and called the _contraderived category of $\mathsf{B}$ in the sense of
Becker_.

An alternative definition of the contraderived category in the sense of Becker
is simply the homotopy category of complexes of injective objects,
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})$. There is an obvious
fully faithful triangulated functor
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(\mathsf{B})$,
constructed as the composition
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(\mathsf{B})$.
For locally presentable abelian categories $\mathsf{B}$ with enough projective
objects, this functor is a triangulated equivalence [90, Corollary 7.4], [91,
Corollary 6.13].

######  Lemma 7.3.

(a) Any Positselski-coacyclic complex in an exact category is Becker-
coacyclic.

(b) Any Positselski-contraacyclic complex in an exact category is Becker-
contraacyclic.

###### Proof.

Part (a) is [58, proof of Theorem 3.5(a)], [90, Lemma 9.1], or [75, Theorem
5.5(a)]. Part (b) is [58, proof of Theorem 3.5(b)], [90, Lemma 7.1], or [75,
Theorem 5.5(b)]. ∎

######  Lemma 7.4.

(a) In an abelian category with enough injective objects, every Becker-
coacyclic complex is acyclic.

(b) In an exact category of finite homological dimension, every acyclic
complex is Becker-coacyclic.

(c) In an abelian category with enough projective objects, every Becker-
contraacyclic complex is acyclic.

(d) In an exact category of finite homological dimension, then every acyclic
complex is Becker-contraacyclic.

###### Proof.

Part (b) follows from Lemmas 7.2(b) and 7.3(a). Part (d) follows from Lemmas
7.2(d) and 7.3(b). Part (a) is [74, Lemma A.2], and part (c) is dual. ∎

######  Proposition 7.5.

(a) Let $\mathsf{A}$ be an exact category of finite homological dimension with
enough injective objects. Then the clases of acyclic, absolutely acyclic, and
Becker-coacyclic complexes in $\mathsf{A}$ coincide. The composition of
triangulated functors
$\operatorname{\mathsf{Hot}}(\mathsf{A}^{\mathsf{inj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{A})\longrightarrow\mathsf{D}^{\mathsf{abs}}(\mathsf{A})=\mathsf{D}^{\mathsf{bco}}(\mathsf{A})=\mathsf{D}(\mathsf{A})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(\mathsf{A}^{\mathsf{inj}})\simeq\mathsf{D}^{\mathsf{abs}}(\mathsf{A})=\mathsf{D}^{\mathsf{bco}}(\mathsf{A})=\mathsf{D}(\mathsf{A}).$

(b) Let $\mathsf{B}$ be an exact category of finite homological dimension with
enough projective objects. Then the classes of acyclic, absolutely acyclic,
and Becker-contraacyclic complexes in $\mathsf{B}$ coincide. The composition
of triangulated functors
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{abs}}(\mathsf{B})=\mathsf{D}^{\mathsf{bctr}}(\mathsf{B})=\mathsf{D}(\mathsf{B})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{abs}}(\mathsf{B})=\mathsf{D}^{\mathsf{bctr}}(\mathsf{B})=\mathsf{D}(\mathsf{B}).$

###### Proof.

One has
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{A})=\operatorname{\mathsf{Ac}}(\mathsf{A})$
and
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{B})=\operatorname{\mathsf{Ac}}(\mathsf{B})$
by Lemma 7.1. Furthermore,
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{A})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bco}}(\mathsf{A})$
and
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{B})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(\mathsf{B})$
by Lemma 7.3. The argument with the totalization of a finite acyclic complex
of complexes similar to [58, proof of Theorem 3.6] or [75, proof of Theorem
5.6] proves that the compositions
$\operatorname{\mathsf{Hot}}(\mathsf{A}^{\mathsf{inj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{A})\longrightarrow\mathsf{D}^{\mathsf{abs}}(\mathsf{A})$
and
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{abs}}(\mathsf{B})$
are triangulated equivalences. Then it follows that
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{A})=\operatorname{\mathsf{Ac}}^{\mathsf{bco}}(\mathsf{A})$
and
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{B})=\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(\mathsf{B})$.
For further discussion, see [76, Theorem 7.8]. ∎

######  Proposition 7.6.

(a) Let $\mathsf{A}$ be an exact category with coproducts and enough injective
objects. Assume that countable coproducts of injective objects have finite
injective dimensions in $\mathsf{A}$. Then the clases of Positselski-coacyclic
and Becker-coacyclic complexes in $\mathsf{A}$ coincide. The composition of
triangulated functors
$\operatorname{\mathsf{Hot}}(\mathsf{A}^{\mathsf{inj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{A})\longrightarrow\mathsf{D}^{\mathsf{pco}}(\mathsf{A})=\mathsf{D}^{\mathsf{bco}}(\mathsf{A})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(\mathsf{A}^{\mathsf{inj}})\simeq\mathsf{D}^{\mathsf{pco}}(\mathsf{A})=\mathsf{D}^{\mathsf{bco}}(\mathsf{A}).$

(b) Let $\mathsf{B}$ be an exact category with products and enough projective
objects. Assume that countable products of injective objects have finite
projective dimensions in $\mathsf{B}$. The composition of triangulated
functors
$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{pctr}}(\mathsf{B})=\mathsf{D}^{\mathsf{bctr}}(\mathsf{B})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(\mathsf{B}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{pco}}(\mathsf{B})=\mathsf{D}^{\mathsf{bctr}}(\mathsf{B}).$

###### Proof.

Existence of coproducts and enough injective objects implies exactness of
coproducts in an exact category, and dually, existence of products enough
projective objects implies exactness of products [75, Remark 5.2]. The
assertions of the proposition are provable similarly to [58, Theorems
3.7–3.8]. For a generalization, see [75, Theorem 5.10]. See also [76, Theorem
7.9]. ∎

Conclusions: as a general rule, all absolutely acyclic, coacyclic, or
contraacyclic complexes tend to be acyclic. However, _we do not know_ how to
prove that all Becker co/contraacyclic complexes in an exact category are
acyclic without making some restrictive assumptions, or what kind of
restrictive assumptions would be optimal for the task. This is not a problem
for abelian categories.

On the other hand, in any exact category of finite homological dimension, all
acyclic complexes are absolutely acyclic. Hence they are also coacyclic and
contraacyclic both in the senses of Positselski and Becker.

###  7.2. Exact subcategories of finite (co)resolution dimension

Let $\mathsf{E}$ be an exact category. A full subcategory
$\mathsf{F}\subset\mathsf{E}$ is said to be _resolving_ if

  * •

the full subcategory $\mathsf{F}$ is closed under extensions and kernels of
admissible epimorphisms in $\mathsf{E}$;

  * •

the full subcategory $\mathsf{F}$ is generating in $\mathsf{E}$ (in the sense
of Section 4.1).

Dually, a full subcategory $\mathsf{C}\subset\mathsf{E}$ is said to be
_coresolving_ if

  * •

the full subcategory $\mathsf{C}$ is closed under extensions and cokernels of
admissible monomorphisms in $\mathsf{E}$;

  * •

the full subcategory $\mathsf{C}$ is cogenerating in $\mathsf{E}$ (in the
sense of Section 4.1).

For example, the left-hand classes of hereditary cotorsion pairs are
resolving, and the right-hand classes are coresolving. Furthermore, any
(co)resolving subcategory is closed under extensions; so it inherits an exact
category structure from the ambient exact category.

The classical concepts of projective, injective, and flat dimensions
generalize naturally to (co)resolving subcategories. Let us assume the exact
category $\mathsf{E}$ to be, at least, weakly idempotent-complete (in the
sense of [11, Section 7]).

Given a resolving subcategory $\mathsf{F}$ in an exact category $\mathsf{E}$,
one says that an object $E\in\mathsf{E}$ has _resolution dimension $\leq d$_
with respect to $\mathsf{F}$ if there exists an exact sequence

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muF_{d}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muF_{d-1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muF_{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muE\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

in $\mathsf{E}$ with $F_{i}\in\mathsf{F}$ for all $0\leq i\leq d$. The
_coresolution dimension_ with respect to a coresolving subcategory is defined
dually.

The (co)resolution dimension of an object with respect to a (co)resolving
subcategory does not depend on the choice of a (co)resolution, in the same
sense as the projective/injective/flat dimension of a module does not depend
on the choice of a (co)resolution. This result can be found in [103,
Proposition 2.3] or [61, Corollary A.5.2]; see also [75, Proposition 6.2].

######  Proposition 7.7.

Let $\mathsf{E}$ be an exact category, $\mathsf{F}\subset\mathsf{E}$ be a
resolving subcategory, and $\mathsf{C}\subset\mathsf{E}$ be a coresolving
subcategory. Then

(a) the inclusion of exact categories $\mathsf{C}\longrightarrow\mathsf{E}$
induces a triangulated equivalence of bounded below derived categories
$\mathsf{D}^{+}(\mathsf{C})\longrightarrow\mathsf{D}^{+}(\mathsf{E})$;

(b) the inclusion of exact categories $\mathsf{F}\longrightarrow\mathsf{E}$
induces a triangulated equivalence of bounded above derived categories
$\mathsf{D}^{-}(\mathsf{F})\longrightarrow\mathsf{D}^{-}(\mathsf{E})$.

###### Proof.

This well-known result goes back to [33, Lemma I.4.6]. See [39, Proposition
13.2.2] or [61, Proposition A.3.1(a)] for some further details. ∎

######  Proposition 7.8.

(a) Let $\mathsf{A}$ be an exact category with exact coproducts and
$\mathsf{C}\subset\mathsf{A}$ be a coresolving subcategory closed under
coproducts. Then the inclusion of exact categories
$\mathsf{C}\longrightarrow\mathsf{A}$ induces a triangulated equivalence of
the Positselski-coderived categories
$\mathsf{D}^{\mathsf{pco}}(\mathsf{C})\longrightarrow\mathsf{D}^{\mathsf{pco}}(\mathsf{A})$.

(b) Let $\mathsf{B}$ be an exact category with exact products and
$\mathsf{F}\subset\mathsf{B}$ be a resolving subcategory closed under
products. Then the inclusion of exact categories
$\mathsf{F}\longrightarrow\mathsf{B}$ induces a triangulated equivalence of
the Positselski-contraderived categories
$\mathsf{D}^{\mathsf{pctr}}(\mathsf{F})\longrightarrow\mathsf{D}^{\mathsf{pctr}}(\mathsf{B})$.

###### Proof.

This result goes back to [16, Proposition and Remark 1.5]. Part (b) in its
stated form can be found in [61, Proposition A.3.1(b)]. For a generalization,
see [75, Theorem 7.11]. ∎

There is not much we can say concerning the comparison of the co/contraderived
categories in the sense of Becker for an exact category and its (co)resolving
subcategory. All we can offer is the following lemma.

######  Lemma 7.9.

(a) Let $\mathsf{A}$ be an exact category and $\mathsf{C}\subset\mathsf{A}$ be
a coresolving subcategory closed under direct summands. Then the classes of
injective objects in $\mathsf{A}$ and $\mathsf{C}$ coincide,
$\mathsf{C}^{\mathsf{inj}}=\mathsf{A}^{\mathsf{inj}}$.

(b) Let $\mathsf{B}$ be an exact category and $\mathsf{F}\subset\mathsf{A}$ be
a resolving subcategory closed under direct summands. Then the classes of
projective objects in $\mathsf{B}$ and $\mathsf{F}$ coincide,
$\mathsf{F}_{\mathsf{proj}}=\mathsf{B}_{\mathsf{proj}}$. ∎

The two propositions and lemma above exhaust our list of results about the
derived/coderived/contraderived categories for _arbitrary_ (co)resolving
subcategories. The assertions below assume _finite (co)resolution dimension_.

######  Proposition 7.10.

Let $\mathsf{E}$ be an exact category, $\mathsf{F}\subset\mathsf{E}$ be a
resolving subcategory, and $\mathsf{C}\subset\mathsf{E}$ be a coresolving
subcategory. Assume that there exists a finite integer $d\geq 0$ such that all
objects of $\mathsf{E}$ have resolution dimensions $\leq d$ with respect to
$\mathsf{F}$ and coresolution dimensions $\leq d$ with respect to
$\mathsf{C}$. Then

(a) for every conventional or absolute derived category symbol
$\star={\mathsf{b}}$, $+$, $-$, $\varnothing$, or ${\mathsf{abs}}$, the
inclusion of exact categories $\mathsf{C}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of the respective derived categories
$\mathsf{D}^{\star}(\mathsf{C})\longrightarrow\mathsf{D}^{\star}(\mathsf{E})$;

(b) for every conventional or absolute derived category symbol
$\star={\mathsf{b}}$, $+$, $-$, $\varnothing$, or ${\mathsf{abs}}$, the
inclusion of exact categories $\mathsf{F}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of derived categories
$\mathsf{D}^{\star}(\mathsf{F})\longrightarrow\mathsf{D}^{\star}(\mathsf{E})$.

###### Proof.

Some cases are covered by Proposition 7.7 and do not need the bounded
(co)resolution dimension assumption. Otherwise, the case $\star=\varnothing$
can be found in [103, Proposition 5.14]. The case $\star={\mathsf{abs}}$ goes
back, essentially, to [57, Theorem 7.2.2(a)]; see also [16, Theorem 1.4]. In
the form stated above, all cases of part (b) can be found in [61, Proposition
A.5.6]. For a generalization, see [75, Theorem 6.6]. ∎

######  Example 7.11.

With the finite (co)resolution dimension conditions dropped, the assertions of
Proposition 7.10 certainly do _not_ stay true. It suffices to consider the
abelian category $\mathsf{E}=R{\operatorname{\mathsf{--mod}}}$ for a ring $R$
of infinite homological dimension, the resolving subcategory of projective
modules $\mathsf{F}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$, the
coresolving subcategory of injective modules
$\mathsf{C}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}$, and
$\star={\mathsf{b}}$ or $\varnothing$.

Indeed, no $R$-module of infinite projective dimension belongs to the
essential image of the functor
$\operatorname{\mathsf{Hot}}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})=\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\longrightarrow\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}})$,
and no $R$-module of infinite injective dimension belongs to the essential
image of the functor
$\operatorname{\mathsf{Hot}}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})=\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})\longrightarrow\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}})$.
On the other hand, the thematic example of an unbounded complex of projective-
injective $R$-modules representing nonzero objects in
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})$
and
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})$,
but vanishing as an object of $\mathsf{D}(R{\operatorname{\mathsf{--mod}}})$,
can be found in [76, formula (2) in Section 7.4].

The following proposition contains a full list of cases for the coderived and
contraderived categories in the sense of Positselski in the context of
(co)resolving subcategories with finite (co)resolution dimension.

######  Proposition 7.12.

Let $\mathsf{E}$ be an exact category, $\mathsf{F}\subset\mathsf{E}$ be a
resolving subcategory, and $\mathsf{C}\subset\mathsf{E}$ be a coresolving
subcategory. Assume that there exists a finite integer $d\geq 0$ such that all
objects of $\mathsf{E}$ have resolution dimensions $\leq d$ with respect to
$\mathsf{F}$ and coresolution dimensions $\leq d$ with respect to
$\mathsf{C}$. Then

(a) assuming that the exact category $\mathsf{E}$ has exact products and the
full subcategory $\mathsf{C}\subset\mathsf{E}$ is closed under products, the
inclusion of exact categories $\mathsf{C}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of the Positselski-contraderived categories
$\mathsf{D}^{\mathsf{pctr}}(\mathsf{C})\longrightarrow\mathsf{D}^{\mathsf{pctr}}(\mathsf{E})$;

(b) assuming that the exact category $\mathsf{E}$ has exact coproducts and the
full subcategory $\mathsf{C}\subset\mathsf{E}$ is closed under coproducts, the
inclusion of exact categories $\mathsf{C}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of the Positselski-coderived categories
$\mathsf{D}^{\mathsf{pco}}(\mathsf{C})\longrightarrow\mathsf{D}^{\mathsf{pco}}(\mathsf{E})$;

(c) assuming that the exact category $\mathsf{E}$ has exact products and the
full subcategory $\mathsf{F}\subset\mathsf{E}$ is closed under products, the
inclusion of exact categories $\mathsf{F}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of the Positselski-contraderived categories
$\mathsf{D}^{\mathsf{pctr}}(\mathsf{F})\longrightarrow\mathsf{D}^{\mathsf{pctr}}(\mathsf{E})$;

(d) assuming that the exact category $\mathsf{E}$ has exact coproducts and the
full subcategory $\mathsf{F}\subset\mathsf{E}$ is closed under coproducts, the
inclusion of exact categories $\mathsf{F}\longrightarrow\mathsf{E}$ induces a
triangulated equivalence of the Positselski-coderived categories
$\mathsf{D}^{\mathsf{pco}}(\mathsf{F})\longrightarrow\mathsf{D}^{\mathsf{pco}}(\mathsf{E})$.

###### Proof.

Parts (b) and (c) are particular cases of Proposition 7.8. Part (d) goes back
to [57, Theorem 7.2.2(a)] and [16, Theorem 1.4]. In the form stated above, it
can be found in [61, Proposition A.5.6]. For a generalization, see [75,
Theorem 6.6]. ∎

Conclusions: as a general rule, the passage to a (co)resolving subcategory
with uniformly bounded (co)resolution dimension tends to not change the
derived, coderived, and contraderived categories. This has been established
for the conventional derived categories and the co/contraderived categories in
the sense of Positselski. For the co/contraderived categories in the sense of
Becker, the formulations and proofs of such assertions remain to be worked
out. When the condition of finite (co)resolution dimension is dropped, the
situation becomes more varied.

###  7.3. Derived categories of flat and projective modules

Let $R$ be an associative ring. Consider the exact category of flat
$R$-modules $\mathsf{E}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ (with
the exact structure inherited from the abelian exact structure of the ambient
abelian category $R{\operatorname{\mathsf{--mod}}}$). Then the full
subcategory of projective $R$-modules
$\mathsf{F}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\subset
R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ is a resolving subcategory in
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$.

The resolution dimensions of objects of
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ with respect to
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ (i. e., the projective
dimensions of flat $R$-modules) are finite when $R$ is a Noetherian
commutative ring of finite Krull dimension [94, Corollaire II.3.2.7], when the
ring $R$ has cardinality $\leq\aleph_{n}$, $n\in\mathbb{Z}_{\geq 0}$ [31,
Théorème 7.10] (cf. [94, Corollaire II.3.3.2]), and for some noncommutative
rings with dualizing complexes [13, first assertion of Proposition 1.5], [64,
Proposition 4.3]. But, generally speaking, the projective dimensions of flat
modules can be infinite. Nevertheless, the following theorem holds.

######  Theorem 7.13.

For any ring $R$, the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ induces an equivalence of
their unbounded derived categories,

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}).$

###### Proof.

This is [51, Proposition 8.1 and Theorem 8.6(i) $\Leftrightarrow$ (iii)].
Notice that, by the definition, the acyclic complexes in the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ are precisely all the acyclic
complexes of flat $R$-modules _with flat $R$-modules of cocycles_ (rather than
arbitrary acyclic complexes of flat $R$-modules). ∎

Thus, even though the assumption of Proposition 7.10 does _not_ hold for the
resolving subcategory
$\mathsf{F}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ in the exact
category $\mathsf{E}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$, the
conclusion of Proposition 7.10(b) still holds true in this case for the
derived category symbols $\star=-$ (by Proposition 7.7(b)) and, much more
nontrivially, $\star=\varnothing$ (by Theorem 7.13). In this sense, one can
say that the resolving subcategory
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ in the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ behaves “as if the resolution
dimensions were finite”.

###  7.4. Becker-contraderived category of flat modules

The following theorem is largely a restatement of Theorem 7.13. Notice that,
by Lemma 7.9(b), the projective objects of the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ are precisely all the
projective $R$-modules,
$(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})_{\mathsf{proj}}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$.
Obviously, there are enough projective objects in
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$.

######  Theorem 7.14.

For any ring $R$, the class of all Becker-contraacyclic complexes in the exact
category $\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ coincides
with the class of all acyclic complexes in this exact category. The
composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}).$

###### Proof.

The first assertion is precisely [51, Theorem 8.6(i) $\Leftrightarrow$ (iii)].
The second assertion is [51, Proposition 8.1]. ∎

This, even though the assumptions of Proposition 7.5(b) do _not_ hold for the
exact category $\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$,
the conclusion of this proposition still partly holds true for this exact
category. In this sense, one can say that the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ behaves “as if its
homological dimension were finite” (see also Section 7.6 below).

###  7.5. Derived categories of cotorsion and all modules

For any ring $R$, the full subcategory of cotorsion $R$-modules
$\mathsf{C}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ is a coresolving
subcategory in the abelian category of $R$-modules
$\mathsf{E}=R{\operatorname{\mathsf{--mod}}}$. By the _cotorsion dimension_ of
an $R$-module one means its coresolution dimension with respect to the
coresolving subcategory
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\subset
R{\operatorname{\mathsf{--mod}}}$.

Let $\mathsf{E}$ be an exact category, $A$, $B\in\mathsf{E}$ be two objects,
and $d\geq-1$ be an integer. One says that the object $A$ has _projective
dimension $\leq d$_ in $\mathsf{E}$ if
$\operatorname{Ext}_{\mathsf{E}}^{d+1}(A,E)=0$ for all objects
$E\in\mathsf{E}$. Dually, one says that the object $B$ has _injective
dimension $\leq d$_ in $\mathsf{E}$ if
$\operatorname{Ext}_{\mathsf{E}}^{d+1}(E,B)=0$ for all objects
$E\in\mathsf{E}$.

We start with a straightforward lemma and corollary.

######  Lemma 7.15.

Let $\mathsf{E}$ be an exact category, $(\mathsf{F},\mathsf{C})$ be a
hereditary cotorsion pair in $\mathsf{E}$, and $d\geq 0$ be an integer. Then
all objects of $\mathsf{F}$ have projective dimensions $\leq d$ in
$\mathsf{E}$ if and only if all objects of $\mathsf{E}$ have coresolution
dimensions $\leq d$ with respect to $\mathsf{C}$. ∎

######  Corollary 7.16.

For any ring $R$ and integer $d\geq 0$, the following two conditions are
equivalent:

  1. (1)

the projective dimensions of all flat $R$-modules do not exceed $d$;

  2. (2)

the cotorsion dimensions of all $R$-modules do not exceed $d$. ∎

So the cotorsion dimensions of modules can be infinite, generally speaking.
Nevertheless, the following theorem holds.

######  Theorem 7.17.

For any ring $R$, the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces an equivalence of their unbounded
derived categories,

$\mathsf{D}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

This is essentially [2, first assertion of Theorem 5.1(2)]. See [92,
Proposition 9.6] for additional details. ∎

Thus, even though the assumption of Proposition 7.10 does _not_ hold for the
coresolving subcategory
$\mathsf{C}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ in the exact
category $\mathsf{E}=R{\operatorname{\mathsf{--mod}}}$, the conclusion of
Proposition 7.10(a) still holds true in this case for the derived category
symbols $\star=+$ (by Proposition 7.7(a)) and, much more nontrivially,
$\star=\varnothing$ (by Theorem 7.17). In this sense, one can say that the
coresolving subcategory $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ in
the abelian category $R{\operatorname{\mathsf{--mod}}}$ behaves “as if the
coresolution dimensions were finite”.

###  7.6. Becker-coderived category of flat modules

An $R$-module is said to be _flat cotorsion_ if it is simultaneously flat and
cotorsion. The full subcategory of flat cotorsion $R$-modules is denoted by
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\subset
R{\operatorname{\mathsf{--mod}}}$.

The flat cotorsion $R$-modules are precisely all the injective objects of the
exact category of flat $R$-modules
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$, and there are enough such
injective objects. The flat cotorsion $R$-modules are also precisely all the
projective objects of the exact category of cotorsion $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$, and there are enough such
projective objects. These assertions follow from the fact that the flat
cotorsion pair in $R{\operatorname{\mathsf{--mod}}}$ is complete (see Theorem
4.4).

######  Theorem 7.18.

For any ring $R$, the class of all Becker-coacyclic complexes in the exact
category $\mathsf{A}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ coincides
with the class of all acyclic complexes in this exact category. The
composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})\longrightarrow\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}).$

###### Proof.

The inclusion
$\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
means that, for any acyclic complex of flat $R$-modules
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with flat modules of
cocycles, and for any complex of flat cotorsion $R$-modules
$Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the complex of abelian
groups
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},Q^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic. More generally, this assertion holds for any complex of (not
necessarily flat) cotorsion $R$-modules
$Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. This is the result of [2,
Theorem 5.3].

It remains to show that for every complex of flat $R$-modules
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ there exists a complex of
flat cotorsion $R$-modules $Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
together with a closed morphism of complexes of $R$-modules
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ whose cone is an acyclic
complex of flat $R$-modules with flat $R$-modules of cocycles. Indeed, by [25,
Definition 3.3 and Corollary 4.10], for any complex of $R$-modules
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, there exists a short exact
sequence of complexes of $R$-modules $0\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$, where
$Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of cotorsion
$R$-modules and $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an
acyclic complex of flat $R$-modules with flat $R$-modules of cocycles. Now if
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of flat
$R$-modules, then $Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
complex of flat cotorsion $R$-modules. Since
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
and the totalization of the short exact sequence $0\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ belongs to
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})\subset\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
(see Lemma 7.1(a)), the cone of the morphism
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ also belongs to the
triangulated subcategory
$\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})\subset\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$.
∎

This, even though the assumptions of Proposition 7.5(a) do _not_ hold for the
exact category $\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$,
the conclusion of this proposition still partly holds true for this exact
category. In this sense, one can say once again that the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ behaves “as if its
homological dimension were finite” (cf. Section 7.4).

###  7.7. Becker-contraderived categories of cotorsion and all modules

As mentioned in the beginning of Section 7.6, the flat cotorsion $R$-modules
are the projective objects of the exact category of cotorsion $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. For any exact category
$\mathsf{E}$, we denote by $\operatorname{\mathsf{Com}}(\mathsf{E})$ the
category of all cochain complexes in $\mathsf{E}$.

######  Theorem 7.19.

For any ring $R$, the class of all Becker-contraacyclic complexes in the exact
category $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ coincides with the
class of all complexes of cotorsion $R$-modules that are Becker-contraacyclic
as complexes in the abelian category $R{\operatorname{\mathsf{--mod}}}$,

$\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})=\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\cap\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

The composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}).$

###### Proof.

It follows from Theorems 7.14 and 7.18 that, for every complex of projective
$R$-modules $P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ there exists a
complex of flat cotorsion $R$-modules
$Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ together with a morphism of
complexes of $R$-modules
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ whose cone is acyclic in the
exact category $R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$. Conversely,
for every complex of flat cotorsion $R$-modules
$Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ there exists a complex of
projective $R$-modules $P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
together with a morphism of complexes of $R$-modules
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
Q^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the same property.
Furthermore, by [2, Theorem 5.3], the complex
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for all complexes
$F^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
and
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$.
Hence the complex
$\operatorname{Hom}_{R}(P^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic if and only if the complex
$\operatorname{Hom}_{R}(Q^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is. This proves the first assertion of the theorem.

To prove the second assertion, notice that the class of all complexes of flat
modules
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$
is deconstructible in the abelian category of all complexes of modules
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}})$ by [101,
Proposition 4.3]. There are enough projective objects in the Grothendieck
abelian category
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}})$, and they
belong to
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$.
By a suitable version of the Eklof–Trlifaj theorem (cf. Theorem 4.3), there
exists a complete cotorsion pair $(\mathsf{F},\mathsf{C})$ in
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}})$ with
$\mathsf{F}=\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$.

In particular, all the contractible two-term complexes of flat modules
$\dotsb\longrightarrow 0\longrightarrow
F\overset{\mathrm{id}}{\longrightarrow}F\longrightarrow 0\
\longrightarrow\dotsb$, $F\in R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$,
belong to $\mathsf{F}$, and it follows that all the objects of $\mathsf{C}$
are complexes of cotorsion $R$-modules (cf. [90, fourth paragraph of the proof
of Theorem 6.6]). Furthermore, all the objects of
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{C}$ are Becker-
contraacyclic as complexes in
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$, since the complex
$\operatorname{Hom}_{R}(G^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for all
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})$,
and in particular for all
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})$,
by [90, Lemma 5.1].

Now let
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$
be a complex of cotorsion $R$-modules, and let $0\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ be a
special precover sequence in
$\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}})$ with
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{C}$ and
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathsf{F}$. Then
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of cotorsion
$R$-modules, hence $G^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
complex of flat cotorsion $R$-modules. Finally, we have
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$,
while the totalization of the short exact sequence $0\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ belongs to
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$.
Hence the cone of the morphism
$G^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belongs to
$\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})$.
This proves the second assertion of the theorem. ∎

######  Lemma 7.20.

For any ring $R$, the composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\allowbreak\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

The composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}})\allowbreak\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$
is a triangulated equivalence

(13)
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$

by [51, Corollary 5.10], [6, Proposition 1.3.6(1)], or [90, Corollary 7.4].
Comparing the results of Theorems 7.14 and 7.18, one comes to a triangulated
equivalence

(14)
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}}).$

Finally, the two equivalences (13) and (14) form a commutative diagram with
the desired triangulated functor
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}^{\mathsf{cot}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$,
because all acyclic complexes in the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ are Becker-contraacyclic in
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$, hence also in
$R{\operatorname{\mathsf{--mod}}}$, by Theorem 7.14. ∎

######  Corollary 7.21.

For any ring $R$, the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces a well-defined triangulated functor
between the Verdier quotient categories
$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$.
The resulting functor is a triangulated equivalence of Becker-contraderived
categories

$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

The first assertion holds, because the inclusion
$\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}\allowbreak(R{\operatorname{\mathsf{--mod}}})$
follows from the first assertion of Theorem 7.19. To deduce the second
assertion, compare the second assertion of Theorem 7.19 with Lemma 7.20. ∎

###  7.8. Periodicity theorems

The most nontrivial aspects of the results of Sections 7.3–7.7 are based on
the class of phenomena called the _periodicity theorems_ , or more
specifically the _flat and projective periodicity_ and the _cotorsion
periodicity_.

Let $L$ be an $R$-module. An $R$-module $M$ is said to be _$L$ -periodic_ if
there is a short exact sequence of $R$-modules $0\longrightarrow
M\longrightarrow L\longrightarrow M\longrightarrow 0$. Periodicity theorems
are results claiming that if a module $L$ belongs to a certain class, a module
$M$ belongs to a certain class, and the exact sequence $0\longrightarrow
M\longrightarrow L\longrightarrow M\longrightarrow 0$ belongs to a certain
class of exact sequences, then the module $M$ actually has to belong to a
certain (more narrow) class of modules.

The _flat and projective periodicity theorem_ claims that if $L$ is a
projective $R$-module and $M$ is a flat $R$-module, then $M$ is actually a
projective $R$-module [8, Theorem 2.5]. The _cotorsion periodicity theorem_
claims that if $L$ is a cotorsion $R$-module, then $M$ is also a cotorsion
$R$-module [2, Theorem 1.2(2) or Proposition 4.8(2)].

Periodicity theorems are closely related to results about modules of cocycles
in acyclic complexes. A general discussion of this connection can be found in
[14, proof of Proposition 7.6], [22, Propositions 1 and 2], and [2,
Proposition 2.4]; see [92, Proposition 8.4] for a generalization. In
particular, in any acyclic complex of projective modules with flat modules of
cocycles, the modules of cocycles are acturally projective [51, Remark 2.15],
while in any acyclic complex of cotorsion modules, the modules of cocycles are
also cotorsion [2, Theorem 5.1(2)].

A general discussion of known periodicity theorems with a long list of such
theorems and connections between them can be found in the introduction to the
paper [3]. In particular, beyond the flat/projective periodicity theorem and
cotorsion periodicity theorem, there is also the fp-injective/injective
periodicity theorem [104, Theorem 5.4 and Corollary 5.5] and the fp-projective
periodicity theorem [96, Example 4.3], [3, Theorems 0.7–0.8 or 4.1–4.2].
Subsequent generalizations can be found in the preprint [78, Theorems 0, A,
and B].

###  7.9. Flat things as direct limits

The classical _Govorov–Lazard theorem_ tells that flat $R$-modules are
precisely the direct limits of projective $R$-modules, or equivalently, of
finitely generated free $R$-modules [27, 44], [26, Corollary 2.22]. This
result can be generalized very widely if one agrees to replace “finitely
generated free/projective modules” by “countably presented flat modules”.

In particular, all at most countable diagrams of flat modules, such as, e. g.,
complexes of flat modules, are direct limits of diagrams/complexes of
countably presented flat modules [81, Corollaries 7.3–7.4]. All acyclic
complexes of flat modules with flat modules of cocycles are direct limits of
such complexes of countably presented flat modules [81, Corollary 7.13].

All flat quasi-coherent sheaves on countably quasi-compact, countably quasi-
separated schemes (e. g., on the usual quasi-compact quasi-separated schemes)
are direct limits of locally countably presented flat quasi-coherent sheaves
[92, Theorems 2.4 and 3.5]. All complexes of flat quasi-coherent sheaves on
such schemes are direct limits of complexes of locally countably presented
flat quasi-coherent sheaves [92, Theorem 4.1]. All acyclic complexes of flat
quasi-coherent sheaves with flat sheaves of cocycles on such schemes are
direct limits of complexes of locally countably presented flat quasi-coherent
sheaves from the same class [92, Theorem 4.2]. The proofs of these assertions
are based on category-theoretic results of [12, Proposition 3.1], [93,
Pseudopullback Theorem 2.2], and the paper [81].

All homotopy flat complexes of flat modules are direct limits of bounded
complexes of finitely generated projective modules [14, Theorem 1.1]. It
follows that all homotopy flat complexes of flat quasi-coherent sheaves on
countably quasi-compact, semi-separated schemes (e. g., on the usual quasi-
compact semi-separated schemes) are direct limits of homotopy flat complexes
of locally countably presented flat quasi-coherent sheaves [92, Theorem 4.5].

Furthermore, all flat quasi-coherent sheaves over certain stacks are direct
limits of locally countably presented ones [80, Theorem 3.1], and so are flat
pro-quasi-coherent pro-sheaves over certain ind-schemes [80, Theorem 10.1].
All acyclic complex of flat quasi-coherent sheaves with flat sheaves of
cocycles over such stacks are direct limits of complexes of locally countably
presented flat quasi-coherent sheaves with flat sheaves of cocycles, and the
same applies to such ind-schemes [80, Corollaries 4.6 and 11.7].

The importance of countably presented flat modules lies in the fact that they
have projective dimensions at most $1$ [26, Corollary 2.23]. The proof of
_quasi-coherent cotorsion periodicity_ in [92, Theorem 9.2] is based on this
fact. Similarly one obtains cotorsion periodicity theorems for quasi-coherent
sheaves over stacks and pro-quasi-coherent pro-sheaves over ind-schemes [80,
Theorems 5.4 and 12.3], as well as certain versions of the “derived $=$
coderived category” theorem for flat quasi-coherent sheaves and flat pro-
quasi-coherent pro-sheaves [80, Theorems 6.5 and 13.2].

_Flat contramodules over topological rings_ is another name for flat pro-
quasi-coherent pro-sheaves over ind-affine ind-schemes (under suitable
assumptions; see [74, Example 3.8(2)]). The exposition in [80] is written in
the language of comodules and contramodules. In fact, the comparison between
the quasi-coherent sheaves over stacks and the comodules over corings is
relatively unproblematic; but (nonflat) pro-quasi-coherent pro-sheaves form a
badly behaved full subcategory in the abelian category of contramodules [74,
Examples 3.1 and 3.8(1)]. So the cotorsion periodicity theorem for ind-affine
ind-schemes [80, Theorem 12.3] should be stated in the language of
contramodules rather than pro-quasi-coherent pro-sheaves (and it is done this
way in [80]). We do _not_ know what, if anything, should a quasi-coherent
cotorsion periodicity theorem for non-ind-affine ind-schemes say.

###  7.10. Periodicity theorems II

The aim of this section is to discuss the nature of periodicity theorems as it
appears in our present understanding.

It appears that both the flat/projective and cotorsion periodicity theorems
stem from the fact that all flat modules are direct limits of (finitely
generated) projective ones. Similarly, both the fp-projective and fp-
injective/injective periodicity theorems stem from the fact that all modules
are direct limits of finitely presented ones.

Concerning the cotorsion and fp-injective/injective periodicities, this
connection is expressed in the form of a general category-theoretic theorem in
[92, Theorem 8.1 and Corollary 8.2]. A common generalization of the cotorsion
and fp-injective/injective periodicity theorems based on these results from
[92] is worked out in [78, Theorem 0(b) in Section 0.1 and Theorem B in
Section 0.3]. The quasi-coherent cotorsion periodicity on quasi-compact semi-
separated schemes [92, Theorem 9.2], as well as generalizations to stacks and
ind-schemes [80, Theorems 5.4 and 12.3], are corollaries of [92, Corollary
8.2] (cf. the last paragraph of Section 7.9).

Concerning the flat/projective and fp-projective periodicities, a common
generalization of both, expressing their interpretation as properties of the
direct limit closure, can be found in [78, Theorem 0(a) in Section 0.1 and
Theorem A in Section 0.2].

A general concept of the direct limit closure of an exact category is worked
out in the preprint [82]. Some related periodicity theorems can be found in
[82, Theorems 7.1, 7.6, and 8.3].

###  7.11. Becker-contraderived category of very flat modules

We start with a very flat version of Theorem 7.13 before passing to a very
flat version of Theorem 7.14.

For any commutative ring $R$, the full subcategory of projective $R$-modules
$\mathsf{F}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}$ is a resolving
subcategory in the exact category of very flat $R$-modules
$\mathsf{E}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$. Moreover, the
projective objects of the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ are precisely all the
projective $R$-modules,
$(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})_{\mathsf{proj}}=R{\operatorname{\mathsf{--mod}}}$,
and there are enough such projective objects.

######  Corollary 7.22.

For any commutative ring $R$ and any conventional derived category symbol
$\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$, the inclusion of exact
categories $R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ induces an equivalence of
their derived categories

$\operatorname{\mathsf{Hot}}^{\star}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})=\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}).$

###### Proof.

Projective dimensions of very flat modules do not exceed $1$ by Theorem
4.5(d). Hence the assertion of the theorem is a particular case of Proposition
7.10(b). ∎

######  Corollary 7.23.

For any commutative ring $R$, the classes of all absolutely acyclic complexes,
Becker-contraacyclic complexes, and acyclic complexes in the exact category
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ coincide. The
composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})\longrightarrow\mathsf{D}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}).$

###### Proof.

The homological dimension of the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ does not exceed $1$, so
Proposition 7.5(b) is applicable. ∎

###  7.12. Derived categories of contraadjusted and all modules

Let $R$ be a commutative ring. Then the full subcategory of contraadjusted
$R$-modules $\mathsf{C}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ is a
coresolving subcategory in the abelian category of $R$-modules
$\mathsf{E}=R{\operatorname{\mathsf{--mod}}}$.

######  Corollary 7.24.

For any commutative ring $R$ and any conventional or absolute derived category
symbol $\star={\mathsf{b}}$, $+$, $-$, $\varnothing$, or ${\mathsf{abs}}$, the
inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces an equivalence of their derived
categories

$\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\simeq\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

Contraadjusted dimensions of $R$-modules do not exceed $1$ by Theorem 4.5(c).
Hence the assertion of the theorem is a particular case of Proposition
7.10(a). ∎

###  7.13. Coderived category of very flat modules

Let $R$ be a commutative ring. An $R$-module is said to be _very flat
contraadjusted_ if it is simultaneously very flat and contraadjusted. The full
subcategory of very flat contraadjusted $R$-modules is denoted by
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}\cap
R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\subset
R{\operatorname{\mathsf{--mod}}}$.

The very flat contraadjusted $R$-modules are precisely all the injective
objects of the exact category of very flat $R$-modules
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$, and there are enough such
injective objects. The very flat contraadjusted $R$-modules are also precisely
all the the projective objects of the exact category of contraadjusted
$R$-modules $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$, and there are
enough such projective objects. These assertions follow from the fact that the
very flat cotorsion pair in $R{\operatorname{\mathsf{--mod}}}$ is complete
(see Theorem 4.5(a)).

######  Corollary 7.25.

For any commutative ring $R$, the classes of all absolutely acyclic complexes,
Positselski-coacyclic complexes, Becker-coacyclic complexes, and acyclic
complexes in the exact category
$\mathsf{A}=R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ coincide. The
composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})\longrightarrow\mathsf{D}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{pco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\\\
\simeq\mathsf{D}^{\mathsf{abs}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{pco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}).$

###### Proof.

The homological dimension of the exact category
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}$ does not exceed $1$, so
Proposition 7.5(a) is applicable and proves all the assertions except the one
concerning Positselski-coacyclic complexes. To deduce the latter one, observe
that the inclusions
$\operatorname{\mathsf{Ac}}^{\mathsf{abs}}(\mathsf{E})\subset\operatorname{\mathsf{Ac}}^{\mathsf{pco}}(\mathsf{E})\subset\operatorname{\mathsf{Ac}}^{\mathsf{bco}}(\mathsf{E})$
hold for any exact category $\mathsf{E}$ by the definition and by Lemma
7.3(a). ∎

###  7.14. Becker-contraderived categories of contraadjusted and all modules

We start with the result about Positselski-contraderived categories before
passing to Becker-contraderived ones.

######  Corollary 7.26.

For any commutative ring $R$, the class of all Positselski-contraacyclic
complexes in the exact category
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$ coincides with the class of
all complexes of contraadjusted $R$-modules that are Positselski-contraacyclic
as complexes in the abelian category $R{\operatorname{\mathsf{--mod}}}$,

$\operatorname{\mathsf{Ac}}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})=\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\cap\operatorname{\mathsf{Ac}}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}).$

The inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces a triangulated equivalence of the
Positselski-contraderived categories

$\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\simeq\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

The inclusion
$\operatorname{\mathsf{Ac}}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\subset\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\cap\operatorname{\mathsf{Ac}}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}})$
holds by the definition, and it follows that the triangulated functor between
the Verdier quotient categories
$\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}})$
induced by the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is well-defined. Since the contraadjusted
dimensions of $R$-modules never exceed $1$, Proposition 7.12(a) tells that
this triangulated functor is an equivalence, proving the second assertion of
the theorem. Then the first assertion of the theorem follows from the second
one. ∎

######  Lemma 7.27.

Let $R$ be a commutative ring. Then, for any acyclic complex of flat
$R$-modules $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with flat
$R$-modules of cocycles, and any complex of contraadjusted $R$-modules
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the complex of morphisms
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic.

###### Proof.

Choose a homotopy injective complex of injective $R$-modules
$J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ together with a quasi-
isomorphism of complexes of $R$-modules
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Then the complex
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic, since $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an
acyclic complex of $R$-modules. Denote the cone of the morphism of complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ by
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Then
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an acyclic complex of
contraadjusted $R$-modules. By Theorem 4.5(c), the $R$-modules of cocycles of
the complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
contraadjusted. Now [25, Lemma 3.9] tells that the complex
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic. Thus the complex
$\operatorname{Hom}_{R}(F^{\text{\smaller\smaller$\scriptstyle\bullet$}},C^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic as well. ∎

We recall from Section 7.13 that the very flat contraadjusted $R$-modules are
the projective objects of the exact category of contraadjusted $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.

######  Theorem 7.28.

For any commutative ring $R$, the class of all Becker-contraacyclic complexes
in the exact category $R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$
coincides with the class of all complexes of contraadjusted $R$-modules that
are Becker-contraacyclic as complexes in the abelian category
$R{\operatorname{\mathsf{--mod}}}$,

$\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})=\operatorname{\mathsf{Com}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\cap\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

The composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}).$

###### Proof.

The argument is similar to the proof of Theorem 7.19, using Corollaries 7.23
and 7.25 instead of Theorems 7.14 and 7.18, respectively. Instead of [2,
Theorem 5.3], use Lemma 7.27. ∎

######  Lemma 7.29.

For any commutative ring $R$, the composition of triangulated functors
$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\longrightarrow\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$
is a triangulated equivalence

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{vfl}}^{\mathsf{cta}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

Similar to the proof of Lemma 7.20. Once again, Corollaries 7.23 and 7.25 need
to be used instead of Theorems 7.14 and 7.18. ∎

######  Corollary 7.30.

For any commutative ring $R$, the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces a well-defined triangulated functor
between the Verdier quotient categories
$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\longrightarrow\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$.
The resulting functor is a triangulated equivalence of Becker-contraderived
categories

$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

###### Proof.

Similar to the proof of Corollary 7.21. Theorem 7.28 and Lemma 7.29 need to be
used instead of Theorem 7.19 and Lemma 7.20. ∎

###  7.15. Summary I

Let us collect here explicit answers to the questions posed in the second
introductory paragraph of Section 7. Let $U$ be an affine scheme and
$R=\mathcal{O}(U)$ a commutative ring.

The inclusion of exact categories
$U{\operatorname{\mathsf{--ctrh}}}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces equivalences of conventional derived
categories

$\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})\simeq\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$

for all symbols $\star={\mathsf{b}}$, $+$, $-$, and $\varnothing$ by Corollary
7.24. The same holds for the absolute derived categories,
$\star={\mathsf{abs}}$.

The same inclusion of exact categories also induces equivalences of the
contraderived categories in the sense of Positselski and Becker,

$\displaystyle\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})$
$\displaystyle\simeq\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}),$
$\displaystyle\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}})$
$\displaystyle\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$

by Corollaries 7.26 and 7.30 (see also Theorem 7.28).

The inclusion of exact categories
$U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces equivalences of conventional derived
categories

$\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$

for the symbols $\star=+$, and much more importantly and nontrivially,
$\star=\varnothing$ (i. e., for the unbounded derived categories). The
assertion for $\star=+$ is a particular case of the very general Proposition
7.7(a). The assertion for $\star=\varnothing$ is Theorem 7.17.

The same inclusion of exact categories also induces an equivalence of the
Becker-contraderived categories

$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$

by Corollary 7.21 (see also Theorem 7.19).

The inclusion of exact categories
$U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}=R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces equivalences of conventional derived
categories

$\operatorname{\mathsf{Hot}}^{+}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})=\mathsf{D}^{+}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})\simeq\mathsf{D}^{+}(R{\operatorname{\mathsf{--mod}}}).$

The triangulated functors

$\operatorname{\mathsf{Hot}}^{\star}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})=\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$

induced by the same inclusion of exact categories are usually _not_
equivalences for $\star={\mathsf{b}}$ or $\star=\varnothing$ (see Example
7.11).

For $\star=-$, this triangulated functor is also _not_ an equivalence for
associative rings $R$ in general [79, Proposition 1.2 together with Corollary
2.3, 4.1, 7.2, or 7.3, or together with Example 8.4]. It would be interesting
to have a counterexample for a commutative ring $R$.

The triangulated functors

$\displaystyle\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})=\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})$
$\displaystyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}^{\mathsf{pctr}}(R{\operatorname{\mathsf{--mod}}}),$
$\displaystyle\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})=\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{inj}})$
$\displaystyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}})$

are also _not_ expected to be equivalences in general. The same
counterexamples from the paper [79] work, providing complexes of injective
modules over certain associative rings that are bounded above and acyclic,
hence contraacyclic in the sense of Positselski [57, Lemma 4.1], [84, Lemma
5.3(b)] (hence also contraacyclic in the sense of Becker) in the abelian
category $R{\operatorname{\mathsf{--mod}}}$, but not contractible.

Let us offer some comment or explanation. Morally, the locally contraadjusted
and locally cotorsion contraherent cosheaves are two contraherent versions or
“dual analogues” of _arbitrary_ quasi-coherent sheaves; while the locally
injective contraherent cosheaves are the dual analogues of _flat_ quasi-
coherent sheaves. So the unbounded derived categories
$\mathsf{D}(X{\operatorname{\mathsf{--ctrh}}})$ and
$\mathsf{D}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ _are_ expected
to be directly comparable to the derived category of quasi-coherent sheaves
$\mathsf{D}(X{\operatorname{\mathsf{--qcoh}}})$ for a good enough scheme $X$;
while the derived category
$\mathsf{D}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$ is _not_
expected to be compared to $\mathsf{D}(X{\operatorname{\mathsf{--qcoh}}})$.
See [61, Theorem 4.6.6], cf. [61, Theorem 5.7.1].

###  7.16. Summary II

The thesis _flat modules are not that much different from projective modules_
in the title of Section 7 is explained/confirmed by Theorems 7.13, 7.14, and
7.18. To elaborate,

  * •

the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ induces an equivalence of the
conventional unbounded derived categories

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})=\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}});$

  * •

the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$ induces an equivalence of the
Becker-coderived categories

$\operatorname{\mathsf{Hot}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})=\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{proj}})\simeq\mathsf{D}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}});$

  * •

the classes of acyclic, Becker-coacyclic, and Becker-contraacyclic complexes
in the exact category $R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}$
coincide,

$\operatorname{\mathsf{Ac}}^{\mathsf{bco}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\operatorname{\mathsf{Ac}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}})=\operatorname{\mathsf{Ac}}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}_{\mathsf{fl}}).$

These are instances of behaviour typical for resolving subcategories of finite
resolution dimension or exact categories of finite homological dimension. Cf.
Corollaries 7.22, 7.23, and 7.25, which are based on Propositions 7.5 and
7.10(b).

The same thesis _flat modules are not that much different from projective
modules_ is also explained/confirmed by Theorem 7.17 and Corollary 7.21. To
repeat,

  * •

the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces an equivalence of the conventional
unbounded derived categories

$\mathsf{D}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}(R{\operatorname{\mathsf{--mod}}});$

  * •

the inclusion of exact categories
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces an equivalence of the Becker-
contraderived categories

$\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}})\simeq\mathsf{D}^{\mathsf{bctr}}(R{\operatorname{\mathsf{--mod}}}).$

These are instances of behaviour typical for coresolving subcategories of
finite coresolution dimension. Cf. Corollaries 7.24 and 7.30, which are based
on Proposition 7.5, Proposition 7.10(a), and Theorem 4.5(c). See Corollary
7.16 for the connection between the projective dimensions of flat modules and
the cotorsion dimensions of arbitrary modules.

###  7.17. Conclusion

For the purposes of the derived, coderived, and contraderived categories,
finite homological dimension is “harmless”. This statement actually refers two
two assertions: firstly, the passage to an exact subcategory of finite
(co)resolution dimension does not change the derived/coderived/contraderived
category; and secondly, for an exact category of finite homological dimension,
the derived and coderived/contraderived category agree whenever they are well-
defined.

The discussion in this section illustrates the thesis that the passage to the
direct limit closure of an exact (sub)category is almost as “harmless” as
finite homological dimension (see Section 7.10). Thus, even though projective
dimensions of flat modules may be infinite, flat modules are “not that much
different” from projective ones.

The typical covers in algebraic geometry are related to commutative ring
homomorphisms $R\longrightarrow S$ making $S$ a flat, but not a projective
$R$-module. The main technical problem of the theory of contraherent cosheaves
consists in this fact, and various specific technical problems tend to arise
from this main one. The discussion in this section suggests that such problems
are solvable.

##  8\. Contraherent Proof of Quasi-Coherent Cotorsion Periodicity

The proof of the quasi-coherent cotorsion periodicity theorem [92, Theorem
9.2] given in the paper [92] (mentioned in Sections 7.9–7.10 above) is
essentially global, based as it is on applying the category-theoretic result
of [92, Corollary 8.2] to the whole category of quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}$. The aim of this section is to suggest a
_local_ proof of quasi-coherent cotorsion periodicity, using reduction to the
affine case established in [2, Theorem 1.2(2), Proposition 4.8(2), or Theorem
5.1(2)]. The concept of a contraherent cosheaf is used in order to accomplish
such a reduction.

###  8.1. Preliminary discussion

The interaction of local and global phenomena is a fundamental aspect of
geometry. Many properties of modules over commutative rings are _local_ in the
sense that they can be checked on an affine open covering
$U=\bigcup_{\alpha=1}^{d}U_{\alpha}$ of an affine scheme $U$ using the
localization functors to pass from $\mathcal{O}(U)$-modules to modules over
the rings $\mathcal{O}(U_{\alpha})$. For example, the projectivity [94, §
II.3.1], [54], flatness, and very flatness [61, Lemma 1.2.6(a)], [77, Example
2.5] are local properties of modules over commutative rings.

The property of a module over commutative ring to be cotorsion is _not_ local,
however [77, Example 2.7]. This fact stands in the way of a direct attempt to
reduce the cotorsion periodicity theorem for quasi-coherent sheaves to the
similar theorem for modules over commutative rings.

As pointed out in [77, Example 3.8], the class of cotorsion modules is
_colocal_ (if the contraadjustedness is presumed). Contraherent cosheaves are
defined by gluing modules over commutative rings using the colocalization
functors. Hence the idea to use contraherent cosheaves in order to obtain a
local proof of the quasi-coherent cotorsion periodicity.

The exposition in this section is based on the discussion of local, colocal,
and antilocal properties in the paper [77], as well as on certain
constructions with contraherent cosheaves worked out in [61, Sections 2.5–2.6
and 4.6]. The class of cotorsion modules over commutative rings is _both
colocal and antilocal_ [77, Examples 3.8, 6.2, and 7.2]; these facts are
important for our argument.

###  8.2. Contraadjusted and cotorsion quasi-coherent sheaves

Throughout the present Section 8, $X$ denotes a quasi-compact semi-separated
scheme. The notation
$\operatorname{Ext}_{X}^{*}({-},{-})=\operatorname{Ext}_{X{\operatorname{\mathsf{--qcoh}}}}^{*}({-},{-})$
stands for the Ext functors computed in the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$. The definition of a _very flat quasi-
coherent sheaf_ was given in Section 6.2.

A quasi-coherent sheaf $\mathcal{C}$ on $X$ is said to be _cotorsion_ if
$\operatorname{Ext}^{1}_{X}(\mathcal{F},\mathcal{C})=0$ for every flat quasi-
coherent sheaf $\mathcal{F}$ on $X$. A quasi-coherent sheaf $\mathcal{C}$ on
$X$ is said to be _contraadjusted_ if
$\operatorname{Ext}^{1}_{X}(\mathcal{F},\mathcal{C})=0$ for all very flat
quasi-coherent sheaves $\mathcal{F}$ on $X$. Similarly to the notation for
modules, we will denoted the classes of flat, very flat, cotorsion, and
contraadjusted quasi-coherent sheaves on $X$ by
$X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{fl}}$,
$X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{vfl}}$,
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$, and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, respectively.

######  Theorem 8.1.

For any quasi-compact semi-separated scheme $X$, the pair of classes of
objects (flat quasi-coherent sheaves, cotorsion quasi-coherent sheaves) is a
hereditary complete cotorsion pair in the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$.

The cotorsion pair from Theorem 8.1 is called the _flat cotorsion pair_ in the
category of quasi-coherent sheaves.

###### Proof.

First of all, it is important that the class of flat quasi-coherent sheaves is
generating in $X{\operatorname{\mathsf{--qcoh}}}$ [48, Section 2.4], [16,
Lemma A.1]. This is where the assumption that the scheme $X$ is quasi-compact
and semi-separated is needed (cf. [99, Theorem 2.2]). Otherwise, existence of
flat covers and cotorsion envelopes in the category of quasi-coherent sheaves
is known for any scheme [21, Theorem 4.2]. Combining these two results, one
comes to the conclusion that the flat cotorsion pair in
$X{\operatorname{\mathsf{--qcoh}}}$ is complete. To show that it is
hereditary, one can notice that condition (1) from Lemma 4.1 holds for flat
quasi-coherent sheaves, because it holds for flat modules.

Alternatively, the results of [61, Lemma 4.1.8, Corollary 4.1.9, Lemma 4.1.10,
and Corollary 4.1.11] are applicable and tell more than the theorem claims.
This argument uses completeness of the flat cotorsion pair in the categories
of modules over commutative rings as a black box (see Theorem 4.4); then the
rest of the construction is more elementary, and no further use of set-
theoretical methods is needed. The construction proceeds from any fixed finite
affine open covering of the scheme $X$. In the context of a finite principal
affine open covering of an affine scheme, one can also find this argument
spelled out in [77, proof of Proposition 4.3]. ∎

######  Theorem 8.2.

For any quasi-compact semi-separated scheme $X$, the pair of classes of
objects (very flat quasi-coherent sheaves, contraadjusted quasi-coherent
sheaves) is a hereditary complete cotorsion pair in the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$.

The cotorsion pair from Theorem 8.2 is called the _very flat cotorsion pair_
in the category of quasi-coherent sheaves.

###### Proof.

As in the previous theorem, first of all one needs to show that the class of
very flat quasi-coherent sheaves is generating in
$X{\operatorname{\mathsf{--qcoh}}}$ [61, Lemma 4.1.1]. Then completeness of
the very flat cotorsion pair in $X{\operatorname{\mathsf{--qcoh}}}$ is
provable by applying a suitable version of the Eklof–Trlifaj theorem (Theorem
4.3) in the abelian category $X{\operatorname{\mathsf{--qcoh}}}$. To show that
this cotorsion pair is hereditary, one deduces property (1) of Lemma 4.1 for
very flat quasi-coherent sheaves from the similar property of very flat
modules (as per Theorem 4.5(a)).

Alternatively, the results of [61, Lemma 4.1.1, Corollary 4.1.2, Lemma 4.1.3,
and Corollary 4.1.4] are available and provide more information than stated in
the theorem. This argument uses completeness of the very flat cotorsion pair
in the categories of modules over commutative rings as a black box (see
Theorem 4.5(a)), similarly to the alternative proof of the previous Theorem
8.1. ∎

The additional information about the flat and very flat cotorsion pairs in
$X{\operatorname{\mathsf{--qcoh}}}$ that one gets from the proofs in [61,
Section 4.1] is the _antilocality_ of the classes of contraadjusted and
cotorsion quasi-coherent sheaves (in the terminology of [77]).

###  8.3. The contraadjusted dimension is finite

The aim of this Section 8 is to present a contraherent/colocal proof of the
following theorem. Another proof can be found in [92, Corollary 9.4].

######  Theorem 8.3.

Let $X$ be a quasi-compact semi-separated scheme, and let
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic
complex of cotorsion quasi-coherent sheaves on $X$. Then the sheaves of
cocycles of the complex
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are also
cotorsion.

Im the present Section 8.3 we prove the following proposition.

######  Proposition 8.4.

Let $X$ be a quasi-compact semi-separated scheme, and let
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic
complex of contraadjusted quasi-coherent sheaves on $X$. Then the sheaves of
cocycles of the complex
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are also
contraadjusted.

The _contraadjusted dimension_ of a quasi-coherent sheaf $\mathcal{M}$ on the
scheme $X$ is defined as its coresolution dimension with respect to the
coresolving subcategory of contraadjusted quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$. (See the beginning of Section 7.2 for the
definition of the coresolution dimension.)

The following lemma can be found in [61, Lemma 4.6.1(a)].

######  Lemma 8.5.

Let $X$ be a quasi-compact semi-separated scheme, and let
$X=\bigcup_{\alpha=1}^{d}U_{\alpha}$ be a finite affine open covering of $X$.
Then the contraadjusted dimensions of quasi-coherent sheaves on $X$ do not
exceed $d$.

###### Proof.

By Theorem 8.2, the pair of full subcategories
$(X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{vfl}}$,
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})$ is a hereditary complete
cotorsion pair in $X{\operatorname{\mathsf{--qcoh}}}$. By Theorem 4.5(d), the
projective dimensions of very flat modules over commutative rings do not
exceed $1$. By [92, Theorem 5.3(b)], it follows that the projective dimensions
of very flat quasi-coherent sheaves on $X$ do not exceed $d$. Now it remains
to apply Lemma 7.15 to the cotorsion pair
$(X{\operatorname{\mathsf{--qcoh}}}_{\mathsf{vfl}}$,
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})$ in
$\mathsf{E}=X{\operatorname{\mathsf{--qcoh}}}$. ∎

###### Proof of Proposition 8.4.

The assertion follows immediately from Lemma 8.5. One has to use the fact that
the coresolution dimension does not depend on the choice of a coresolution;
see the discussion and references in the beginning of Section 7.2. ∎

###  8.4. Exact categories of contraherent cosheaves

We refer to Section 5.3 for the definition of the category of locally
contraadjusted contraherent cosheaves $X{\operatorname{\mathsf{--ctrh}}}$. The
definitions of the full subcategories of locally cotorsion contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and locally
injective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$,

$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\subset
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--ctrh}}},$

can be found in Section 5.4.

It was mentioned in Remark 5.4 and Section 5.4 that the categories
$X{\operatorname{\mathsf{--ctrh}}}$,
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$, and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ have natural exact
structures. Let us define these exact category structures now.

A short sequence of contraherent cosheaves
$0\longrightarrow\mathfrak{P}\longrightarrow\mathfrak{Q}\longrightarrow\mathfrak{R}\longrightarrow
0$ on $X$ is said to be admissible exact in
$X{\operatorname{\mathsf{--ctrh}}}$ if, for every affine open subscheme
$U\subset X$, the short sequence of (contraadjusted) $\mathcal{O}(U)$-modules
$0\longrightarrow\mathfrak{P}[U]\longrightarrow\mathfrak{Q}[U]\longrightarrow\mathfrak{R}[U]\longrightarrow
0$ is exact in $\mathcal{O}(U){\operatorname{\mathsf{--mod}}}$. The admissible
short exact sequences in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$
and $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ are defined similarly.

It is clear from the definition that the full subcategories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ are closed under extensions
and cokernels of admissible monomorphisms in
$X{\operatorname{\mathsf{--ctrh}}}$ (because the full subcategories of
cotorsion and injective $\mathcal{O}(U)$-modules are closed under extensions
and cokernels of monomorphisms in
$\mathcal{O}(U){\operatorname{\mathsf{--mod}}}$). A more nontrivial assertion
is that, on a quasi-compact semi-separated scheme $X$, the full subcategories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ are coresolving. In other
words, every contraherent cosheaf on $X$ is an admissible subobject of a
locally injective contraherent cosheaf. This is the result of [61, Lemma
4.2.2(a)]; see Theorem 8.6 below for a stronger assertion.

It follows that, on a quasi-compact semi-separated scheme $X$, the functors
$\operatorname{Ext}$ computed in the exact categories
$X{\operatorname{\mathsf{--ctrh}}}$,
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$, and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$ agree with each other. We
will denote this functor by
$\operatorname{Ext}^{X,*}({-},{-})=\operatorname{Ext}_{X{\operatorname{\mathsf{--ctrh}}}}^{*}({-},{-})$.

###  8.5. Antilocal contraherent cosheaves

A contraherent cosheaf $\mathfrak{P}$ on the scheme $X$ is called _antilocal_
(“colocally projective” in the terminology of [61, Section 4.2]) if
$\operatorname{Ext}^{X,1}(\mathfrak{P},\mathfrak{J})=0$ for every locally
injective contraherent cosheaf $\mathfrak{J}$ on $X$ (cf. [61, Corollary
4.2.3(a)]). See Section 5.2 for the terminological discussion.

We will denote the full subcategory of antilocal contraherent cosheaves by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}\subset
X{\operatorname{\mathsf{--ctrh}}}$. (The notation
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is used instead in the
preprint [61].) The full subcategory of antilocal locally cotorsion
contraherent cosheaves will be denoted by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{al}}=X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--ctrh}}}$.

It follows from the next theorem that
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$ is a resolving subcategory
in $X{\operatorname{\mathsf{--ctrh}}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{al}}$ is a
resolving subcategory in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.

######  Theorem 8.6.

For any quasi-compact semi-separated scheme $X$, the pair of classes of
objects (antilocal contraherent cosheaves, locally injective contraherent
cosheaves) is a hereditary complete cotorsion pair in the exact category
$X{\operatorname{\mathsf{--ctrh}}}$.

The cotorsion pair from Theorem 8.6 is called the _locally injective cotorsion
pair_ in the category of contraherent cosheaves.

###### Proof.

This is [61, Corollary 4.2.5(a–b)]. Part (c) of the same corollary provides
additional information, viz., a description of antilocal contraherent
cosheaves on $X$. Furthermore, [61, Corollary 4.2.6(c)] provides a description
of antilocal locally cotorsion contraherent cosheaves on $X$. ∎

For completeness of the exposition, let us also give the following definition.
A contraherent cosheaf $\mathfrak{F}$ on the scheme $X$ is called _antilocally
flat_ (“colocally flat” in the terminology of [61, Section 4.3]) if
$\operatorname{Ext}^{X,1}(\mathfrak{F},\mathfrak{Q})=0$ for every locally
cotorsion contraherent cosheaf $\mathfrak{Q}$ on $X$ (cf. [61, Corollary
4.3.2(a)]).

######  Theorem 8.7.

For any quasi-compact semi-separated scheme $X$, the pair of classes of
objects (antilocally flat contraherent cosheaves, locally cotorsion
contraherent cosheaves) is a hereditary complete cotorsion pair in the exact
category $X{\operatorname{\mathsf{--ctrh}}}$.

The cotorsion pair from Theorem 8.7 is called the _antilocally flat cotorsion
pair_ in the category of contraherent cosheaves.

###### Proof.

This is [61, Corollary 4.3.4(a–b)]. Part (c) of the same corollary provides
additional information, viz., a description of antilocally flat contraherent
cosheaves on $X$. ∎

The descriptions of the classes of antilocal (locally contraadjusted)
contraherent cosheaves, antilocal locally cotorsion contraherent cosheaves,
and antilocally flat contraherent cosheaves mentioned in the proofs of
Theorems 8.6–8.7 can be expressed by saying that these classes of contraherent
cosheaves are _antilocal_ (just as their names suggest) in the terminology of
[77].

###  8.6. Underived naïve co-contra correspondence

A _co-contra correspondence_ is an equivalence between the coderived category
of a comodule-like category and the contraderived category of a contramodule-
like category. A _naïve co-contra correspondence_ is an equivalence between
the conventional derived category of a comodule-like category and the
conventional derived category of a contramodule-like category. We refer to the
introductions to the papers [63, 67] for a detailed discussion of the
philosophy of co-contra correspondence.

In the context of quasi-coherent sheaves and contraherent cosheaves over a
quasi-compact semi-separated scheme $X$, the _underived naïve co-contra
correspondence_ is an equivalence of exact categories [61, Lemma 4.6.7]

(15) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}.$

Furthermore, it is important for us that the category equivalence (15)
restricts to an equivalence between the full subcategories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{al}}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$, providing an equivalence of
exact categories [61, proof of Corollary 4.6.8(a)]

(16) $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\simeq
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{al}}.$

The equivalence of categories (15) is given by the respective restrictions of
the partially adjoint functors

$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},{-})\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}$

and

$\mathcal{O}_{X}\odot_{X}{-}\,\colon X{\operatorname{\mathsf{--ctrh}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--qcoh}}}$

defined in [61, Sections 2.5 and 2.6]. It is instructive to look into the
constructions of the co-contra correspondence functors for corings over
associative rings [57, Sections 0.2.6 and 5.1], [62, Section 3.4], which
served as an inspiration for the more complicated constructions for sheaves
and cosheaves over schemes.

###  8.7. Details of the constructions

Let us briefly sketch the constructions of the functors
$\operatorname{\mathfrak{Hom}}_{X}$ and $\odot_{X}$ here.

The functor of contraherent $\operatorname{\mathfrak{Hom}}$ from a quasi-
coherent sheaf $\mathcal{M}$ to a quasi-coherent sheaf $\mathcal{P}$ on $X$ is
defined by the rule

$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})[U]=\operatorname{Hom}_{X}(j_{U}{}_{*}j_{U}^{*}\mathcal{M},\mathcal{P})$

for all affine open subschemes $U\subset X$. Here
$\operatorname{Hom}_{X}({-},{-})=\operatorname{Hom}_{X{\operatorname{\mathsf{--qcoh}}}}({-},{-})$
is the notation for the groups of morphisms in the category of quasi-coherent
sheaves $X{\operatorname{\mathsf{--qcoh}}}$, while $j_{U}\colon
U\longrightarrow X$ is the identity open immersion morphism. Given two
embedded affine open subschemes $V\subset U\subset X$, there is a natural
morphism $j_{U}{}_{*}j_{U}^{*}\mathcal{M}\longrightarrow
j_{V}{}_{*}j_{V}^{*}\mathcal{M}$ of quasi-coherent sheaves on $X$, inducing a
homomorphism of $\mathcal{O}(U)$-modules
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})[V]\longrightarrow\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})[U]$.

Similarly to the construction of the contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}(\mathcal{M},\mathfrak{P})$ in Section
5.5, the contraherence condition (i) from Section 5.3 holds for
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})$ for any pair of
quasi-coherent sheaves $\mathcal{M}$ and $\mathcal{P}$ on $X$. The
contraadjustedness condition (ii) for
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})$ holds under
certain assumptions on $\mathcal{M}$ and $\mathcal{P}$, as per [61, Section
2.5]. As a result, the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{P})$ is well-defined
in the following cases:

  1. (1)

if $\mathcal{F}$ is a very flat quasi-coherent sheaf and $\mathcal{P}$ is a
contraadjusted quasi-coherent sheaf on $X$, then
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{F},\mathcal{P})$ is a locally
contraadjusted contraherent cosheaf;

  2. (2)

if $\mathcal{F}$ is a flat quasi-coherent sheaf and $\mathcal{P}$ is a
cotorsion quasi-coherent sheaf, then
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{F},\mathcal{P})$ is a locally
cotorsion contraherent cosheaf;

  3. (3)

if $\mathcal{M}$ is an arbitrary quasi-coherent sheaf and $\mathcal{J}$ is an
injective quasi-coherent sheaf, then
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{J})$ is a locally
cotorsion contraherent cosheaf;

  4. (4)

if $\mathcal{F}$ is a flat quasi-coherent sheaf and $\mathcal{J}$ is an
injective quasi-coherent sheaf, then
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{F},\mathcal{J})$ is a locally
injective contraherent cosheaf.

So $\operatorname{\mathfrak{Hom}}_{X}\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{op}}\times
X{\operatorname{\mathsf{--qcoh}}}\dasharrow X{\operatorname{\mathsf{--ctrh}}}$
is a partially defined functor of two arguments (cf. the discussion of
partially defined functors taking values in exact categories in Section 5.6).

For the purposes of the naïve co-contra correspondence, we are interested in
cases (1–2) with the structure sheaf $\mathcal{F}=\mathcal{O}_{X}$. Then one
has

(17)
$\displaystyle\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},{-})\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ $\displaystyle\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}$ (18)
$\displaystyle\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},{-})\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ $\displaystyle\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}.$

Both these functors are exact by construction (because the quasi-coherent
sheaves $j_{U}{}_{*}j_{U}^{*}\mathcal{O}_{X}$ are very flat on $X$).

For any quasi-coherent sheaf $\mathcal{M}$ and contraherent cosheaf
$\mathfrak{P}$ on $X$, the _contratensor product_
$\mathcal{M}\odot_{\mathfrak{X}}\mathfrak{P}$ is constructed as the
(nondirected) colimit of the following diagram of quasi-coherent sheaves on
$X$, indexed by the poset of all affine open subschemes $U\subset X$ ordered
by inclusion. To every affine open subscheme $U\subset X$, the quasi-coherent
sheaf
$j_{U}{}_{*}j_{U}^{*}\mathcal{M}\otimes_{\mathcal{O}_{X}(U)}\mathfrak{P}[U]$
is assigned. Here the tensor product sign $\otimes_{\mathcal{O}_{X}(U)}$ means
the tensor product of an object $j_{U}{}_{*}j_{U}^{*}\mathcal{M}\in
X{\operatorname{\mathsf{--qcoh}}}$ endowed with an action of the ring
$\mathcal{O}_{X}(U)$ and a usual $\mathcal{O}_{X}(U)$-module
$\mathfrak{P}[U]$. For any pair of embedded affine open subschemes $V\subset
U\subset X$, there is a natural isomorphism
$j_{V}{}_{*}j_{V}^{*}\mathcal{M}\simeq
j_{U}{}_{*}j_{U}^{*}\mathcal{M}\otimes_{\mathcal{O}_{X}(U)}\mathcal{O}_{X}(V)$
of quasi-coherent sheaves on $X$, which allows to construct the morphisms in
the diagram [61, Section 2.6].

The adjunction isomorphism [61, formula (20) in Section 2.6]

$\operatorname{Hom}_{X}(\mathcal{M}\odot_{X}\mathfrak{P},\>\mathcal{J})\simeq\operatorname{Hom}^{X}(\mathfrak{P},\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{J}))$

holds for all quasi-coherent sheaves $\mathcal{M}$ and $\mathcal{J}$ and all
contraherent cosheaves $\mathfrak{P}$ on $X$ for which the contraherent
cosheaf $\operatorname{\mathfrak{Hom}}_{X}(\mathcal{M},\mathcal{J})$ is well-
defined (as per items (1–4) above). Here
$\operatorname{Hom}^{X}({-},{-})=\operatorname{Hom}_{X{\operatorname{\mathsf{--ctrh}}}}({-},{-})$
is the notation for the groups of morphisms in the category of contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}$. Taking
$\mathcal{M}=\mathcal{O}_{X}$ and making $\mathcal{J}$ range over the
injective quasi-coherent sheaves on $X$, one concludes from item (4) that the
restriction of the functor $\mathcal{O}_{X}\odot_{X}{-}$ to the full
subcategory of antilocal contraherent cosheaves is an exact functor

(19) $\mathcal{O}_{X}\odot_{X}{-}\,\colon
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--qcoh}}}$

[61, proof of Lemma 4.6.7].

###  8.8. Details of the arguments: antilocality

In order to finish the proofs of the exact category equivalences (15) and
(16), it still needs to be explained that the essential image of the functor
(17) is contained in the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}\subset
X{\operatorname{\mathsf{--ctrh}}}$, the essential image of the functor (19) is
contained in the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, and the functor (19) takes the full
subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{al}}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$ into the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
X{\operatorname{\mathsf{--qcoh}}}$. It also needs to be explained why the
resulting pair of adjoint functors between the categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$ is a category equivalence.

The most important property of the classes of objects involved on which these
proofs are based in the _antilocality_ , and the most important property of
the functors involved is the _compatibility with the direct limits_.

First of all, for any affine morphism of quasi-compact semi-separated schemes
$f\colon Y\longrightarrow X$, there is the exact functor of direct image of
quasi-coherent sheaves $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ taking
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ into
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ and
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ into
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ [61, Section 2.5]. Dual-
analogously, there is the exact functor of direct image of contraherent
cosheaves $f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ [61, Sections 2.3 and 3.3] taking
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ into
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$ into
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$ [61, Section 4.2].

The direct image functors $f_{*}$ and $f_{*}$ form a commutative square
diagram with the functors
$\operatorname{\mathfrak{Hom}}_{Y}(\mathcal{O}_{Y},{-})$ and
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},{-})$ [61, formula (44) in
Section 3.8], as well as with the functors $\mathcal{O}_{Y}\odot_{Y}{-}$ and
$\mathcal{O}_{X}\odot_{X}{-}$ [61, formula (46) in Section 3.8].

The antilocality property of classes of quasi-coherent sheaves and
contraherent cosheaves applies in the following setting. Let
$X=\bigcup_{\alpha=1}^{d}U_{\alpha}$ be a finite affine open covering of a
quasi-compact semi-separated scheme $X$. Then the contraadjusted quasi-
coherent sheaves on $X$ are precisely the direct summands of finitely iterated
extensions of the direct images of contraadjusted quasi-coherent sheaves with
respect to the open immersions $U_{\alpha}\longrightarrow X$ [61, Corollary
4.1.4(c)]. Similarly, the cotorsion quasi-coherent sheaves on $X$ are
precisely the direct summands of finitely iterated extensions of the direct
images of cotorsion quasi-coherent sheaves from the open subschemes
$U_{\alpha}\subset X$ [61, Corollary 4.1.11(c)].

Furthermore, the antilocal (locally contraadjusted) contraherent cosheaves on
$X$ are precisely the direct summands of finitely iterated extensions of the
direct images of contraherent cosheaves from $U_{\alpha}$ [61, Corollary
4.2.5(c)]. Similarly, the antilocal locally cotorsion contraherent cosheaves
on $X$ are precisely the direct summands of finitely iterated extensions of
the direct images of locally cotorsion contraherent cosheaves from
$U_{\alpha}$ [61, Corollary 4.2.6(c)].

The proofs of the exact category equivalences (15) and (16) are based on these
observations. See [61, Lemma 4.6.7 and proof of Corollary 4.6.8(a)] for
further details.

###  8.9. Proof of quasi-coherent cotorsion periodicity

Now we can present our proof of the quasi-coherent cotorsion periodicity
theorem.

###### Proof of Theorem 8.3.

We are given an acyclic complex of cotorsion quasi-coherent sheaves
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on a quasi-
compact semi-separated scheme $X$. Obviously, any cotorsion quasi-coherent
sheaf is contraadjusted. By Proposition 8.4, the sheaves of cocycles of the
complex $\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are, at
least, contraadjusted.

Now the equivalence of exact categories (15) implies that the complex of
antilocal contraherent cosheaves
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic in the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$. Furthermore, by formula
(16), the antilocal contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{n})$ is
locally cotorsion for every $n\in\mathbb{Z}$.

For every affine open subscheme $U\subset X$, we have a complex of
$\mathcal{O}(U)$-modules
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})[U]$.
By the definition of the exact structure on
$X{\operatorname{\mathsf{--ctrh}}}$, which is inherited by the resolving
subcategory $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}\subset
X{\operatorname{\mathsf{--ctrh}}}$, the complex of $\mathcal{O}(U)$-modules
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})[U]$
is acyclic.

By the cotorsion periodicity theorem for modules over rings [2, Theorem
5.1(2)], any acyclic complex of cotorsion $\mathcal{O}(U)$-modules has
cotorsion $\mathcal{O}(U)$-modules of cocycles; so the
$\mathcal{O}(U)$-modules of cocycles of the complex
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})[U]$
are cotorsion. Now it is clear that the complex
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})[U]$
is acyclic in the exact category of cotorsion $\mathcal{O}(U)$-modules
$\mathcal{O}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.

As this holds for every affine open subscheme $U\subset X$, we can conclude
that the complex of contraherent cosheaves
$\operatorname{\mathfrak{Hom}}_{X}(\mathcal{O}_{X},\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic not only in the exact category of antilocal locally contraadjusted
contraherent cosheaves $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}$, but
also in the exact category of antilocal locally cotorsion contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{al}}^{\mathsf{lct}}$.
Hence the formula (16) tells that the complex
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic in the
exact category of cotorsion quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$. The latter assertion means
that the quasi-coherent sheaves of cocycles of the complex
$\mathcal{C}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are cotorsion, as
desired. ∎

###  8.10. Conclusion

One can say, and indeed we argued in Sections 1–3, that contraherent cosheaves
are an important device for globalizing contramodules and contraderived
categories. Then one needs contraadjusted and/or cotorsion modules over
commutative rings in order to define the contraherent cosheaves and work with
them.

Alternatively, one may be interested in cotorsion modules for some other,
unrelated reasons. Then one may run into the problem that the class of
cotorsion modules is not preserved by the localization functors in commutative
algebra, and the class of cotorsion quasi-coherent sheaves on schemes is not
local.

If one still wants to develop a local point of view on cotorsion modules, one
needs to use the colocalization functors instead of the localizations. In this
context, the contraherent cosheaves are indispensable.

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