneself
to (co)constructible cosheaves, for which the projective limits defining the
costalks may be stabilizing and hence exact. This is apparently the approach
taken in the notable recent preprint [13] (see the discussion with further
references in [13, Section 3.5]). The present work is essentially based on the
observation that one does not really need the (co)stalks in the quasi-
coherent/contraherent (co)sheaf theory, as _the functors of (co)sections over
affine open subschemes are already exact on the (co)sheaf categories_.

Let us explain the main definition in some detail. A quasi-coherent sheaf
${\mathcal{F}}$ on a scheme $X$ can be simply defined as a correspondence
assigning to every affine open subscheme $U\subset X$ an
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ and to every pair of embedded
affine open subschemes $V\subset U\subset X$ an isomorphism of
${\mathcal{O}}_{X}(V)$-modules

${\mathcal{F}}(V)\simeq{\mathcal{O}}_{X}(V)\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{F}}(U).$

The obvious compatibility condition for three embedded affine open subschemes
$W\subset V\subset U\subset X$ needs to be imposed.

Analogously, a contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$ is a
correspondence assigning to every affine open subscheme $U\subset X$ an
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ and to every pair of
embedded affine open subschemes $V\subset U\subset X$ an isomorphism of
${\mathcal{O}}_{X}(V)$-modules

${\mathfrak{P}}[V]\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U]).$

The difference with the quasi-coherent case is that the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{O}}_{X}(V)$ is always flat, but not
necessarily projective. So to make one’s contraherent cosheaves well-behaved,
one has to impose the additional Ext-vanishing condition

$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}^{1}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])=0$

for all affine open subschemes $V\subset U\subset X$. Notice that the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{O}}_{X}(V)$ always has projective
dimension not exceeding $1$, so the condition on $\operatorname{Ext}^{1}$ is
sufficient.

This nonprojectivity problem is the reason why one does not have an abelian
category of contraherent cosheaves. Imposing the Ext-vanishing requirement
allows to obtain, at least, an exact one. Given a rule assigning to affine
open subschemes $U\subset X$ the ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{P}}[U]$ together with the isomorphisms for embedded affine open
subschemes $V\subset U$ as above, and assuming that the
$\operatorname{Ext}^{1}$-vanishing condition holds, one can show that
${\mathfrak{P}}$ satisfies the cosheaf axioms for coverings of affine open
subschemes of $X$ by other affine open subschemes. Then a general result from
[25] says that ${\mathfrak{P}}$ extends uniquely from the base of affine open
subschemes to a cosheaf of ${\mathcal{O}}_{X}$-modules defined, as it should
be, on all the open subsets of $X$.

A module $P$ over a commutative ring $R$ is called _contraadjusted_ if
$\operatorname{Ext}^{1}_{R}(R[s^{-1}],P)\allowbreak=0$ for all $s\in R$. This
is equivalent to the vanishing of $\operatorname{Ext}^{1}_{R}(S,P)$ for all
the $R$-algebras $S$ of functions on the affine open subschemes of
$\operatorname{Spec}R$. More generally, a left module $P$ over a (not
necessarily commutative) ring $R$ is said to be _cotorsion_ if
$\operatorname{Ext}^{1}_{R}(F,P)=0$ (or equivalently,
$\operatorname{Ext}^{>0}_{R}(F,P)=0$) for any flat left $R$-module $F$. It has
been proven that cotorsion modules are “numerous enough” (see [17, 6]); one
can prove the same for contraadjusted modules in the similar way.

Any quotient module of a contraadjusted module is contraadjusted, so the
“contraadjusted dimension” of any module does not exceed $1$; while the
cotorsion dimension of a module may be infinite if the projective dimensions
of flat modules are. This makes the contraadjusted modules useful when working
with schemes that are not necessarily Noetherian of finite Krull dimension.

Modules of the complementary class to the contraadjusted ones (in the same way
as the flats are complementary to the cotorsion modules) we call _very flat_.
All very flat modules have projective dimensions not exceeding $1$. The wide
applicability of contraadjusted and very flat modules (cf. [53, Remark 2.6])
implies the importance of very flat morphisms of schemes, and we initiate the
study of these (though our results in this direction are still far from what
one would hope for).

The above-discussed phenomenon of dependence of the category of contramodules
over the coring ${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ over the ring
$A={\mathcal{O}}(T)$ on the affine covering $T\longrightarrow X$ used to
construct it manifests itself in our approach in the unexpected predicament of
the _contraherence property_ of a cosheaf of ${\mathcal{O}}_{X}$-modules
_being not local_. So we have to deal with the _locally contraherent
cosheaves_ , and the necessity to control the extension of this locality
brings the coverings back.

Once an open covering in restriction to which our cosheaf becomes contraherent
is safely fixed, though, many other cosheaf properties that we consider in
this paper become indeed local. And any locally contraherent cosheaf on a
quasi-compact semi-separated scheme has a finite left Čech resolution by
contraherent cosheaves.

One difference between homological theories developed in the settings of exact
and abelian categories is that whenever a functor between abelian categories
isn’t exact, a similar functor between exact categories will tend to have a
shrinked domain. Any functor between abelian categories that has an everywhere
defined left or right derived functor will tend to be everywhere defined
itself, if only because one can always pass to the degree-zero cohomology of
the derived category objects. Not so with exact categories, in which complexes
may have no cohomology objects in general.

Hence the (sometimes annoying) necessity to deal with multitudes of domains of
definitions of various functors in our exposition. On the other hand, a
functor with the domain consisting of adjusted objects is typically exact on
the exact subcategory where it is defined.

Another difference between the theory of comodules and contramodules over
corings as developed in [50] and our present setting is that in _loc. cit._ we
considered corings ${\mathcal{C}}$ over base rings $A$ of finite homological
dimension. On the other hand, the ring $A={\mathcal{O}}(T)$ constructed above
has infinite homological dimension in most cases, while the coring
${\mathcal{C}}={\mathcal{O}}(T\times_{X}T)$ can be said to have “finite
homological dimension relative to $A$”. For this reason, while the comodule-
contramodule correspondence theorem [50, Theorem 5.4] was stated for the
derived categories of the second kind, one of the most general of our co-
contra correspondence results in this paper features an equivalence of the
conventional derived categories.

One application of this equivalence is a new construction of the extraordinary
inverse image functor $f^{!}$ on the derived categories of quasi-coherent
sheaves for any morphism of quasi-compact semi-separated schemes $f\colon
Y\longrightarrow X$. To be more precise, this is a construction of what we
call _Neeman’s extraordinary inverse image functor_ , that is the right
adjoint functor $f^{!}$ to the derived direct image functor
${\mathbb{R}}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
on the derived categories of quasi-coherent sheaves (see discussion below). In
fact, we construct a _right derived_ functor ${\mathbb{R}}f^{!}$, rather than
just a triangulated functor $f^{!}$, as it was usually done before [29, 45].

To a morphism $f$ one assigns the direct and inverse image functors
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ and $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ between the abelian categories of quasi-
coherent sheaves on $X$ and $Y$; the functor $f^{*}$ is left adjoint to the
functor $f_{*}$. To the same morphism, one also assigns the direct image
functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ between the exact categories
of colocally projective contraherent cosheaves and the inverse image functor
$f^{!}\colon X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ between the exact categories
of locally injective locally contraherent cosheaves on $X$ and $Y$. The
functor $f^{!}$ is “partially” right adjoint to the functor $f_{!}$.

Passing to the derived functors, one obtains the adjoint functors
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
between the (conventional unbounded) derived categories of quasi-coherent
cosheaves. One also obtains the adjoint functors
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
between the derived categories of contraherent cosheaves on $X$ and $Y$.

The derived co-contra correspondence (for the conventional derived categories)
provides equivalences of triangulated categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
transforming the direct image functor ${\mathbb{R}}f_{*}$ into the direct
image functor ${\mathbb{L}}f_{!}$. So the two inverse image functors
${\mathbb{L}}f^{*}$ and ${\mathbb{R}}f^{!}$ can be viewed as the adjoints on
the two sides to the same triangulated functor of direct image. This finishes
our construction of the triangulated functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
right adjoint to ${\mathbb{R}}f_{*}$.

As usually in homological algebra, the tensor product and Hom-type operations
on the quasi-coherent sheaves and contraherent cosheaves play an important
role in our theory. First of all, under appropriate adjustness assumptions one
can assign a contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ to a
quasi-coherent sheaf ${\mathcal{F}}$ and a contraherent cosheaf
${\mathfrak{P}}$ over a scheme $X$. This operation is the analogue of the
tensor product of quasi-coherent sheaves in the contraherent world.

Secondly, to a quasi-coherent sheaf ${\mathcal{F}}$ and a cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ over a scheme $X$ one can assign
a cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$. Under our duality-
analogy, this corresponds to taking the sheaf of
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}$ from a quasi-coherent
sheaf to a sheaf of ${\mathcal{O}}_{X}$-modules. When the scheme $X$ is
Noetherian, the sheaf ${\mathcal{F}}$ is coherent, and the cosheaf
${\mathfrak{P}}$ is contraherent, the cosheaf
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$ is contraherent, too.
Under some other assumptions one can apply the (derived or underived)
_contraherator_ functor ${\mathbb{L}}\operatorname{\mathfrak{C}}$ or
$\operatorname{\mathfrak{C}}$ to the cosheaf
${\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathfrak{P}}$ to obtain the
(complex of) contraherent cosheaves
${\mathcal{F}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{P}}$
or ${\mathcal{F}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{P}}$.
These are the analogues of the quasi-coherent internal Hom functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}$
on the quasi-coherent sheaves, which can be obtained by applying the coherator
functor $\operatorname{\mathcal{Q}}$ [61, Appendix B] to the
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({-},{-})$
sheaf.

The remaining two operations are harder to come by. Modelled after the
comodule-contramodule correspondence functors $\Phi_{\mathcal{C}}$ and
$\Psi_{\mathcal{C}}$ from [50], they play a similarly crucial role in our
present co-contra correspondence theory. Given a quasi-coherent sheaf
${\mathcal{F}}$ and a cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$
on a quasi-separated scheme $X$, one constructs a quasi-coherent sheaf
${\mathcal{F}}\odot_{X}{\mathfrak{P}}$ on $X$. Given two quasi-coherent
sheaves ${\mathcal{F}}$ and ${\mathcal{P}}$ on a quasi-separated scheme, under
certain adjustness assuptions one can constuct a contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},{\mathcal{P}})$.

_Derived categories of the second kind_ , whose roots go back to the work of
Husemoller, Moore, and Stasheff on two kinds of differential derived functors
[31] and Hinich’s paper about DG-coalgebras [30], were introduced in their
present form in the author’s monograph [50] and memoir [51]. The most
important representatives of this class of derived category constructions are
known as the _coderived_ and the _contraderived_ categories; the difference
between them consists in the use of the closure with respect to infinite
direct sums in one case and with respect to infinite products in the other.
Here is a typical example of how they occur.

According to Iyengar and Krause [33], the homotopy category of complexes of
projective modules over a Noetherian commutative ring with a dualizing complex
is equivalent to the homotopy category of complexes of injective modules. This
theorem was extended to semi-separated Noetherian schemes with dualizing
complexes by Neeman [47] and Murfet [42] in the following form: the derived
category ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ of
the exact category of flat quasi-coherent sheaves on such a scheme $X$ is
equivalent to the homotopy category of injective quasi-coherent sheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$. These
results are known as the _covariant Serre–Grothendieck duality_ theory.

One would like to reformulate this equivalence so that it connects certain
derived categories of modules/sheaves, rather than just subcategories of
resolutions. In other words, it would be nice to have some procedure assigning
complexes of projective, flat, and/or injective modules/sheaves to arbitrary
complexes.

In the case of modules, the homotopy category of projectives is identified
with the contraderived category of the abelian category of modules, while the
homotopy category of injectives is equivalent to the coderived category of the
same abelian category. Hence the Iyengar–Krause result is interpreted as an
instance of the “co-contra correspondence”—in this case, an equivalence
between the coderived and contraderived categories of the same abelian
category.

In the case of quasi-coherent sheaves, however, only a half of the above
picture remains true. The homotopy category of injectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ is still
equivalent to the coderived category of quasi-coherent sheaves
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$. But the
attempt to similarly describe the derived category of flats runs into the
problem that the infinite products of quasi-coherent sheaves are not exact, so
the contraderived category construction does not make sense for them.

This is where the contraherent cosheaves come into play. The covariant
Serre–Grothendieck duality for a nonaffine (but semi-separated) scheme with a
dualizing complex is an equivalence of _four_ triangulated categories, rather
than just two. In addition to the derived category of flat quasi-coherent
sheaves ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
the homotopy category of injective quasi-coherent sheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$, there are
also the homotopy category of projective contraherent cosheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$ and the
derived category of locally injective contraherent cosheaves
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.

Just as the homotopy category of injectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ is equivalent
to the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$, the homotopy
category of projectives
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$ is identified
with the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$ of
contraherent cosheaves. The equivalence between the two “injective” categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ and
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$ does not
depend on the dualizing complex, and neither does the equivalence between the
two “projective” (or “flat”) categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{fl}})$ and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$. The
equivalences connecting the “injective” categories with the “projective” ones
do.

So far, our discussion in this introduction was essentially limited to semi-
separated schemes. Such a restriction of generality is not really that natural
or desirable. Indeed, the affine plane with a double point (which is not semi-
separated) is no less a worthy object of study than the line with a double
point (which is). There is a problem, however, that appears to stand in the
way of a substantial development of the theory of contraherent cosheaves on
the more general kinds of schemes.

The problem is that even when the quasi-coherent sheaves on schemes remain
well-behaved objects, the complexes of such sheaves may be no longer so. More
precisely, the trouble is with the derived functors of direct image of
complexes of quasi-coherent sheaves, which do not always have good properties
(such as locality along the base, etc.) Hence the common wisdom that for the
more complicated schemes $X$ one is supposed to consider complexes of sheaves
of ${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves, rather
than complexes of quasi-coherent sheaves as such (see, e. g., [56]). As we do
not know what is supposed to be either a second kind or a contraherent
analogue of the construction of the derived category of complexes of sheaves
of ${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves, we have
to restrict our exposition to, approximately, those situations where the
derived category of the abelian category of quasi-coherent sheaves on $X$ is
still a good category to work in.

There are, basically, two such situations [61, Appendix B]: (1) the quasi-
compact semi-separated schemes and (2) the Noetherian or, sometimes, locally
Noetherian schemes (which, while always quasi-separated, do not have to be
semi-separated). Accordingly, our exposition largely splits in two streams
corresponding to the situations (1) and (2), where different techniques are
applicable. The main difference in the generality level with the quasi-
coherent case is that in the contraherent context one often needs also to
assume one’s schemes to have finite Krull dimension, in order to use Raynaud
and Gruson’s homological dimension results [55].

In particular, we prove the equivalence of the conventional derived categories
of quasi-coherent sheaves and contraherent cosheaves not only for quasi-
compact semi-separated schemes, but also, separately, for all Noetherian
schemes of finite Krull dimension. As to the covariant Serre–Grothendieck
duality theorem, it remains valid in the case of a non-semi-separated
Noetherian scheme with a dualizing complex in the form of an equivalence
between two derived categories of the second kind: the coderived category of
quasi-coherent sheaves
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$
and the contraderived category of contraherent cosheaves
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.

Before describing another application of our theory, let us have a more
detailed discussion of the extraordinary inverse images of quasi-coherent
sheaves. There are, in fact, _two_ different functors going by the name of
“the functor $f^{!}$” in the literature. One of them, which we name after
Neeman, is simply the functor right adjoint to the derived direct image
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$,
which we were discussing above. Neeman proved its existence for an arbitrary
morphism of quasi-compact semi-separated schemes, using the techniques of
compact generators and Brown representability [45]. Similar arguments apply in
the case of an arbitrary morphism of Noetherian schemes $f\colon
Y\longrightarrow X$.

The other functor, which we call _Deligne’s extraordinary inverse image_ and
denote (to avoid ambiguity) by $f^{+}$, coincides with Neeman’s functor in the
case of a proper morphism $f$. In the case when $f$ is an open embedding, on
the other hand, the functor $f^{+}$ coincides with the conventional
restriction (inverse image) functor $f^{*}$ (which is left adjoint to
${\mathbb{R}}f_{*}$, rather than right adjoint). More generally, in the case
of a smooth morphism $f$ the functor $f^{+}$ only differs from $f^{*}$ by a
dimensional shift and a top form bundle twist. This is the functor that was
constructed in Hartshorne’s book [29] and Deligne’s appendix to it [14] (hence
the name). It is Deligne’s, rather than Neeman’s, extraordinary inverse image
functor that takes a dualizing complex on $X$ to a dualizing complex on $Y$,
i. e.,
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{+}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

It is an important idea apparently due to Gaitsgory [21] that Deligne’s
extraordinary inverse image functor actually acts between the coderived
categories of quasi-coherent sheaves, rather than between their conventional
derived categories. To be sure, Hartshorne and Deligne only construct their
functor for bounded below complexes (for which there is no difference between
the coderived and derived categories). Gaitsgory’s “ind-coherent sheaves” are
closely related to our coderived category of quasi-coherent sheaves.

Concerning Neeman’s right adjoint functor $f^{!}$, it can be shown to exist on
both the derived and (in the case of Noetherian schemes) the coderived
categories. The problem arises when one attempts to define the functor $f^{+}$
by decomposing a morphism $f\colon Y\longrightarrow X$ into proper morphisms
$g$ and open embeddings $h$ and subsequently composing the functors $g^{!}$
for the former with the functors $h^{*}$ for the latter. It just so happens
that the functor
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$
obtained in this way depends on the chosen decomposition of the morphism $f$.
This was demonstrated by Neeman in his counterexample [45, Example 6.5].

The arguments of Deligne [14], if extended to the conventional unbounded
derived categories, break down on a rather subtle point: while it is true that
the restriction of an injective quasi-coherent sheaf to an open subscheme of a
Noetherian scheme remains injective, the restriction of a homotopy injective
complex of quasi-coherent sheaves to such a subscheme may no longer be
homotopy injective. On the other hand, Deligne computes the Hom into the
object produced by his extraordinary inverse image functor from an arbitrary
bounded complex of coherent sheaves, which is essentially sufficient to make a
functor between the coderived categories well-defined, as these are compactly
generated by bounded complexes of coherents.

The above discussion of the functor $f^{+}$ is to be compared with the remark
that the conventional derived inverse image functor ${\mathbb{L}}f^{*}$ is not
defined on the coderived categories of quasi-coherent sheaves (but only on
their derived categories), except in the case of a morphism $f$ of finite flat
dimension [53, 21]. On the other hand, the conventional inverse image $f^{*}$
is perfectly well defined on the derived categories of flat quasi-coherent
sheaves (where one does not even need to resolve anything in order to
construct a triangulated functor). We show that the functor
$f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is transformed into the functor
$f^{+}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$
by the above-described equivalence of triangulated categories.

Let us finally turn to the connection between Deligne’s extrardinary inverse
image functor and our contraherent cosheaves. Given a morphism of Noetherian
schemes $f\colon Y\longrightarrow X$, just as the direct image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
has a right adjoint functor
$f^{!}\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$,
so does the direct image functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
has a left adjoint functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$.
This is the contraherent analogue of Neeman’s extraordinary inverse image
functor for quasi-coherent sheaves.

Now assume that $f$ is a morphism of finite type and the scheme $X$ has a
dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$; set
${\mathcal{D}}_{Y}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=f^{+}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
As we mentioned above, the choice of the dualizing complexes induces
equivalences of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and similarly for $Y$. Those equivalences of categories transform the functor
$f^{*}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(Y{\operatorname{\mathsf{--ctrh}}})$
into a certain functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(Y{\operatorname{\mathsf{--qcoh}}})$.
It is the latter functor that turns out to be isomorphic to Deligne’s
extraordinary inverse image functor (which we denote here by $f^{+}$).

Two proofs of this result, working on slightly different generality levels,
are given in this paper. One applies to “compactifiable” separated morphisms
of finite type between semi-separated Noetherian schemes and presumes
comparison with a Deligne-style construction of the functor $f^{+}$, involving
a decomposition of the morphism $f$ into an open embedding followed by a
proper morphism [14]. The other one is designed for “embeddable” morphisms of
finite type between Noetherian schemes and the comparison with a Hartshorne-
style construction of $f^{+}$ based on a factorization of $f$ into a finite
morphism followed by a smooth one [29].

To end, let us describe some prospects for future research and applications of
contraherent cosheaves. One of such expected applications is related to the
${\boldsymbol{\mathcal{D}}}$–${\boldsymbol{\Omega}}$ duality theory. Here
${\boldsymbol{\mathcal{D}}}$ stands for the sheaf of rings of differential
operators on a smooth scheme and ${\boldsymbol{\Omega}}$ denotes the de Rham
DG-algebra. The derived ${\boldsymbol{\mathcal{D}}}$–${\boldsymbol{\Omega}}$
duality, as formulated in [51, Appendix B] (see also [57] for a further
development), happens on two sides. On the “co” side, the functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({\boldsymbol{\Omega}},{-})$
takes right ${\boldsymbol{\mathcal{D}}}$-modules to right DG-modules over
${\boldsymbol{\Omega}}$, and there is the adjoint functor
${-}\otimes_{{\mathcal{O}}_{X}}{\boldsymbol{\mathcal{D}}}$. These functors
induce an equivalence between the derived category of
${\boldsymbol{\mathcal{D}}}$-modules and the coderived category of DG-modules
over ${\boldsymbol{\Omega}}$.

On the “contra” side, over an affine scheme $X$ the functor
${\boldsymbol{\Omega}}(X)\otimes_{{\mathcal{O}}(X)}{-}$ takes left
${\boldsymbol{\mathcal{D}}}$-modules to left DG-modules over
${\boldsymbol{\Omega}}$, and the adjoint functor is
$\operatorname{Hom}_{{\mathcal{O}}(X)}({\boldsymbol{\mathcal{D}}}(X),{-})$.
The induced equivalence is between the derived category of
${\boldsymbol{\mathcal{D}}}(X)$-modules and the contraderived category of DG-
modules over ${\boldsymbol{\Omega}}(X)$. One would like to extend the “contra”
side of the story to nonaffine schemes using the contraherent
cosheafification, as opposed to the quasi-coherent sheafification on the “co”
side. The need for the contraherent cosheaves arises, once again, because the
contraderived category construction does not make sense for quasi-coherent
sheaves. The contraherent cosheaves are designed for being plugged into it.

Another direction in which we would like to extend the theory presented below
is that of Noetherian formal schemes, and more generally, ind-schemes of ind-
finite or even ind-infinite type. The idea is to define an exact category of
contraherent cosheaves of contramodules over a formal scheme that would serve
as the natural “contra”-side counterpart to the abelian category of quasi-
coherent torsion sheaves on the “co” side. (For a taste of the contramodule
theory over complete Noetherian rings, the reader is referred to [54, Appendix
B].)

Moreover, one would like to have a “semi-infinite” version of the homological
formalism of quasi-coherent sheaves and contraherent cosheaves (see Preface to
[50] for the related speculations). A possible setting might be that of an
ind-scheme flatly fibered over an ind-Noetherian ind-scheme with quasi-compact
schemes as the fibers. One would define the semiderived categories of quasi-
coherent sheaves and contraherent cosheaves on such an ind-scheme as mixtures
of the co/contraderived categories along the base ind-scheme and the
conventional derived categories along the fibers.

The idea is to put two our (present) co-contra correspondence theorems “on top
of” one another. That is, join the equivalence of conventional derived
categories of quasi-coherent sheaves and contraherent cosheaves on a quasi-
compact semi-separated scheme together with the equivalence between the
coderived category of quasi-coherent (torsion) sheaves and the contraderived
category of contraherent cosheaves (of contramodules) on an (ind-)Noetherian
(ind-)scheme with a dualizing complex into a single “semimodule-
semicontramodule correspondence” theorem claiming an equivalence between the
two semiderived categories.

What we have in mind here is a theory that might bear the name of a “semi-
infinite homological theory of doubly infinite-dimensional algebraic
varieties”, or _semi-infinite algebraic geometry_. In addition to the above-
described correspondence between the complexes of sheaves and cosheaves, this
theory would also naturally feature a double-sided derived functor of
semitensor (or “mixed tensor-cotensor”) product of complexes of quasi-coherent
torsion sheaves (cf. the discussion of cotensor products of quasi-coherent
sheaves in [53, Section B.2]), together with its contraherent analogue.

Furthermore, one might hope to join the de Rham DG-module and the contramodule
stories together in a single theory by considering contraherent cosheaves of
DG-contramodules over the de Rham–Witt complex, with an eye to applications to
crystalline sheaves and the $p$-adic Hodge theory. As compared so such high
hopes, our real advances in this paper are quite modest.

The very possibility of having a meaningful theory of contraherent cosheaves
on a scheme rests on there being enough contraadjusted or cotorsion modules
over a commutative or associative ring. The contemporary-style set-theoretical
proofs of these results [17, 6, 19, 58, 5] only seem to work in abelian (or
exact) categories with exact functors of filtered inductive limits. Thus the
first problem one encounters when trying to build up the theories of
contraherent cosheaves of contramodules is the necessity of developing
applicable techniques for constructing flat, cotorsion, contraadjusted, and
very flat contramodules to be used in the resolutions.

In the present version of the paper, we partially overcome this obstacle by
constructing enough flat, cotorsion, contraadjusted, and very flat
contramodules over a Noetherian ring in the adic topology, enough flat and
cotorsion contramodules over a pro-Noetherian topological ring of totally
finite Krull dimension, and also enough contraadjusted and very flat
contramodules over the projective limit of a sequence of commutative rings and
surjective morphisms between them with finitely generated kernel ideals. The
constructions are based on the existence of enough objects of the respective
classes in the conventional categories of modules, which is used as a given
fact, and also on the older, more explicit construction [63] of cotorsion
resolutions of modules over commutative Noetherian rings of finite Krull
dimension.

This provides the necessary background for possible definitions of locally
contraadjusted or locally cotorsion contraherent cosheaves of contramodules
over Noetherian formal schemes or ind-Noetherian ind-schemes of totally finite
Krull dimension, and also of locally contraadjusted contraherent cosheaves
over ind-schemes of a more general nature, such as, e. g., the
projectivization of an infinite-dimensional discrete vector space, or the
total space of such projective space’s cotangent bundle. The latter might be a
typical kind of example for which one would like to have the “semi-infinite
algebraic geometry” worked out. And among the ind-Noetherian ind-schemes of
totally finite Krull dimension there are, e. g., the spectra of the (sheaves
of) rings of Witt vectors of algebraic varieties in finite characteristic.

This kind of work, intended to prepare ground for future theory building, is
presently relegated to appendices. There we also construct the derived co-
contra correspondence (an equivalence between the coderived category of
torsion modules and the contraderived category of contramodules) over affine
Noetherian formal schemes and ind-affine ind-Noetherian ind-schemes with
dualizing complexes. Another appendix is devoted to the co-contra
correspondence over noncommutative semi-separated stacks (otherwise known as
flat corings [36]).

Among the many people whose questions, remarks, and suggestions contributed to
the present research, I should mention Sergey Arkhipov, Roman Bezrukavnikov,
Alexander Polishchuk, Sergey Rybakov, Alexander Efimov, Amnon Neeman, Henning
Krause, Mikhail Bondarko, Alexey Gorodentsev, Paul Bressler, Vladimir Hinich,
Greg Stevenson, Michael Finkelberg, and Pierre Deligne. Parts of this paper
were written when I was visiting the University of Bielefeld, and I want to
thank Collaborative Research Center 701 and Prof. Krause for the invitation.
This work was supported in part by RFBR grants.

##  1\. Contraadjusted and Cotorsion Modules

###  1.1. Contraadjusted and very flat modules

Let $R$ be a commutative ring. We will say that an $R$-module $P$ is
_contraadjusted_ if the $R$-module $\operatorname{Ext}_{R}^{1}(R[r^{-1}],P)$
vanishes for every element $r\in R$. An $R$-module $F$ is called _very flat_
if one has $\operatorname{Ext}_{R}^{1}(F,P)=0$ for every contraadjusted
$R$-module $P$.

By the definition, any injective $R$-module is contraadjusted and any
projective $R$-module is very flat. Notice that the projective dimension of
the $R$-module $R[r^{-1}]$ never exceeds $1$, as it has a natural two-term
free resolution
$0\longrightarrow\bigoplus_{n=0}^{\infty}R\longrightarrow\bigoplus_{n=0}^{\infty}R\longrightarrow
R[r^{-1}]\longrightarrow 0$. It follows that any quotient module of a
contraadjusted module is contraadjusted, and one has
$\operatorname{Ext}_{R}^{>0}(F,P)=0$ for any very flat $R$-module $F$ and
contraadjusted $R$-module $P$.

Computing the $\operatorname{Ext}^{1}$ in terms of the above resolution, one
can more explicitly characterize contraadjusted $R$-modules as follows. An
$R$-module $P$ is contraadjusted if and only if for any sequence of elements
$p_{0}$, $p_{1}$, $p_{2}$, $\dotsc\in P$ and $r\in R$ there exists a (not
necessarily unique) sequence of elements $q_{0}$, $q_{1}$, $q_{2}$, $\dotsc\in
P$ such that $q_{i}=p_{i}+rq_{i+1}$ for all $i\ge 0$.

Furthermore, the projective dimension of any very flat module is equal to $1$
or less. Indeed, any $R$-module $M$ has a two-term right resolution by
contraadjusted modules, which can be used to compute
$\operatorname{Ext}^{*}_{R}(F,M)$ for a very flat $R$-module $F$. (The
converse assertion is _not_ true, however; see Example 1.7.7 below.)

It is also clear that the classes of contraadjusted and very flat modules are
closed under extensions. Besides, the class of very flat modules is closed
under the passage to the kernel of a surjective morphism (i. e., the kernel of
a surjective morphism of very flat $R$-modules is very flat). In addition, we
notice that the class of contraadjusted $R$-modules is closed under infinite
products, while the class of very flat $R$-modules is closed under infinite
direct sums.

######  Theorem 1.1.1.

(a) Any $R$-module can be embedded into a contraadjusted $R$-module in such a
way that the quotient module is very flat.

(b) Any $R$-module admits a surjective map onto it from a very flat $R$-module
such that the kernel is contraadjusted.

###### Proof.

Both assertions follow from the results of Eklof and Trlifaj [17, Theorem 10].
It suffices to point out that all the $R$-modules of the form $R[r^{-1}]$ form
a set rather than a proper class. For the reader’s convenience and our future
use, parts of the argument from [17] are reproduced below. ∎

The following definition will be used in the sequel. Let ${\mathsf{A}}$ be an
abelian category with exact functors of inductive limit, and let
${\mathsf{C}}\subset{\mathsf{A}}$ be a class of objects. An object
$X\in{\mathsf{A}}$ is said to be a _transfinitely iterated extension_ of
objects from ${\mathsf{C}}$ if there exist a well-ordered set $\Gamma$ and a
family of subobjects $X_{\gamma}\subset X$, $\gamma\in\Gamma$, such that
$X_{\delta}\subset X_{\gamma}$ whenever $\delta<\gamma$, the union (inductive
limit) $\varinjlim_{\gamma\in\Gamma}X_{\gamma}$ of all $X_{\gamma}$ coincides
with $X$, and the quotient objects
$X_{\gamma}/\varinjlim_{\delta<\gamma}X_{\delta}$ belong to ${\mathsf{C}}$ for
all $\gamma\in\Gamma$ (cf. Section 4.1).

By [17, Lemma 1], any transfinitely iterated extension of the $R$-modules
$R[r^{-1}]$, with arbitrary $r\in R$, is a very flat $R$-module. Proving a
converse assertion will be one of our goals. The following lemma is a
particular case of [17, Theorem 2].

######  Lemma 1.1.2.

Any $R$-module can be embedded into a contraadjusted $R$-module in such a way
that the quotient module is a transfinitely iterated extension of the
$R$-modules $R[r^{-1}]$.

###### Proof.

The proof is a set-theoretic argument based on the fact that the Cartesian
square of any infinite cardinality $\lambda$ is equicardinal to $\lambda$. In
our case, let $\lambda$ be any infinite cardinality no smaller than the
cardinality of the ring $R$. For an $R$-module $L$ of the cardinality not
exceeding $\lambda$ and an $R$-module $M$ of the cardinality $\mu$, the set
$\operatorname{Ext}^{1}_{R}(L,M)$ has the cardinality at most $\mu^{\lambda}$,
as one can see by computing the $\operatorname{Ext}^{1}$ in terms of a
projective resolution of the first argument. In particular; set
$\aleph=2^{\lambda}$; then for any $R$-module $M$ of the cardinality not
exceeding $\aleph$ and any $r\in R$ the cardinality of the set
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],M)$ does not exceed $\aleph$, either.

Let $\bethl$ be the smallest cardinality that is larger than $\aleph$ and let
$\Delta$ be the smallest ordinal of the cardinality $\bethl$. Notice that the
natural map
$\varinjlim_{\delta\in\Delta}\operatorname{Ext}^{*}_{R}(L,Q_{\delta})\longrightarrow\operatorname{Ext}^{*}_{R}(L,\>\varinjlim_{\delta\in\Delta}Q_{\delta})$
is an isomorphism for any $R$-module $L$ of the cardinality not exceeding
$\lambda$ (or even $\aleph$) and any inductive system of $R$-modules
$Q_{\delta}$ indexed by $\Delta$. Indeed, the functor of filtered inductive
limit of abelian groups is exact and the natural map
$\varinjlim_{\delta\in\Delta}\operatorname{Hom}_{R}(L,Q_{\delta})\longrightarrow\operatorname{Hom}_{R}(L,\>\varinjlim_{\delta\in\Delta}Q_{\delta})$
is an isomorphism for any $R$-module $L$ of the cardinality not exceeding
$\aleph$. The latter assertion holds because the image of any map of sets
$L\longrightarrow\Delta$ is contained in a proper initial segment
$\\{\delta^{\prime}\mid\delta^{\prime}<\delta\\}\subset\Delta$ for some
$\delta\in\Delta$.

We proceed by induction on $\Delta$ constructing for every element
$\delta\in\Delta$ an $R$-module $P_{\delta}$ and a well-ordered set
$\Gamma_{\delta}$. For any $\delta^{\prime}<\delta$, we will have an embedding
of $R$-modules $P_{\delta^{\prime}}\longrightarrow P_{\delta}$ (such that the
three embeddings form a commutative diagram for any three elements
$\delta^{\prime\prime}<\delta^{\prime}<\delta$) and an ordered embedding
$\Gamma_{\delta^{\prime}}\subset\Gamma_{\delta}$ (making
$\Gamma_{\delta^{\prime}}$ an initial segment of $\Gamma_{\delta}$).
Furthermore, for every element $\gamma\in\Gamma_{\delta}$, a particular
extension class
$c(\gamma,\delta)\in\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta})$,
where $r(\gamma)\in R$, will be defined. For every $\delta^{\prime}<\delta$
and $\gamma\in\Gamma_{\delta^{\prime}}$, the class $c(\gamma,\delta)$ will be
equal to the image of the class $c(\gamma,\delta^{\prime})$ with respect to
the natural map
$\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta^{\prime}})\longrightarrow\operatorname{Ext}^{1}_{R}(R[r(\gamma)^{-1}],P_{\delta})$
induced by the embedding $P_{\delta^{\prime}}\longrightarrow P_{\delta}$.

At the starting point $0\in\Delta$, the module $P_{0}$ is our original
$R$-module $M$ and the set $\Gamma_{0}$ is the disjoint union of all the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],M)$ with $r\in R$, endowed with an
arbitrary well-ordering. The elements $r(\gamma)$ and the classes
$c(\gamma,0)$ for $\gamma\in\Gamma_{0}$ are defined in the obvious way.

Given $\delta=\delta^{\prime}+1\in\Delta$ and assuming that the $R$-module
$P_{\delta^{\prime}}$ and the set $\Gamma_{\delta^{\prime}}$ have been
constructed already, we produce the module $P_{\delta}$ and the set
$\Gamma_{\delta}$ as follows. It will be clear from the construction below
that $\delta^{\prime}$ is always smaller than the well-ordering type of the
set $\Gamma_{\delta^{\prime}}$. So there is a unique element
$\gamma_{\delta^{\prime}}\in\Gamma_{\delta^{\prime}}$ corresponding to the
ordinal $\delta^{\prime}$ (i. e., such that the well-ordering type of the
subset all the elements in $\Gamma_{\delta^{\prime}}$ that are smaller than
$\gamma_{\delta^{\prime}}$ is equivalent to $\delta^{\prime}$).

Define the $R$-module $P_{\delta}$ as the middle term of the extension
corresponding to the class
$c(\gamma_{\delta^{\prime}},\delta^{\prime})\in\operatorname{Ext}^{1}_{R}(R[r(\gamma_{\delta^{\prime}})^{-1}],P_{\delta^{\prime}})$.
There is a natural embedding $P_{\delta^{\prime}}\longrightarrow P_{\delta}$,
as required. Set $\Gamma_{\delta}$ to be the disjoint union of
$\Gamma_{\delta^{\prime}}$ and the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ with $r\in R$, well-ordered
so that $\Gamma_{\delta^{\prime}}$ is an initial segment, while the well-
ordering of the remaining elements is chosen arbitrarily. The elements
$r(\gamma)$ for $\gamma\in\Gamma_{\delta^{\prime}}$ have been defined already
on the previous steps and the classes $c(\gamma,\delta)$ for such $\gamma$ are
defined in the unique way consistent with the previous step, while for the
remaining $\gamma\in\Gamma_{\delta}\setminus\Gamma_{\delta^{\prime}}$ these
elements and classes are defined in the obvious way.

When $\delta$ is a limit ordinal, set
$P_{\delta}=\varinjlim_{\delta^{\prime}<\delta}P_{\delta}^{\prime}$. Let
$\Gamma_{\delta}$ be the disjoint union of
$\bigcup_{\delta^{\prime}<\delta}\Gamma_{\delta^{\prime}}$ and the sets
$\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ with $r\in R$, well-ordered
so that $\bigcup_{\delta^{\prime}<\delta}\Gamma_{\delta^{\prime}}$ is an
initial segment. The elements $r(\gamma)$ and $c(\gamma,\delta)$ for
$\gamma\in\Gamma_{\delta}$ are defined as above.

Arguing by transfinite induction, one easily concludes that the cardinality of
the $R$-module $P_{\delta}$ never exceeds $\aleph$ for $\delta\in\Delta$, and
neither does the cardinality of the set $\Gamma_{\delta}$. It follows that the
well-ordering type of the set
$\Gamma=\bigcup_{\delta\in\Delta}\Gamma_{\delta}$ is equal to $\Delta$. So for
every $\gamma\in\Gamma$ there exists $\delta\in\Delta$ such that
$\gamma=\gamma_{\delta}$.

Set $P=\varinjlim_{\delta\in\Delta}P_{\delta}$. By construction, there is a
natural embedding of $R$-modules $M\longrightarrow P$ and the cokernel is a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$. As every
class $c\in\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta})$ corresponds to an
element $\gamma\in\Gamma_{\delta}$, has the corresponding ordinal
$\delta^{\prime}\in\Delta$ such that $\gamma=\gamma_{\delta^{\prime}}$, and
dies in $\operatorname{Ext}^{1}_{R}(R[r^{-1}],P_{\delta^{\prime}+1})$, we
conclude that $\operatorname{Ext}^{1}_{R}(R[r^{-1}],P)=0$. ∎

######  Lemma 1.1.3.

Any $R$-module admits a surjective map onto it from a transfinitely iterated
extension of the $R$-modules $R[r^{-1}]$ such that the kernel is
contraadjusted.

###### Proof.

The proof follows the second half of the proof of Theorem 10 in [17].
Specifically, given an $R$-module $M$, pick a surjective map onto it from a
free $R$-module $L$. Denote the kernel by $K$ and embed it into a
contraadjusted $R$-module $P$ so that the quotient module $Q$ is a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$. Then the
fibered coproduct $F$ of the $R$-modules $L$ and $P$ over $K$ is an extension
of the $R$-modules $Q$ and $L$. It also maps onto $M$ surjectively with the
kernel $P$. ∎

Both assertions of Theorem 1.1.1 are now proven.

######  Corollary 1.1.4.

An $R$-module is very flat if and only if it is a direct summand of a
transfinitely iterated extension of the $R$-modules $R[r^{-1}]$.

###### Proof.

The “if” part has been explained already; let us prove “only if”. Given a very
flat $R$-module $F$, present it as the quotient module of a transfinitely
iterated extension $E$ of the $R$-modules $R[r^{-1}]$ by a contraadjusted
$R$-module $P$. Since $\operatorname{Ext}^{1}_{R}(F,P)=0$, we can conclude
that $F$ is a direct summand of $E$. ∎

In particular, we have proven that any very flat $R$-module is flat.

######  Corollary 1.1.5.

(a) Any very flat $R$-module can be embedded into a contraadjusted very flat
$R$-module in such a way that the quotient module is very flat.

(b) Any contraadjusted $R$-module admits a surjective map onto it from a very
flat contraadjusted $R$-module such that the kernel is contraadjusted.

###### Proof.

Follows from Theorem 1.1.1 and the fact that the classes of contraadjusted and
very flat $R$-modules are closed under extensions. ∎

###  1.2. Affine geometry of contraadjusted and very flat modules

The results of this section form the module-theoretic background of our main
definitions and constructions in Sections 2–3.

######  Lemma 1.2.1.

(a) The class of very flat $R$-modules is closed with respect to the tensor
products over $R$.

(b) For any very flat $R$-module $F$ and contraadjusted $R$-module $P$, the
$R$-module $\operatorname{Hom}_{R}(F,P)$ is contraadjusted.

###### Proof.

One approach is to prove both assertions simultaneously using the adjunction
isomorphism
$\operatorname{Ext}_{R}^{1}(F\otimes_{R}G,\>P)\simeq\operatorname{Ext}_{R}^{1}(G,\operatorname{Hom}_{R}(F,P))$,
which clearly holds for any $R$-module $G$, any very flat $R$-module $F$, and
contraadjusted $R$-module $P$, and raising the generality step by step. Since
$R[r^{-1}]\otimes_{R}R[s^{-1}]\simeq R[(rs)^{-1}]$, it follows that the
$R$-module $\operatorname{Hom}_{R}(R[s^{-1}],P)$ is contraadjusted for any
contraadjusted $R$-module $P$ and $s\in R$. Using the same adjunction
isomorphism, one then concludes that the $R$-module $R[s^{-1}]\otimes_{R}G$ is
very flat for any very flat $R$-module $G$. From this one can deduce in full
generality the assertion (b), and then the assertion (a).

Alternatively, one can use the full strength of Corollary 1.1.4 and check that
the tensor product of two transfinitely iterated extensions of flat modules is
a transfinitely iterated extension of the pairwise tensor products. Then
deduce (b) from (a). ∎

######  Lemma 1.2.2.

Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings. Then

(a) any contraadjusted $S$-module is also a contraadjusted $R$-module in the
$R$-module structure obtained by the restriction of scalars via $f$;

(b) if $F$ is a very flat $R$-module, then the $S$-module $S\otimes_{R}F$
obtained by the extension of scalars via $f$ is also very flat;

(c) if $F$ is a very flat $R$-module and $Q$ is a contraadjusted $S$-module,
then $\operatorname{Hom}_{R}(F,Q)$ is also a contraadjusted $S$-module;

(d) if $F$ is a very flat $R$-module and $G$ is a very flat $S$-module, then
$F\otimes_{R}G$ is also a very flat $S$-module.

###### Proof.

Part (a): one has
$\operatorname{Ext}_{R}^{*}(R[r^{-1}],P)\simeq\operatorname{Ext}_{S}^{*}(S[f(r)^{-1}],P)$
for any $R$-module $P$ and $r\in R$. Part (b) follows from part (a), or
alternatively, from Corollary 1.1.4. To prove part (c), notice that
$\operatorname{Hom}_{R}(F,Q)\simeq\operatorname{Hom}_{S}(S\otimes_{R}F,\>Q)$
and use part (b) together with Lemma 1.2.1(b) (applied to the ring $S$).
Similarly, part (d) follows from part (b) and Lemma 1.2.1(a). ∎

######  Lemma 1.2.3.

Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings such
that the localization $S[s^{-1}]$ is a very flat $R$-module for any element
$s\in S$. Then

(a) the $S$-module $\operatorname{Hom}_{R}(S,P)$ obtained by the coextension
of scalars via $f$ is contraadjusted for any contraadjusted $R$-module $P$;

(b) any very flat $S$-module is also a very flat $R$-module (in the $R$-module
structure obtained by the restriction of scalars via $f$);

(c) the $S$-module $\operatorname{Hom}_{R}(G,P)$ is contraadjusted for any
very flat $S$-module $G$ and contraadjusted $R$-module $P$.

###### Proof.

Part (a): one has
$\operatorname{Ext}_{S}^{1}(S[s^{-1}],\operatorname{Hom}_{R}(S,P))\simeq\operatorname{Ext}_{R}^{1}(S[s^{-1}],P)$
for any $R$-module $P$ such that $\operatorname{Ext}_{R}^{1}(S,P)=0$ and any
$s\in S$. Part (b) follows from part (a), or alternatively, from Corollary
1.1.4. Part (c): one has
$\operatorname{Ext}_{S}^{1}(S[s^{-1}],\operatorname{Hom}_{R}(G,P))\simeq\operatorname{Ext}_{R}^{1}(G\otimes_{S}S[s^{-1}],\>P)$.
By Lemma 1.2.1(a), the $S$-module $G\otimes_{S}S[s^{-1}]$ is very flat; by
part (b), it is also a very flat $R$-module; so the desired vanishing follows.
∎

######  Lemma 1.2.4.

Let $R\longrightarrow S$ be a homomorphism of commutative rings such that the
related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Then $S$ is a very flat $R$-module.

###### Proof.

The open subset $\operatorname{Spec}S\subset\operatorname{Spec}R$, being
quasi-compact, can be covered by a finite number of principal affine open
subsets $\operatorname{Spec}R[r_{\alpha}^{-1}]\subset\operatorname{Spec}R$,
where $\alpha=1$, …, $N$. The Čech sequence

(1) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}R[r_{\alpha}^{-1}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}R[(r_{\alpha}r_{\beta})^{-1}]\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muR[(r_{1}\dotsm
r_{N})^{-1}]\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu0$

is an exact sequence of $S$-modules, since is its localization by every
element $r_{\alpha}$ is exact. It remains to recall that the class of very
flat $R$-modules is closed under the passage to the kernels of surjective
morphisms. ∎

######  Corollary 1.2.5.

The following assertions hold in the assumptions of Lemma 1.2.4.

(a) The $S$-module $S\otimes_{R}F$ is very flat for any very flat $R$-module
$F$.

(b) An $S$-module $G$ is very flat if and only if it is very flat as an
$R$-module.

(c) The $S$-module $\operatorname{Hom}_{R}(S,P)$ is contraadjusted for any
contraadjusted $R$-module $P$.

(d) An $S$-module $Q$ is contraadjusted if and only if it is contraadjusted as
an $R$-module.

###### Proof.

Part (a) is a particular case of Lemma 1.2.2(b). Part (b): if $G$ is a very
flat $S$-module, then it is also very flat as an $R$-module by Lemma 1.2.3(b)
and Lemma 1.2.4. Conversely, if $G$ is very flat as an $R$-module, then
$G\simeq S\otimes_{R}G$ is also a very flat $S$-module by part (a).

Part (c) follows from Lemma 1.2.3(a) and Lemma 1.2.4. Part (d): if $Q$ is a
contraadjusted $S$-module, then it is also contraadjusted as an $R$-module by
Lemma 1.2.2(a). Conversely, for any $S$-module $Q$ there are natural
isomorphisms of $S$-modules
$Q\simeq\operatorname{Hom}_{S}(S,Q)\simeq\operatorname{Hom}_{S}(S\otimes_{R}S,\>Q)\simeq\operatorname{Hom}_{R}(S,Q)$;
and if $Q$ is contraadjusted as an $R$-module, then it is also a
contraadjusted $S$-module by part (c). ∎

######  Lemma 1.2.6.

Let $R\longrightarrow S_{\alpha}$, $\alpha=1$, …, $N$, be a collection of
homomorphisms of commutative rings for which the corresponding collection of
morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Then

(a) an $R$-module $F$ is very flat if and only if all the $S_{\alpha}$-modules
$S_{\alpha}\otimes_{R}F$ are very flat;

(b) for any contraadjusted $R$-module $P$, the Čech sequence

(2) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(S_{1}\otimes_{R}\dotsb\otimes_{R}S_{N},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}\operatorname{Hom}_{R}(S_{\alpha}\otimes_{R}S_{\beta},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}\operatorname{Hom}_{R}(S_{\alpha},\>P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is an exact sequence of $R$-modules.

###### Proof.

Part (a): by Corollary 1.2.5(a-b), all the $R$-modules
$S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}}\otimes_{R}F$ are
very flat whenever the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$ are very
flat. For any $R$-module $F$ the Čech sequence

(3) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muF\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}S_{\alpha}\otimes_{R}F\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}S_{\alpha}\otimes_{R}S_{\beta}\otimes_{R}F\\\
\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS_{1}\otimes_{R}\dotsb\otimes_{R}S_{N}\otimes_{R}F\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is an exact sequence of $R$-modules (since its localization at any prime ideal
of $R$ is). It remains to recall that the class of very flat $R$-modules is
closed with respect to the passage to the kernels of surjections.

Part (b): the exact sequence of $R$-modules

(4) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muR\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}S_{\alpha}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}S_{\alpha}\otimes_{R}S_{\beta}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS_{1}\otimes_{R}\dotsb\otimes_{R}S_{N}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is composed from short exact sequences of very flat $R$-modules, so the
functor $\operatorname{Hom}_{R}({-},P)$ into a contraadjusted $R$-module $P$
preserves its exactness. ∎

######  Lemma 1.2.7.

Let $f_{\alpha}\colon S\longrightarrow T_{\alpha}$, $\alpha=1$, …, $N$, be a
collection of homomorphisms of commutative rings for which the corresponding
collection of morphisms of affine schemes
$\operatorname{Spec}T_{\alpha}\longrightarrow\operatorname{Spec}S$ is a finite
open covering, and let $R\longrightarrow S$ be a homomorphism of commutative
rings. Then

(a) if all the $R$-modules
$T_{\alpha_{1}}\otimes_{S}\dotsb\otimes_{S}T_{\alpha_{k}}$ are very flat, then
the $R$-module $S$ is very flat;

(b) the $R$-modules $T_{\alpha}[t_{\alpha}^{-1}]$ are very flat for all
$t_{\alpha}\in T_{\alpha}$, $1\le\alpha\le N$, if and only if the $R$-module
$S[s^{-1}]$ is very flat for all $s\in S$.

###### Proof.

Part (a) follows from the Čech exact sequence (cf. (4))

$\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}T_{\alpha}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}T_{\alpha}\otimes_{S}T_{\beta}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT_{1}\otimes_{S}\dotsb\otimes_{S}T_{N}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$

To prove the “if” assertion in (b), notice that
$\operatorname{Spec}T_{\alpha}[t_{\alpha}^{-1}]$ as an open subscheme in
$\operatorname{Spec}S$ can be covered by a finite number of principal affine
open subschemes $\operatorname{Spec}S[s^{-1}]$, and the intersections of these
are also principal open affines. The “only if” assertion follows from part (a)
applied to the covering of the affine scheme $\operatorname{Spec}S[s^{-1}]$ by
the affine open subschemes
$\operatorname{Spec}T_{\alpha}[f_{\alpha}(s)^{-1}]$, together with the fact
that the intersection of every subset of these open affines can be covered by
a finite number of principal affine open subschemes of one of the schemes
$\operatorname{Spec}T_{\alpha}$. ∎

######  Lemma 1.2.8.

Let $f_{\alpha}\colon S\longrightarrow T_{\alpha}$, $\alpha=1$, …, $N$, be a
collection of homomorphisms of commutative rings for which the corresponding
collection of morphisms of affine schemes
$\operatorname{Spec}T_{\alpha}\longrightarrow\operatorname{Spec}S$ is a finite
open covering, and let $R\longrightarrow S$ be a homomorphism of commutative
rings. Let $F$ be an $S$-module. Then

(a) if all the $R$-modules
$T_{\alpha_{1}}\otimes_{S}\dotsb\otimes_{S}T_{\alpha_{k}}\otimes_{S}F$ are
very flat, then the $R$-module $F$ is very flat;

(b) the $R$-modules $T_{\alpha}[t_{\alpha}^{-1}]\otimes_{S}F$ are very flat
for all $t_{\alpha}\in T_{\alpha}$, $1\le\alpha\le N$, if and only if the
$R$-module $F[s^{-1}]$ is very flat for all $s\in S$.

###### Proof.

Just as in the previous lemma, part (a) follows from the Čech exact sequence
(3) constructed for the collection of morphisms of rings $S\longrightarrow
T_{\alpha}$ and the $S$-module $F$. The proof of part (b) is similar to that
of Lemma 1.2.7(b). ∎

###  1.3. Cotorsion modules

Let $R$ be an associative ring. A left $R$-module $P$ is said to be
_cotorsion_ [63, 19] if $\operatorname{Ext}^{1}_{R}(F,P)=0$ for any flat left
$R$-module $F$, or equivalently, $Ext^{>0}_{R}(F,P)=0$ for any flat left
$R$-module $F$. Clearly, the class of cotorsion left $R$-modules is closed
under extensions and the passage to the cokernels of embeddings, and also
under infinite products.

The following theorem, previously known essentially as the “flat cover
conjecture”, was proven by Eklof–Trlifaj [17] and Bican–Bashir–Enochs [6] (cf.
our Theorem 1.1.1). The case of a Noetherian commutative ring $R$ of finite
Krull dimension was previously treated by Xu [63] (cf. Lemma 1.3.9 below).

######  Theorem 1.3.1.

(a) Any $R$-module can be embedded into a cotorsion $R$-module in such a way
that the quotient module is flat.

(b) Any $R$-module admits a surjective map onto it from a flat $R$-module such
that the kernel is cotorsion. ∎

The following results concerning cotorsion (and injective) modules are similar
to the results about contraadjusted modules presented in Section 1.2. With a
possible exception of the last lemma, all of these are very well known.

######  Lemma 1.3.2.

Let $R$ be a commutative ring. Then

(a) for any flat $R$-module $F$ and cotorsion $R$-module $P$, the $R$-module
$\operatorname{Hom}_{R}(F,P)$ is cotorsion;

(b) for any $R$-module $M$ and any injective $R$-module $J$, the $R$-module
$\operatorname{Hom}_{R}(M,J)$ is cotorsion;

(c) for any flat $R$-module $M$ and any injective $R$-module $J$, the
$R$-module $\operatorname{Hom}_{R}(F,J)$ is injective.

###### Proof.

One has
$\operatorname{Ext}_{R}^{1}(G,\operatorname{Hom}_{R}(F,P))\simeq\operatorname{Ext}_{R}^{1}(F\otimes_{R}G,\>P)$
for any $R$-modules $F$, $G$, and $P$ such that
$\operatorname{Ext}_{R}^{1}(F,P)=0=\operatorname{Tor}^{R}_{1}(F,G)$. All the
three assertions follow from this simple observation. ∎

Our next lemma is a generalization of Lemma 1.3.2 to the noncommutative case.

######  Lemma 1.3.3.

Let $R$ and $S$ be associative rings. Then

(a) for any $R$-flat $R$-$S$-bimodule $F$ and any cotorsion left $R$-module
$P$, the left $S$-module $\operatorname{Hom}_{R}(F,P)$ is cotorsion;

(b) for any $R$-$S$-bimodule $M$ and injective left $R$-module $J$, the left
$S$-module $\operatorname{Hom}_{R}(M,J)$ is cotorsion;

(c) for any $S$-flat $R$-$S$-bimodule $F$ and any injective left $R$-module
$J$, the left $S$-module $\operatorname{Hom}_{R}(F,J)$ is injective.

###### Proof.

One has
$\operatorname{Ext}_{S}^{1}(G,\operatorname{Hom}_{R}(F,P))\simeq\operatorname{Ext}_{R}^{1}(F\otimes_{S}G,\>P)$
for any $R$-$S$-bimodule $F$, left $S$-module $G$, and left $R$-module $P$
such that $\operatorname{Ext}_{R}^{1}(F,P)=0=\operatorname{Tor}^{S}_{1}(F,G)$.
Besides, the tensor product $F\otimes_{S}G$ is flat over $R$ if $F$ is flat
over $R$ and $G$ is flat over $S$. This proves (a); and (b-c) are even easier.
∎

######  Lemma 1.3.4.

Let $f\colon R\longrightarrow S$ be a homomorphism of associative rings. Then

(a) any cotorsion left $S$-module is also a cotorsion left $R$-module in the
$R$-module structure obtained by the restriction of scalars via $f$;

(b) the left $S$-module $\operatorname{Hom}_{R}(S,J)$ obtained by coextension
of scalars via $f$ is injective for any injective left $S$-module $J$.

###### Proof.

Part (a): one has
$\operatorname{Ext}_{R}^{1}(F,Q)\simeq\operatorname{Ext}_{S}^{1}(S\otimes_{R}F,\>Q)$
for any flat left $R$-module $F$ and any left $S$-module $Q$. Part (b) is left
to reader. ∎

######  Lemma 1.3.5.

Let $f\colon R\longrightarrow S$ be an associative ring homomorphism such that
$S$ is a flat left $R$-module in the induced $R$-module structure. Then

(a) the left $S$-module $\operatorname{Hom}_{R}(S,P)$ obtained by coextension
of scalars via $f$ is cotorsion for any cotorsion left $R$-module $P$;

(b) any injective right $S$-module is also an injective right $R$-module in
the $R$-module structure obtained by the restriction of scalars via $f$.

###### Proof.

Part (a): one has
$\operatorname{Ext}_{S}^{1}(F,\operatorname{Hom}_{R}(S,P))\simeq\operatorname{Ext}_{R}^{1}(F,P)$
for any left $R$-module $P$ such that $\operatorname{Ext}_{R}^{1}(S,P)=0$ and
any left $S$-module $F$. In addition, in the assumptions of Lemma any flat
left $S$-module $F$ is also a flat left $R$-module. ∎

######  Lemma 1.3.6.

Let $R\longrightarrow S_{\alpha}$ be a collection of commutative ring
homomorphisms such that the corresponding collection of morphisms of affine
schemes $\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is
an open covering. Then

(a) a contraadjusted $R$-module $P$ is cotorsion if and only if all the
contraadjusted $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$ are
cotorsion;

(b) a contraadjusted $R$-module $J$ is injective if and only if all the
contraadjusted $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},J)$ are
injective.

###### Proof.

Part (a): the assertion “only if” follows from Lemma 1.3.5(a). To prove “if”,
use the Čech exact sequence (2) from Lemma 1.2.6(b). By Lemmas 1.3.4(a) and
1.3.5(a), all the terms of the sequence, except perhaps the rightmost one, are
cotorsion $R$-modules, and since the class of cotorsion $R$-modules is closed
under the cokernels of embeddings, it follows that the rightmost term is
cotorsion as well.

Part (b) is proven in the similar way using parts (b) of Lemmas 1.3.4–1.3.5. ∎

Let $R$ be a Noetherian commutative ring, ${\mathfrak{p}}\subset R$ be a prime
ideal, $R_{\mathfrak{p}}$ denote the localization of $R$ at ${\mathfrak{p}}$,
and $\widehat{R}_{\mathfrak{p}}$ be the completion of the local ring
$R_{\mathfrak{p}}$. We recall the notion of a _contramodule_ over a
topological ring, and in particular over a complete Noetherian local ring,
defined in [54, Section 1 and Appendix B], and denote the abelian category of
contramodules over a topological ring $T$ by
$T{\operatorname{\mathsf{--contra}}}$.

The restriction of scalars with respect to the natural ring homomorphism
$R\longrightarrow\widehat{R}_{\mathfrak{p}}$ provides an exact forgetful
functor
$\widehat{R}_{\mathfrak{p}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$. It follows from [54, Theorem B.1.1(1)] that
this functor is fully faithful. The following proposition is a particular case
of the assertions of [54, Propositions B.10.1 and B.9.1].

######  Proposition 1.3.7.

(a) Any $\widehat{R}_{\mathfrak{p}}$-contramodule is a cotorsion $R$-module.

(b) Any free/projective $\widehat{R}_{\mathfrak{p}}$-contramodule is a flat
cotorsion $R$-module.

###### Proof.

In addition to the cited results from [54], take into account Lemma 1.3.4(a)
and the fact that any flat $R_{\mathfrak{p}}$-module is also a flat
$R$-module. ∎

The following theorem is a restatement of the main result of Enochs’ paper
[18].

######  Theorem 1.3.8.

Let $R$ be a Noetherian commutative ring. Then an $R$-module is flat and
cotorsion if and only if it is isomorphic to an infinite product
$\prod_{\mathfrak{p}}F_{\mathfrak{p}}$ of free contramodules
$F_{\mathfrak{p}}=\widehat{R}_{\mathfrak{p}}[[X_{\mathfrak{p}}]]=\varprojlim_{n}\widehat{R}_{\mathfrak{p}}/{\mathfrak{p}}^{n}[X]$
over the complete local rings $\widehat{R}_{\mathfrak{p}}$. Here
$X_{\mathfrak{p}}$ are some sets, and the direct product is taken over all
prime ideals ${\mathfrak{p}}\subset R$. ∎

The following explicit construction of an injective morphism with flat
cokernel from a flat $R$-module $G$ to a flat cotorsion $R$-module
$\operatorname{\mathrm{FC}}_{R}(G)$ was given in the book [63]. For any prime
ideal ${\mathfrak{p}}\subset R$, consider the localization
$G_{\mathfrak{p}}=R_{\mathfrak{p}}\otimes_{R}G$ of the $R$-module $G$ at
${\mathfrak{p}}$, and take its ${\mathfrak{p}}$-adic completion
$\widehat{G}_{\mathfrak{p}}=\varprojlim_{n}G_{\mathfrak{p}}/{\mathfrak{p}}^{n}G_{\mathfrak{p}}$.
By [54, Lemma 1.3.2 and Corollary B.8.2(b)], the $R$-module
$\widehat{G}_{\mathfrak{p}}$ is a free
$\widehat{R}_{\mathfrak{p}}$-contramodule. Furthermore, there is a natural
$R$-module map $G\longrightarrow\widehat{G}_{\mathfrak{p}}$. Set
$\operatorname{\mathrm{FC}}_{R}(G)=\prod_{\mathfrak{p}}\widehat{G}_{\mathfrak{p}}$.

######  Lemma 1.3.9.

The natural $R$-module morphism
$G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$ is injective, and its
cokernel $\operatorname{\mathrm{FC}}_{R}(G)/G$ is a flat $R$-module.

###### Proof.

The following argument can be found in [63, Proposition 4.2.2 and Lemma
3.1.6]. Any $R$-module morphism from $G$ to an
$R_{\mathfrak{p}}/{\mathfrak{p}}^{n}$-module, and hence also to a projective
limit of such modules, factorizes through the morphism
$G\longrightarrow\widehat{G}_{\mathfrak{p}}$. Consequently, in view of Theorem
1.3.8 any morphism from $G$ to a flat cotorsion $R$-module factorizes through
the morphism $G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$.

Now one could apply Theorem 1.3.1(a), but it is more instructive to argue
directly as follows. Let $E$ be an injective cogenerator of the abelian
category of $R$-modules; e. g.,
$E=\operatorname{Hom}_{\mathbb{Z}}(R,{\mathbb{Q}}/{\mathbb{Z}})$. The
$R$-module $\operatorname{Hom}_{R}(G,E)$ being injective by Lemma 1.3.2(c),
the $R$-module $\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$ is flat
and cotorsion by part (b) of the same lemma and by Lemma 1.6.1(b) below.

The natural morphism
$G\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$ is
injective, and moreover, for any finitely generated/presented $R$-module $M$
the induced map
$G\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)\otimes_{R}M\simeq\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G\otimes_{R}M,\>E),\>E)$
is injective, too. The map
$G\longrightarrow\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(G,E),E)$
factorizes through the map
$G\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)$, and it follows that the
map
$G\otimes_{R}M\longrightarrow\operatorname{\mathrm{FC}}_{R}(G)\otimes_{R}M$ is
injective as well. ∎

###  1.4. Exact categories of contraadjusted and cotorsion modules

Let $R$ be a commutative ring. As full subcategories of the abelian category
of $R$-modules closed under extensions, the categories of contraadjusted and
very flat $R$-modules have natural exact category structures. In the exact
category of contraadjusted $R$-modules every morphism has a cokernel, which
is, in addition, an admissible epimorphism.

In the exact category of contraadjusted $R$-modules the functors of infinite
product are everywhere defined and exact; they also agree with the infinite
products in the abelian category of $R$-modules. In the exact category of very
flat $R$-modules, the functors of infinite direct sum are everywhere defined
and exact, and agree with the infinite direct sums in the abelian category of
$R$-modules.

It is clear from Corollary 1.1.5(b) that there are enough projective objects
in the exact category of very flat $R$-modules; these are precisely the very
flat contraadjusted $R$-modules. Similarly, by Corollary 1.1.5(a) in the exact
category of very flat $R$-modules there are enough injective objects; these
are also precisely the very flat contraadjusted modules.

Denote the exact category of contraadjusted $R$-modules by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ and the exact category of
very flat $R$-modules by $R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$.
The tensor product of two very flat $R$-modules is an exact functor of two
arguments $R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$. The $\operatorname{Hom}_{R}$
from a very flat $R$-module into a contraadjusted $R$-module is an exact
functor of two arguments
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ (where
${\mathsf{C}}^{\mathsf{op}}$ denotes the opposite category to a category
${\mathsf{C}}$).

For any homomorphism of commutative rings $f\colon R\longrightarrow S$, the
restriction of scalars with respect to $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$. The extension of scalars
$F\longmapsto S\otimes_{R}F$ is an exact functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$.

For any homomorphism of commutative rings $f\colon R\longrightarrow S$
satisfying the condition of Lemma 1.2.3, the restriction of scalars with
respect to $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}}$. The coextension of scalars
$P\longmapsto\operatorname{Hom}_{R}(S,P)$ is an exact functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muS{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$. In particular, these
assertions hold for any homomorphism of commutative rings $R\longrightarrow S$
satysfying the assumption of Lemma 1.2.4.

######  Lemma 1.4.1.

Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is an open
covering. Then

(a) a pair of homomorphisms of contraadjusted $R$-modules $K\longrightarrow
L\longrightarrow M$ is a short exact sequence if and only if such are the
induced sequences of contraadjusted $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},K)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},L)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},M)$
for all $\alpha$;

(b) a homomorphism of contraadjusted $R$-modules $P\longrightarrow Q$ is an
admissible epimorphism in $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ if
and only if the induced homomorphisms of contraadjusted $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
are admissible epimorphisms in
$S_{\alpha}{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ for all $\alpha$.

###### Proof.

Part (a): the “only if” assertion follows from Lemma 1.2.4. For the same
reason, if the sequences
$0\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},K)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},L)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},M)\longrightarrow
0$ are exact, then so are the sequences obtained by applying the functors
$\operatorname{Hom}_{R}(S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}},\>{-})$,
$k\ge 1$, to the sequence $K\longrightarrow L\longrightarrow M$. Now it
remains to make use of Lemma 1.2.6(b) in order to deduce exactness of the
original sequence $0\longrightarrow K\longrightarrow L\longrightarrow
M\longrightarrow 0$.

Part (b): it is clear from the very right segment of the exact sequence (2)
that surjectivity of the maps
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
implies surjectivity of the map $P\longrightarrow Q$. It remains to check that
the kernel of the latter morphism is a contraadjusted $R$-module. Denote this
kernel by $K$. Since the morphisms
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
are admissible epimorphisms, so are all the morphisms obtained by applying the
coextension of scalars with respect to the ring homomorphisms
$R\longrightarrow S_{\alpha_{1}}\otimes_{R}\dotsb\otimes_{R}S_{\alpha_{k}}$,
$k\ge 1$, to the morphism $P\longrightarrow Q$.

Now Lemma 1.2.6(b) applied to both sides of the morphism $P\longrightarrow Q$
provides a termwise surjective morphism of finite exact sequences of
$R$-modules. The corresponding exact sequence of kernels has $K$ as its
rightmost nontrivial term, while by Lemma 1.2.2(a) all the other terms are
contraadjusted $R$-modules. It follows that the $R$-module $K$ is also
contraadjusted. ∎

Let $R$ be an associative ring. As a full subcategory of the abelian category
of $R$-modules closed under extensions, the category of cotorsion left
$R$-modules has a natural exact category structure.

The functors of infinite product are everywhere defined and exact in this
exact category, and agree with the infinite products in the abelian category
of $R$-modules. Similarly, the category of flat $R$-modules has a natural
exact category structure with exact functors of infinite direct sum.

It follows from Theorem 1.3.1 that there are enough projective objects in the
exact category of cotorsion $R$-modules; these are precisely the flat
cotorsion $R$-modules. Similarly, there are enough injective objects in the
exact category of flat $R$-modules, and these are also precisely the flat
cotorsion $R$-modules.

Denote the exact category of cotorsion left $R$-modules by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ and the exact category of
flat left $R$-modules by $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}$. The
abelian category of left $R$-modules will be denoted simply by
$R{\operatorname{\mathsf{--mod}}}$, and the additive category of injective
$R$-modules (with the trivial exact category structure) by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}$.

For any commutative ring $R$, the $\operatorname{Hom}_{R}$ from a flat
$R$-module into a cotorsion $R$-module is an exact functor of two arguments
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. Analogously, the
$\operatorname{Hom}_{R}$ from an arbitrary $R$-module into an injective
$R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}})^{\mathsf{op}}\times
R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. The functors
$\operatorname{Hom}$ over a noncommutative ring $R$ mentioned in Lemma 1.3.3
have similar exactness properties.

For any associative ring homomorphism $f\colon R\longrightarrow S$, the
restriction of scalars via $f$ is an exact functor
$S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. For any associative ring
homomorphism $f\colon R\longrightarrow S$ making $S$ a flat left $R$-module,
the coextension of scalars $P\longmapsto\operatorname{Hom}_{R}(S,P)$ is an
exact functor $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.

######  Lemma 1.4.2.

Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is an open
covering. Then

(a) a pair of morphisms of cotorsion $R$-modules $K\longrightarrow
L\longrightarrow M$ is a short exact sequence if and only if such are the
sequences of cotorsion $S_{\alpha}$-modules
$\operatorname{Hom}(S_{\alpha},K)\longrightarrow\operatorname{Hom}(S_{\alpha},L)\longrightarrow\operatorname{Hom}(S_{\alpha},M)$
for all $\alpha$;

(b) a morphism of cotorsion $R$-modules $P\longrightarrow Q$ is an admissible
epimorphism if and only if such are the morphisms of cotorsion
$S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)\longrightarrow\operatorname{Hom}_{R}(S_{\alpha},Q)$
for all $\alpha$.

###### Proof.

Part (a) follows from Lemma 1.4.1(a); part (b) can be proven in the way
similar to Lemma 1.4.1(b). ∎

###  1.5. Very flat and cotorsion dimensions

Let $R$ be a commutative ring. By analogy with the definition of the flat
dimension of a module, define the _very flat dimension_ of an $R$-module $M$
as the minimal length of its very flat left resolution.

Clearly, the very flat dimension of an $R$-module $M$ is equal to the supremum
of the set of all integers $d$ for which there exists a contraadjusted
$R$-module $P$ such that $\operatorname{Ext}_{R}^{d}(M,P)\neq 0$. The very
flat dimension of a module cannot differ from its projective dimension by more
than $1$.

Similarly, the _cotorsion dimension_ of a left module $M$ over an associative
ring $R$ is conventionally defined as the minimal length of its right
resolution by cotorsion $R$-modules. The cotorsion dimension of a left
$R$-module $M$ is equal to the supremum of the set of all integers $d$ for
which there exists a flat left $R$-module $F$ such that
$\operatorname{Ext}_{R}^{d}(F,M)\neq 0$.

Both the very flat and the cotorsion dimensions of a module do not depend on
the choice of a particular very flat/cotorsion resolution in the same sense as
the familiar projective, flat, and injective dimensions do not (see Corollary
A.5.2 for the general assertion of this kind).

######  Lemma 1.5.1.

Let $R\longrightarrow S$ be a morphism of associative rings. Then any left
$S$-module $Q$ of cotorsion dimension $\le d$ over $S$ has cotorsion dimension
$\le d$ over $R$.

###### Proof.

Follows from Lemma 1.3.4(a). ∎

######  Lemma 1.5.2.

Let $R\longrightarrow S$ be a morphism of associative rings such that $S$ is a
left $R$-module of flat dimension $\le D$. Then

(a) any left $S$-module $G$ of flat dimension $\le d$ over $S$ has flat
dimension $\le d+D$ over $R$;

(b) any right $S$-module $Q$ of injective dimension $\le d$ over $S$ has
injective dimension $\le\nobreak d+D$ over $R$.

###### Proof.

Part (a) follows from the spectral sequence
$\operatorname{Tor}^{S}_{p}(\operatorname{Tor}^{R}_{q}(M,S),G)\Longrightarrow\operatorname{Tor}^{R}_{p+q}(M,G)$,
which holds for any right $R$-module $M$. Part (b) follows from the spectral
sequence
$\operatorname{Ext}_{S^{\mathrm{op}}}^{p}(\operatorname{Tor}^{R}_{q}(N,S),Q)\Longrightarrow\operatorname{Ext}_{R^{\mathrm{op}}}^{p+q}(N,Q)$,
which holds for any right $R$-module $N$ (where $S^{\mathsf{op}}$ and
$R^{\mathsf{op}}$ denote the rings opposite to $S$ and $R$). ∎

######  Lemma 1.5.3.

(a) Let $R\longrightarrow S$ be a morphism of commutative rings such that $S$
is an $R$-module of very flat dimension $\le D$. Then any $S$-module $G$ of
very flat dimension $\le d$ over $S$ has very flat dimension $\le d+1+D$ over
$R$.

(b) Let $R\longrightarrow S$ be a morphism of commutative rings such that
$S[s^{-1}]$ is an $R$-module of very flat dimension $\le D$ for any element
$s\in S$. Then any $S$-module $G$ of very flat dimension $\le d$ over $S$ has
very flat dimension $\le d+D$ over $R$.

###### Proof.

Part (a): the $S$-module $G$ has a left projective resolution of length $\le
d+1$, and any projective $S$-module has very flat dimension $\le D$ over $R$,
which implies the desired assertion (see Corollary A.5.5(a)).

Part (b): by Corollary 1.1.4, the $S$-module $G$ has a left resolution of
length $\le d$ by direct summands of transfinitely iterated extensions of the
$S$-modules $S[s^{-1}]$. Hence it suffices to show that the very flat
dimension of $R$-modules is not raised by the transfinitely iterated
extension.

More generally, we claim that one has $\operatorname{Ext}^{n}_{R}(M,P)=0$
whenever a module $M$ over an associative ring $R$ is a transfinitely iterated
extension of $R$-modules $M_{\alpha}$ and
$\operatorname{Ext}^{n}_{R}(M_{\alpha},P)=0$ for all $\alpha$. The case $n=0$
is easy; the case $n=1$ is the result of [17, Lemma 1]; and the case $n>1$ is
reduced to $n=1$ by replacing the $R$-module $P$ with an $R$-module $Q$
occuring at the rightmost end of a resolution $0\longrightarrow
P\longrightarrow J^{0}\longrightarrow\dotsb\longrightarrow
J^{n-2}\longrightarrow Q\longrightarrow 0$ with injective $R$-modules $J^{i}$.
∎

######  Lemma 1.5.4.

Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes is a finite open covering. Then

(a) the flat dimension of an $R$-module $F$ is equal to the supremum of the
flat dimensions of the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$;

(b) the very flat dimension of an $R$-module $F$ is equal to the supremum of
the very flat dimensions of the $S_{\alpha}$-modules $S_{\alpha}\otimes_{R}F$;

(c) the cotorsion dimension of a contraadjusted $R$-module $P$ is equal to the
supremum of the cotorsion dimensions of the contraadjusted
$S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$;

(d) the injective dimension of a contraadjusted $R$-module $P$ is equal to the
supremum of the injective dimensions of the contraadjusted
$S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$.

###### Proof.

Part (b) follows easily from Lemma 1.2.6(a), and the proof of part (a) is
similar. Parts (c-d) analogously follow from Lemma 1.3.6(a-b). ∎

The following lemma will be needed in Section 4.10.

######  Lemma 1.5.5.

(a) Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings
and $P$ be an $R$-module such that $\operatorname{Ext}^{1}_{R}(S[s^{-1}],P)=0$
for all elements $s\in S$. Then the $S$-module $\operatorname{Hom}_{R}(S,P)$
is contraadjusted.

(b) Let $f\colon R\longrightarrow S$ be a homomorphism of associative rings
and $P$ be a left $R$-module such that $\operatorname{Ext}^{1}_{R}(G,P)=0$ for
all flat left $S$-modules $G$. Then the $S$-module
$\operatorname{Hom}_{R}(S,P)$ is cotorsion.

###### Proof.

See the proofs of Lemmas 1.2.3(a) and 1.3.5(a). ∎

The following theorem is due to Raynaud and Gruson [55, Corollaire II.3.2.7].

######  Theorem 1.5.6.

Let $R$ be a commutative Noetherian ring of Krull dimension $D$. Then the
projective dimension of any flat $R$-module does not exceed $D$. Consequently,
the very flat dimension of any flat $R$-module also does not exceed $D$. ∎

######  Corollary 1.5.7.

Let $R$ be a commutative Noetherian ring of Krull dimension $D$. Then the
cotorsion dimension of any $R$-module does not exceed $D$.

###### Proof.

For any associative ring $R$, the supremum of the projective dimensions of
flat left $R$-modules and the supremum of the cotorsion dimensions of
arbitrary left $R$-modules are equal to each other. Indeed, both numbers are
equal to the supremum of the set of all integers $d$ for which there exist a
flat left $R$-module $F$ and a left $R$-module $P$ such that
$\operatorname{Ext}_{R}^{d}(F,P)\neq 0$. ∎

###  1.6. Coherent rings, finite morphisms, and coadjusted modules

Recall that an associative ring $R$ is called _left coherent_ if all its
finitely generated left ideals are finitely presented. Finitely presented left
modules over a left coherent ring $R$ form an abelian subcategory in
$R{\operatorname{\mathsf{--mod}}}$ closed under kernels, cokernes, and
extensions. The definition of a right coherent associative ring is similar.

######  Lemma 1.6.1.

Let $R$ and $S$ be associative rings. Then

(a) Assuming that the ring $R$ is left Noetherian, for any $R$-injective
$R$-$S$-bimodule $J$ and any flat left $S$-module $F$ the left $R$-module
$J\otimes_{S}F$ is injective.

(b) Assuming that the ring $S$ is right coherent, for any $S$-injective
$R$-$S$-bimodule $I$ and any injective left $R$-module $J$ the left $S$-module
$\operatorname{Hom}_{R}(I,J)$ is flat.

###### Proof.

Part (a) holds due to the natural isomorphism
$\operatorname{Hom}_{R}(M,\>J\otimes_{S}F)\simeq\operatorname{Hom}_{R}(M,J)\otimes_{S}F$
for any finitely presented left module $M$ over an associative ring $R$, any
$R$-$S$-bimodule $J$, and any flat left $S$-module $F$. Part (b) follows from
the natural isomorphism
$N\otimes_{S}\operatorname{Hom}_{R}(I,J)\simeq\operatorname{Hom}_{R}(\operatorname{Hom}_{S^{\mathrm{op}}}(N,I),J))$
for any finitely presented right module $N$ over an associative ring $S$, any
$R$-$S$-bimodule $I$, and any injective left $R$-module $J$ (where
$S^{\mathrm{op}}$ denotes the ring opposite to $S$).

Here we use the facts that injectivity of a left module $I$ over a left
Noetherian ring $R$ is equivalent to exactness of the functor
$\operatorname{Hom}_{R}({-},I)$ on the category of finitely generated left
$R$-modules, while flatness of a left module $F$ over a right coherent ring
$S$ is equivalent to exactness of the functor ${-}\otimes_{S}F$ on the
category of finitely presented right $S$-modules (cf. the proof of the next
Lemma 1.6.2). ∎

######  Lemma 1.6.2.

Let $R$ and $S$ be associative rings such that $S$ is left coherent. Let $F$
be a left $R$-module of finite projective dimension, $P$ be an $S$-flat
$R$-$S$-bimodule such that $\operatorname{Ext}_{R}^{>0}(F,P)=0$, and $M$ be a
finitely presented left $S$-module. Then one has
$\operatorname{Ext}_{R}^{>0}(F,\>P\otimes_{S}M)=0$, the natural map of abelian
groups
$\operatorname{Hom}_{R}(F,P)\otimes_{S}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$
is an isomorphism, and the right $S$-module $\operatorname{Hom}_{R}(F,P)$ is
flat.

###### Proof.

Let $L_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow M$ be a
left resolution of $M$ by finitely generated projective $S$-modules. Then
$P\otimes_{S}L_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
P\otimes_{S}M$ is a left resolution of the $R$-module $P\otimes_{S}M$ by
$R$-modules annihilated by $\operatorname{Ext}^{>0}(F,{-})$. Since the
$R$-module $F$ has finite projective dimension, it follows that
$\operatorname{Ext}^{>0}(F,\>P\otimes_{S}M)=0$.

Consequently, the functor
$M\longmapsto\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$ is exact on the
abelian category of finitely presented left $S$-modules $M$. Obviously, the
functor $M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{R}M$ is right exact.
Since the morphism of functors
$\operatorname{Hom}_{R}(F,P)\otimes_{S}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{S}M)$
is an isomorphism for finitely generated projective $S$-modules $M$, we can
conclude that it is an isomorphism for all finitely presented left
$S$-modules.

Now we have proven that the functor
$M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{S}M$ is exact on the abelian
category of finitely presented left $S$-modules. Since any left $S$-module is
a filtered inductive limit of finitely presented ones and the inductive limits
commute with tensor products, it follows that the $S$-module
$\operatorname{Hom}_{R}(F,P)$ is flat. ∎

######  Corollary 1.6.3.

Let $R$ be a commutative ring. Then

(a) for any finitely generated $R$-module $M$ and any contraadjusted
$R$-module $P$, the $R$-module $M\otimes_{R}P$ is contraadjusted;

(b) if the ring $R$ is coherent, then for any very flat $R$-module $F$ and any
flat contraadjusted $R$-module $P$, the $R$-module
$\operatorname{Hom}_{R}(F,P)$ is flat and contraadjusted;

(c) in the situation of (b), for any finitely presented $R$-module $M$ the
natural morphism of $R$-modules
$\operatorname{Hom}_{R}(F,P)\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$
is an isomorphism.

###### Proof.

Part (a) immediately follows from the facts that the class of contraadjusted
$R$-modules is closed under finite direct sums and quotients. Part (b) is
provided Lemma 1.6.2 together with Lemma 1.2.1(b), and part (c) is also Lemma
1.6.2. ∎

######  Corollary 1.6.4.

Let $R$ be either a coherent commutative ring such that any flat $R$-module
has finite projective dimension, or a Noetherian commutative ring. Then

(a) for any finitely presented $R$-module $M$ and any flat cotorsion
$R$-module $P$, the $R$-module $M\otimes_{R}P$ is cotorsion;

(b) for any flat $R$-module $F$ and flat cotorsion $R$-module $P$, the
$R$-module $\operatorname{Hom}_{R}(F,P)$ is flat and cotorsion;

(c) in the situation of (a) and (b), the natural morphism of $R$-modules
$\operatorname{Hom}_{R}(F,P)\allowbreak\otimes_{R}M\longrightarrow\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$
is an isomorphism.

###### Proof.

In the former set of assumptions about the ring $R$, parts (a) and (c) follow
from Lemma 1.6.2, and part (b) is provided by the same Lemma together with
Lemma 1.3.2(a). In the latter case, the argument is based on Proposition 1.3.7
and Theorem 1.3.8. Part (a): since the functor $M\otimes_{R}{-}$ preserves
infinite products, it suffices to show that the $R$-module
$M\otimes_{R}Q_{\mathfrak{p}}$ has an
$\widehat{R}_{\mathfrak{p}}$-contramodule structure for any
$\widehat{R}_{\mathfrak{p}}$-contramodule $Q_{\mathfrak{p}}$. Indeed, the full
subcategory
$\widehat{R}_{\mathfrak{p}}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed with respect to finite direct sums
and cokernels.

Part (c): the morphism in question is clearly an isomorphism for a finitely
generated projective $R$-module $M$. Hence it suffices to show that the
functor $M\longmapsto\operatorname{Hom}_{R}(F,\>P\otimes_{R}M)$ is right
exact. In fact, it is exact, since the functor $\operatorname{Hom}_{R}(F,{-})$
preserves exactness of short sequences of
$\widehat{R}_{\mathfrak{p}}$-contramodules. Therefore, the functor
$M\longmapsto\operatorname{Hom}_{R}(F,P)\otimes_{R}M$ is also exact, and we
have proven part (b) as well. ∎

######  Corollary 1.6.5.

(a) Let $R\longrightarrow S$ be a homomorphism of commutative rings such that
the related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Assume that the ring $R$ is coherent. Then the $S$-module
$\operatorname{Hom}_{R}(S,P)$ is flat and contraadjusted for any flat
contraadjusted $R$-module $P$.

(b) Let $R\longrightarrow S_{\alpha}$ be a collection of homomorphisms of
commutative rings for which the corresponding collection of morphisms of
affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Assume that either the ring $R$ is Noetherian and an $R$-module
$P$ is cotorsion, or the ring $R$ is Noetherian of finite Krull dimension and
an $R$-module $P$ is contraadjusted. Then the $R$-module $P$ is flat if and
only if all the $S_{\alpha}$-modules $\operatorname{Hom}_{R}(S_{\alpha},P)$
are flat.

###### Proof.

Part (a): the $S$-module $\operatorname{Hom}_{R}(S,P)$ is contraadjusted by
Corollary 1.2.5(c). The $R$-module $\operatorname{Hom}_{R}(S,P)$ is flat by
Lemma 1.2.4 and Corollary 1.6.3(b). Since
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding, it follows that $\operatorname{Hom}_{R}(S,P)$ is also flat as an
$S$-module (cf. Corollary 1.2.5(b)).

The “only if” assertion in part (b) is provided by part (a). The proof of the
“if” is postponed to Section 5. The cotorsion case will follow from Corollary
5.1.4, while the finite Krull dimension case will be covered by Corollary
5.2.2(b). ∎

Let $R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}$ denote the abelian
category of finitely presented left modules over a left coherent ring $R$. For
a coherent commutative ring $R$, the tensor product of a finitely presented
$R$-module with a flat contraadjusted $R$-module is an exact functor
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ (where the exact category
structure on $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ is induced from
$R{\operatorname{\mathsf{--mod}}}$). The $\operatorname{Hom}_{R}$ from a very
flat $R$-module into a flat contraadjusted $R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{vfl}})^{\mathsf{op}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.

Let $R$ be either a coherent commutative ring such that any flat $R$-module
has finite projective dimension, or a Noetherian commutative ring. Then the
tensor product of a finitely presented $R$-module with a flat cotorsion
$R$-module is an exact functor
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fp}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. The $\operatorname{Hom}_{R}$
from a flat $R$-module to a flat cotorsion $R$-module is an exact functor
$(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}})^{\mathsf{op}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. Here the additive category
of flat cotorsion $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ is endowed with a trivial
exact category structure.

######  Lemma 1.6.6.

Let $R$ be a commutative ring and $I\subset R$ be an ideal. Then

(a) an $R/I$-module $Q$ is a contraadjusted $R/I$-module if and only if it is
a contraadjusted $R$-module;

(b) the $R/I$-module $P/IP$ is contraadjusted for any contraadjusted
$R$-module $P$;

(c) assuming that the ring $R$ is coherent and the ideal $I$ is finitely
generated, for any very flat $R$-module $F$ and flat contraadjusted $R$-module
$P$ the natural morphism of $R/I$-modules
$\operatorname{Hom}_{R}(F,P)/I\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{R/I}(F/IF,\>P/IP)$
is an isomorphism.

###### Proof.

Part (a): the characterization of contraadjusted modules given in the
beginning of Section 1.1 shows that the contraadjustness property of a module
depends only on its abelian group structure and the operators by which the
ring acts in it (rather than on the ring indexing such operators).
Alternatively, the “only if” assertion is a particular case of Lemma 1.2.2(a),
and one can deduce the “if” from the observation that any element $\bar{r}\in
R/I$ can be lifted to an element $r\in R$ so that one has an isomorphism of
$R/I$-modules $R/I[\bar{r}^{-1}]\simeq R/I\otimes_{R}R[r^{-1}]$.

Part (b): the $R$-module $P/IP$ is contraadjusted as a quotient module of a
contraadjusted $R$-module. By part (a), $P/IP$ is also a contraadjusted
$R/I$-module. Part (c): by Corollary 1.6.3(c), the natural morphism of
$R$-modules
$\operatorname{Hom}_{R}(F,P)/I\operatorname{Hom}_{R}(F,P)\allowbreak\longrightarrow\operatorname{Hom}_{R}(F,\>P/IP)$
is an isomorphism. ∎

Recall that a morphism of Noetherian rings $R\longrightarrow S$ is called
_finite_ if $S$ is a finitely generated $R$-module in the induced $R$-module
structure.

######  Lemma 1.6.7.

Let $R\longrightarrow S$ be a finite morphism of Noetherian commutative rings.
Then

(a) the $S$-module $S\otimes_{R}P$ is flat and cotorsion for any flat
cotorsion $R$-module $P$;

(b) assuming that the ring $S$ has finite Krull dimension, an $S$-module $Q$
is a cotorsion $S$-module if and only if it is a cotorsion $R$-module;

(c) for any flat $R$-module $F$ and flat cotorsion $R$-module $P$, the natural
morphism of $S$-modules
$S\otimes_{R}\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{S}(S\otimes_{R}F,\>S\otimes_{R}P)$
is an isomorphism.

###### Proof.

The proof of parts (a-b) is based on Theorem 1.3.8. Given a prime ideal
${\mathfrak{p}}\subset R$, consider all the prime ideals
${\mathfrak{q}}\subset S$ whose full preimage in $R$ coincides with
${\mathfrak{p}}$. Such ideals form a nonempty finite set, and there are no
inclusions between them [40, Theorems 9.1 and 9.3, and Exercise 9.3]. Let us
denote these ideals by ${\mathfrak{q}}_{1}$, …, ${\mathfrak{q}}_{m}$. By [40,
Theorems 9.4(i), 8.7, and 8.15], we have
$S\otimes_{R}\widehat{R}_{\mathfrak{p}}\simeq\widehat{S}_{{\mathfrak{q}}_{1}}\oplus\dotsb\oplus\widehat{S}_{{\mathfrak{q}}_{m}}$.

Since the functor $S\otimes_{R}{-}$ preserves infinite products, in order to
prove (a) it suffices to show that the $S$-module
$S\otimes_{R}F_{\mathfrak{p}}$ is a finite direct sum of certain free
$\widehat{S}_{{\mathfrak{q}}_{i}}$-contramodules $F_{{\mathfrak{q}}_{i}}$.
This can be done either by noticing that $F_{\mathfrak{p}}$ is a direct
summand of an infinite product of copies of $\widehat{R}_{\mathfrak{p}}$ (see
[54, Section 1.3]), or by showing that the natural map
$S\otimes_{R}\widehat{R}_{\mathfrak{p}}[[X]]\longrightarrow\bigoplus_{i=1}^{m}\widehat{S}_{{\mathfrak{q}}_{i}}[[X]]$
is an isomorphism for any set $X$ (see [54, proof of the first assertion of
Proposition B.9.1]). In addition to the assertion of part (a), we have also
proven that any flat cotorsion $S$-module $Q$ is a direct summand of an
$S$-module $S\otimes_{R}P$ for a certain flat cotorsion $R$-module $P$.

The “only if” assertion in part (b) is a particular case of Lemma 1.3.4(a).
Let us prove the “if”. According to Corollary 1.5.7, the cotorsion dimension
of any $R$-module is finite. By Theorem 1.3.1(a), it follows that any flat
$S$-module admits a finite right resolution by flat cotorsion $S$-modules (cf.
the dual version of Corollary A.5.3). Hence it suffices to prove that
$\operatorname{Ext}_{S}^{>0}(G,Q)=0$ for a flat cotorsion $S$-module $G$. This
allows us to assume that $G=S\otimes_{R}F$, where $F$ is a flat (cotorsion)
$R$-module. It remains to recall the $\operatorname{Ext}$ isomorphism from the
proof of Lemma 1.3.4(a).

Part (c): by Corollary 1.6.4(c), the natural morphism of $R$-modules
$S\otimes_{R}\operatorname{Hom}_{R}(F,P)\longrightarrow\operatorname{Hom}_{R}(F,\>S\otimes_{R}P)$
is an isomorphism. ∎

######  Lemma 1.6.8.

Let $R$ be a commutative ring and $I\subset R$ be a finitely generated
nilpotent ideal. Then

(a) an $R$-module $P$ is contraadjusted if and only if the $R/I$-module $P/IP$
is contraadjusted;

(b) a flat $R$-module $F$ is very flat if and only if the $R/I$-module $F/IF$
is very flat;

(c) assuming that the ring $R$ is Noetherian, a flat $R$-module $P$ is
cotorsion if and only if the $R/I$-module $P/IP$ is cotorsion;

(d) assuming that the ring $R$ is coherent and any flat $R$-module has finite
projective dimension, a flat $R$-module $P$ is cotorsion whenever the
$R/I$-module $P/IP$ is cotorsion.

###### Proof.

Part (a): the “only if” assertion is a particular case of Lemma 1.6.6(b). To
prove the “if”, notice that in our assumptions about $I$ the $R$-module $P$
has a finite decreasing filtration by its submodules $I^{n}P$. Furthermore,
the successive quotients $I^{n}P/I^{n+1}P$ are the targets of the natural
surjective homomorphisms of $R/I$-modules
$I^{n}/I^{n+1}\otimes_{R/I}P/IP\longrightarrow I^{n}P/I^{n+1}P$. Since the
$R/I$-module $I^{n}/I^{n+1}$ is finitely generated, the $R/I$-module
$I^{n}P/I^{n+1}P$ is a quotient module of a finite direct sum of copies of the
$R/I$-module $P/IP$. It remains to use the facts that the class of
contraadjusted modules over a given commutative ring is closed under
extensions and quotients, together with the result of Lemma 1.6.6(a).

Part (b): the “only if” is a particular case of Lemma 1.2.2(b); let us prove
the “if”. Let $P$ be a contraadjusted $R$-module; we have to show that
$\operatorname{Ext}^{1}_{R}(F,P)=0$. According to the above proof of part (a),
the $R$-module $P$ has a finite filtration whose successive quotients are
contraadjusted $R/I$-modules with the $R$-module structures obtained by
restriction of scalars. So it suffices to check that
$\operatorname{Ext}^{1}_{R}(F,Q)=0$ for any contraadjusted $R/I$-module $Q$.
Now, the $R$-module $F$ being flat, one has
$\operatorname{Ext}^{1}_{R}(F,Q)=\operatorname{Ext}^{1}_{R/I}(F/IF,\>Q)$ for
any $R/I$-module $Q$ (see the proof of Lemma 1.3.4(a)).

Parts (c-d): in “only if” assertion in (c) is a particular case of Lemma
1.6.7(a). To prove the “if”, notice the isomorphisms of $R/I$-modules
$I^{n}P/I^{n+1}P\simeq I^{n}/I^{n+1}\otimes_{R}P\simeq
I^{n}/I^{n+1}\otimes_{R/I}P/IP$ (the former of which holds, since the
$R$-module $P$ is flat). The $R/I$-module $P/IP$ being flat and cotorsion, the
$R/I$-modules $I^{n}/I^{n+1}\otimes_{R/I}P/IP$ are cotorsion by Corollary
1.6.4(a), the $R$-modules $I^{n}/I^{n+1}\otimes_{R/I}P/IP$ are cotorsion by
Lemma 1.3.4(a), and the $R$-module $P$ is cotorsion, since the class of
cotorsion $R$-modules is closed under extensions. ∎

Let $R$ be a commutative ring and $I\subset R$ be an ideal. Then the reduction
$P\longmapsto P/IP$ of a flat contraadjusted $R$-module $P$ modulo $I$ is an
exact functor $R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R/I{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R/I{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.

Let $f\colon R\longrightarrow S$ be a finite morphism of Noetherian
commutative rings. Then the extension of scalars $P\longmapsto S\otimes_{R}P$
of a flat cotorsion $R$-module $P$ with respect to $f$ is an additive functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
S{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ (between additive categories
naturally endowed with trivial exact category structures).

Let $R$ be a commutative ring. We will say that an $R$-module $K$ is
_coadjusted_ if the functor of tensor product with $K$ over $R$ preserves the
class of contraadjusted $R$-modules. By Corollary 1.6.3(a), any finitely
generated $R$-module is coadjusted.

An $R$-module $K$ is coadjusted if and only if the $R$-module $K\otimes_{R}P$
is contraadjusted for every flat (or very flat) contraadjusted $R$-module $P$.
Indeed, by Corollary 1.1.5(b), any contraajusted $R$-module is a quotient
module of a very flat contraadjusted $R$-module; so it remains to recall that
any quotient module of a contraadjusted $R$-module is contraadjusted.

Clearly, any quotient module of a coadjusted $R$-module is coadjusted.
Furthermore, the class of coadjusted $R$-modules is closed under extensions.
One can see this either by applying the above criterion of coadjustness in
terms of tensor products with flat contraadjusted $R$-modules, or
straightforwardly from the right exactness property of the functor of tensor
product together with the facts that the class of contraadjusted $R$-modules
is closed under quotients and extensions.

Consequently, there is the induced exact category structure on the full
subcategory of coadjusted $R$-modules in the abelian category
$R{\operatorname{\mathsf{--mod}}}$. We denote this exact category by
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{coa}}$. The tensor product of a
coadjusted $R$-module with a flat contraadjusted $R$-module is an exact
functor
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{coa}}\times(R{\operatorname{\mathsf{--mod}}}^{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.

Over a Noetherian commutative ring $R$, any injective module $J$ is
coadjusted. Indeed, for any $R$-module $P$, the tensor product $J\otimes_{R}P$
is a quotient module of an infinite direct sum of copies of $J$, which means a
quotient module of an injective module, which is contraadjusted. Hence any
quotient module of an injective module is coadjusted, too, as is any extension
of such modules.

######  Lemma 1.6.9.

(a) Let $f\colon R\longrightarrow S$ be a homomorphism of commutative rings
such that the related morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is an open
embedding. Then the $S$-module $S\otimes_{R}K$ obtained by the extension of
scalars via $f$ is coadjusted for any coadjusted $R$-module $K$.

(b) Let $f_{\alpha}\colon R\longrightarrow S_{\alpha}$ be a collection of
homomorphisms of commutative rings for which the corresponding collection of
morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Then an $R$-module $K$ is coadjusted if and only if all the
$S_{\alpha}$-modules $S_{\alpha}\otimes_{R}K$ are coadjusted.

###### Proof.

Part (a): any contraadjusted $S$-module $Q$ is also contraadjusted as an
$R$-module, so the tensor product $(S\otimes_{R}K)\otimes_{S}Q\simeq
K\otimes_{R}Q$ is a contraadjusted $R$-module. By Corollary 1.2.5(d), it is
also a contraadjusted $S$-module.

Part (b): the “only if” assertion is provided by part (a); let us prove “if”.
Let $P$ be a contraadjusted $R$-module. Applying the functor $K\otimes_{R}{-}$
to the Čech exact sequence (2) from Lemma 1.2.6(b), we obtain a sequence of
$R$-modules that is exact at its rightmost nontrivial term. So it suffices to
show that the $R$-modules $K\otimes_{R}\operatorname{Hom}_{R}(S_{\alpha},P)$
are contraadjusted.

Now one has
$K\otimes_{R}\operatorname{Hom}_{R}(S_{\alpha},P)\simeq(S_{\alpha}\otimes_{R}K)\otimes_{S_{\alpha}}\operatorname{Hom}_{R}(S_{\alpha},P)$,
the $S_{\alpha}$-module $\operatorname{Hom}_{R}(S_{\alpha},P)$ is
contraadjusted by Corollary 1.2.5(c), and the restriction of scalars from
$S_{\alpha}$ to $R$ preserves contraadjustness by Lemma 1.2.2(a). ∎

###  1.7. Very flat morphisms of schemes

A morphism of schemes $f\colon Y\longrightarrow X$ is called _very flat_ if
for any two affine open subschemes $V\subset Y$ and $U\subset X$ such that
$f(V)\subset U$ the ring of regular functions ${\mathcal{O}}(V)$ is a very
flat module over the ring ${\mathcal{O}}(U)$. By Lemma 1.2.4, any embedding of
an open subscheme is a very flat morphism.

According to Lemmas 1.2.6(a) and 1.2.7(b), the property of a morphism to be
very flat is local in both the source and the target schemes. A morphism of
affine schemes $\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is
very flat if and only if the morphism of commutative rings $R\longrightarrow
S$ satisfies the condition of Lemma 1.2.3. By Lemma 1.2.3(b), the composition
of very flat morphisms of schemes is a very flat morphism.

It does not seem to follow from anything that the base change of a very flat
morphism of schemes should be a very flat morphism. Here is a partial result
in this direction.

######  Lemma 1.7.1.

The base change of a very flat morphism with respect to any locally closed
embedding or universal homeomorphism of schemes is a very flat morphism.

###### Proof.

Essentially, given a very flat morphism $f\colon Y\longrightarrow X$ and a
morphism of schemes $g\colon x\longrightarrow X$, in order to conclude that
the morphism $f^{\prime}\colon y=x\times_{x}Y\longrightarrow x$ is very flat
it suffices to know that the morphism $g^{\prime}\colon y\longrightarrow Y$ is
injective and the topology of $y$ is induced from the topology of $Y$.

Indeed, the very flatness condition being local, one can assume all the four
schemes to be affine. Now any affine open subscheme $v\subset y$ is the full
preimage of a certain open subscheme $V\subset Y$. Covering $V$ with open
affines if necessary and using the locality again, one can assume that $V$ is
affine, too. Finally, if the ${\mathcal{O}}(X)$-module ${\mathcal{O}}(V)$ is
very flat, then by Lemma 1.2.2(b) so is the ${\mathcal{O}}(x)$-module
${\mathcal{O}}(v)={\mathcal{O}}(x\times_{X}V)$. ∎

A quasi-coherent sheaf ${\mathcal{F}}$ over a scheme $X$ is called _very flat_
if the ${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ is very flat for any
affine open subscheme $U\subset X$. According to Lemma 1.2.6(a), very flatness
of a quasi-coherent sheaf over a scheme is a local property. By Lemma
1.2.2(b), the inverse image of a very flat quasi-coherent sheaf under any
morphism of schemes is very flat. By Lemma 1.2.3(b), the direct image of a
very flat quasi-coherent sheaf under a very flat affine morphism of schemes is
very flat.

More generally, given a morphism of schemes $f\colon Y\longrightarrow X$, a
quasi-coherent sheaf ${\mathcal{F}}$ on $Y$ is said to be _very flat over $X$_
if for any affine open subschemes $U\subset X$ and $V\subset Y$ such that
$f(V)\subset U$ the module of sections ${\mathcal{F}}(V)$ is very flat over
the ring ${\mathcal{O}}_{X}(U)$. According to Lemmas 1.2.6(a) and 1.2.8(b),
the property of very flatness of ${\mathcal{F}}$ over $X$ is local in both $X$
and $Y$. By Lemma 1.2.3(b), if the scheme $Y$ is very flat over $X$ and a
quasi-coherent sheaf ${\mathcal{F}}$ is very flat on $Y$, then ${\mathcal{F}}$
is also very flat over $X$.

The following conjecture looks natural. Some evidence in its support is
gathered below in this section.

######  Conjecture 1.7.2.

Any flat morphism of finite type between Noetherian schemes is very flat.

It is well-known that any flat morphism of finite type between Noetherian
schemes (or of finite presentation between arbitrary schemes) is an open map
[28, Théorème 2.4.6]. We will see below that the similar result about very
flat morphisms does not require any finiteness conditions at all.

Given a commutative ring $R$ and an $R$-module $M$, we define its support
$\operatorname{Supp}M\subset\operatorname{Spec}R$ as the set of all prime
ideals ${\mathfrak{p}}\subset R$ for which the tensor product
$k({\mathfrak{p}})\otimes_{R}M$, where $k({\mathfrak{p}})$ denotes the residue
field of the ideal ${\mathfrak{p}}$, does not vanish.

######  Lemma 1.7.3.

Let $R$ be a commutative ring without nilpotent elements and $F$ be a nonzero
very flat $R$-module. Then the support of $F$ contains a nonempty open subset
in $\operatorname{Spec}R$.

###### Proof.

By Corollary 1.1.4, any very flat $R$-module $F$ is a direct summand of a
transfinitely iterated extension $M=\varinjlim_{\alpha}M_{\alpha}$ of certain
$R$-modules of the form $R[s_{\alpha}^{-1}]$, where $s_{\alpha}\in R$. In
particular, $F$ is an $R$-submodule in $M$; consider the minimal index
$\alpha$ for which the intersection $F\cap M_{\alpha}\subset M$ is nonzero.
Denote this intersection by $G$ and set $s=s_{\alpha}$; then $G$ is a nonzero
submodule in $F$ and in $R[s^{-1}]$ simultaneously.

Hence the localization $G[s^{-1}]$ is a nonzero ideal in $R[s^{-1}]$. The
support $\operatorname{Supp}G[s^{-1}]\subset\operatorname{Spec}R[s^{-1}]$ of
the $R[s^{-1}]$-module $G[s^{-1}]$ is equal to the intersection of
$\operatorname{Supp}G\subset\operatorname{Spec}R$ with
$\operatorname{Spec}R[s^{-1}]\subset\operatorname{Spec}R$. By right exactness
of the tensor product functor, the subset $\operatorname{Supp}G[s^{-1}]$
contains the complement $V(s,G)\subset\operatorname{Spec}R[s^{-1}]$ to the
support of the quotient module $R[s^{-1}]/G[s^{-1}]$ in
$\operatorname{Spec}R[s^{-1}]$. The latter is a closed subset in
$\operatorname{Spec}R[s^{-1}]$ corresponding to the ideal $G[s^{-1}]$; if
there are no nilpotents in $R$ then the open subset
$V(s,G)\subset\operatorname{Spec}R[s^{-1}]$ is nonempty.

Let us show that the support of $F$ contains $V(s,G)$. Let ${\mathfrak{p}}\in
V(s,G)\subset\operatorname{Spec}R$ be a prime ideal in $R$ and
$k({\mathfrak{p}})$ be its residue field. Then the map
$k({\mathfrak{p}})\otimes_{R}G=k({\mathfrak{p}})\otimes_{R}G[s^{-1}]\longrightarrow
k({\mathfrak{p}})\otimes_{R}R[s^{-1}]=k({\mathfrak{p}})$ is surjective by the
above argument, the map $k({\mathfrak{p}})\otimes_{R}M_{\alpha}\longrightarrow