$\mathfrak{R}$-contramodule $\mathfrak{F}$ is $\kappa$-presentable as an
object of $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ (or of
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$) if and only
if the flat $\mathfrak{R}/\mathfrak{I}_{n}$-module
$F_{n}=\mathfrak{F}/(\mathfrak{I}_{n}\rightthreetimes\mathfrak{F})$ is
$\kappa$-presentable for every $n\geq 1$.

###### Proof.

The first “only if” assertion follows easily from the “only if” assertion of
Proposition 9.3. To prove the first “if”, use the description of the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ provided by
Lemma 8.6(a) together with the fact that $\aleph_{1}$-directed colimits
commute with countable limits in the category of abelian groups. Then the
remaining assertions (concerning flat contramodules) follow from Lemma 9.4. ∎

######  Lemma 9.6.

Let $\mathfrak{R}$ be a complete, separated topological ring, and let
$\mathfrak{R}\supset\mathfrak{I}_{1}\supset\mathfrak{I}_{2}\supset\mathfrak{I}_{3}\supset\dotsb$
be a countable base of neighborhoods of zero in $\mathfrak{R}$ consisting of
open _two-sided_ ideals $\mathfrak{I}_{n}\subset\mathfrak{R}$, $n\geq 1$. Then
the projective dimension of a flat $\mathfrak{R}$-contramodule $\mathfrak{F}$
(as an object of $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$) is equal to
the supremum of the projective dimensions of the flat
$\mathfrak{R}/\mathfrak{I}_{n}$-modules
$F_{n}=\mathfrak{F}/(\mathfrak{I}_{n}\rightthreetimes\mathfrak{F})$ (as
objects of $\mathfrak{R}/\mathfrak{I}_{n}{\operatorname{\mathsf{--Mod}}}$),
$n\geq 1$.

###### Proof.

Let
$\mathfrak{P}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\mathfrak{F}$
be a projective resolution of $\mathfrak{F}$ in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ and $\mathfrak{F}_{i}$ be the
image of the differential $\mathfrak{P}_{i}\longrightarrow\mathfrak{P}_{i-1}$
(so $\mathfrak{F}=\mathfrak{F}_{0}$). Applying Lemma 8.4(a) iteratively and
keeping in mind that projective contramodules are flat, one shows that the
$\mathfrak{R}$-contramodules $\mathfrak{F}_{i}$ are flat for all $i\geq 0$.
Now Lemma 8.4(b) tells that the functors
$\mathfrak{Q}\longmapsto(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{Q}=\mathfrak{Q}/(\mathfrak{I}_{n}\rightthreetimes\mathfrak{Q})$
preserve exactness of the short exact sequences
$0\longrightarrow\mathfrak{F}_{i}\longrightarrow\mathfrak{P}_{i}\longrightarrow\mathfrak{F}_{i-1}\longrightarrow
0$. Therefore,
$(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{P}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a projective resolution of the $\mathfrak{R}/\mathfrak{I}_{n}$-module
$F_{n}=(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{F}$, and
the $\mathfrak{R}/\mathfrak{I}_{n}$-module
$(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{F}_{i}$ is the
image of the differential
$(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{P}_{i}\longrightarrow(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{P}_{i-1}$.
It remains to use the characterization of projective
$\mathfrak{R}$-contramodules provided by Lemma 8.7 in order to deduce the
desired assertion. ∎

######  Corollary 9.7.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then any countably presentable flat $\mathfrak{R}$-contramodule has
projective dimension at most $1$ in the abelian category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

###### Proof.

Recall that all countably presentable flat modules (over a discrete ring $R$)
have projective dimension at most $1$ in $R{\operatorname{\mathsf{--Mod}}}$,
by Lemma 1.3. Then it remains to compare Lemma 9.5 (for flat contramodules and
$\kappa=\aleph_{1}$) with Lemma 9.6. ∎

######  Question 9.8.

One would expect the assertion of Corollary 9.7 to hold for left contramodules
over any complete, separated topological ring $\mathfrak{R}$ with a countable
base of neighborhoods of zero consisting of open _right_ ideals. But we do not
know whether this is true. In fact, we don’t even know the answer to Question
11.2 below, which would seem to be easier.

##  10\. Flat Contramodules as Directed Colimits

The following theorem describing flat $\mathfrak{R}$-contramodules is the
second main result of this paper. It should be compared with the discussion of
(the failure of) the conventional finite, rather than countable Govorov–Lazard
theorem for flat contramodules (in the absense of a countable base of
neighborhoods of zero in the topological ring) in the paper [39, Section 10
and Example 13.4]. Theorem 10.1 is also a contramodule analogue of [44,
Theorems 2.4 and 3.5] and Theorem 3.1 above.

We refer to Lemmas 9.4 and 9.5 for an explication of what is meant by a
countably presentable flat $\mathfrak{R}$-contramodule in Theorem 10.1 (i. e.,
countably presentable in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ or
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$, as described
in Propositions 9.1(a) and 9.3).

######  Theorem 10.1.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ of flat left
$\mathfrak{R}$-contramodules is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ are precisely
all the countably presentable flat $\mathfrak{R}$-contramodules. Consequently,
every flat $\mathfrak{R}$-contramodule is an $\aleph_{1}$-directed colimit of
countably presentable flat $\mathfrak{R}$-contramodules.

###### Proof.

The full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ is closed
under directed colimits in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.
Therefore, all countably presentable flat $\mathfrak{R}$-contramodules are
$\aleph_{1}$-presentable in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$.

The full assertion of the theorem is obtained by applying Theorem 1.6 for
$\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$. The argument is based on the
description of the category of flat $\mathfrak{R}$-contramodules provided by
Lemma 8.6(b) and the description of countably presentable separated/flat
$\mathfrak{R}$-countramodules provided by Lemma 9.5.

Let
$\mathfrak{R}\supset\mathfrak{I}_{1}\supset\mathfrak{I}_{2}\supset\mathfrak{I}_{3}\supset\dotsb$
be a countable base of neighborhoods of zero in $\mathfrak{R}$ consisting of
open two-sided ideals. Let $\mathsf{K}=\mathsf{L}$ be the Cartesian product of
the categories of flat left modules over the rings
$\mathfrak{R}/\mathfrak{I}_{n}$, taken over the integers $n\geq 1$; so
$\mathsf{K}=\mathsf{L}=\prod_{n=1}^{\infty}(\mathfrak{R}/\mathfrak{I}_{n}{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}})$.
Consider the following pair of parallel functors $F_{1}$,
$F_{2}\colon\mathsf{K}\rightrightarrows\mathsf{L}$. The functor $F_{1}$ takes
a collection of flat modules
$(K_{n}\in\mathfrak{R}/\mathfrak{I}_{n}{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}})_{n\geq
1}$ to the collection of flat modules
$L_{n}=\mathfrak{R}/\mathfrak{I}_{n}\otimes_{\mathfrak{R}/\mathfrak{I}_{n+1}}K_{n+1}$.
The functor $F_{2}$ is the identity functor.

Then Lemma 8.6(b) tells that the isomorpher category $\mathsf{C}$ is
equivalent to the category of flat left $\mathfrak{R}$-contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$. The
categories
$\mathfrak{R}/\mathfrak{I}_{n}{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$
are $\aleph_{1}$-accessible by Lemma 1.2, and their Cartesian product
$\prod_{n=1}^{\infty}(\mathfrak{R}/\mathfrak{I}_{n}{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}})$
is $\aleph_{1}$-accessible by Proposition 1.4. Theorem 1.6 is applicable, and
it tells that the category $\mathsf{C}$ is $\aleph_{1}$-accessible. Comparing
the description of the $\aleph_{1}$-presentable objects of $\mathsf{C}$
provided by Theorem 1.6 with the description of the $\aleph_{1}$-presentable
flat $\mathfrak{R}$-contramodules provided by Lemma 9.5 one can see that these
two classes of objects coincide, as desired. ∎

Let us also provide a description of arbitrary complexes of flat
$\mathfrak{R}$-contramodules as directed colimits.

######  Proposition 10.2.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the category
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}})$
of complexes of flat left $\mathfrak{R}$-contramodules is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}})$
are precisely all the complexes of countably presentable flat
$\mathfrak{R}$-contramodules. Consequently, every complex of flat
$\mathfrak{R}$-contramodules is an $\aleph_{1}$-directed colimit of complexes
of countably presentable flat $\mathfrak{R}$-contramodules.

###### Proof.

All complexes of countably presentable $\mathfrak{R}$-contramodules are
$\aleph_{1}$-presentable as objects of
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}})$ by Proposition
9.1(b). Since the full subcategory
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}})$
is closed under directed colimits in
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}})$, it follows
that all complexes of countably presentable flat $\mathfrak{R}$-contramodules
are $\aleph_{1}$-presentable in
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}})$.

The full assertion of the proposition is obtained by combining the results of
Theorems 10.1 and 1.9. The argument is similar to the proof of Proposition
3.3. ∎

######  Remark 10.3.

One can restrict the results of this section to commutative topological rings
$\mathfrak{R}$, interpret commutative topological rings as ind-affine ind-
schemes (as per Example 8.9, and then generalize to non-ind-affine ind-
schemes. To avoid a detailed discussion of the basics of the ind-scheme theory
(see [34, Section 1.2] and the references therein), let us say that a _strict
ind-quasi-compact ind-quasi-separated $\aleph_{0}$-ind-scheme_ $\mathfrak{X}$
can be defined as an ind-object in the category of quasi-compact quasi-
separated schemes representable by a countable directed diagram of closed
immersions of quasi-compact quasi-separated schemes $X_{1}\longrightarrow
X_{2}\longrightarrow X_{3}\longrightarrow\dotsb$.

An “$\mathcal{O}^{p}$-module on $\mathfrak{X}$” (in the terminology of [3,
Section 7.11.3]) or a _pro-quasi-coherent pro-sheaf on $\mathfrak{X}$_ in our
preferred terminology of [34, Section 3.1] can be then defined as a sequence
of quasi-coherent sheaves $M_{n}$ of the schemes $X_{n}$ together with
isomorphisms of quasi-coherent sheaves $M_{n}\simeq i_{n}^{*}M^{n+1}$ on
$X_{n}$, where $i_{n}\colon X_{n}\longrightarrow X_{n+1}$ denote the closed
immersion morphisms ($n\geq 1$). A pro-quasi-coherent pro-sheaf
$(F_{n})_{n\geq 1}$ is said to be _flat_ if the quasi-coherent sheaf $F_{n}$
on $X_{n}$ is flat for every $n\geq 1$. The flat pro-quasi-coherent pro-
sheaves form a well-behaved (exact) full subcategory in a badly behaved
additive category of arbitrary pro-quasi-coherent pro-sheaves on
$\mathfrak{X}$ (see Example 8.9 and Remark 8.5).

The argument similar to the proof of Theorem 10.1 and based on the result of
[44, Theorem 2.4] shows that the category of flat pro-quasi-coherent pro-
sheaves on $\mathfrak{X}$ is $\aleph_{1}$-accessible, and the flat pro-quasi-
coherent pro-sheaves $(F_{n})_{n\geq 1}$ with locally countably presentable
flat quasi-coherent sheaves $F_{n}$ on $X_{n}$ are the
$\aleph_{1}$-presentable objects of this category. The flat pro-quasi-coherent
pro-sheaf version of Proposition 10.2 also holds, with the same proof. The
same applies to the results of Proposition 11.3 and Corollary 11.4 below.

##  11\. Exact Sequences of Flat Contramodules as Directed Colimits

Let us start with a contramodule analogue/generalization of Lemma 4.1.

######  Lemma 11.1.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the kernel of any surjective morphism from a countably
presentable flat $\mathfrak{R}$-contramodule to a countably presentable flat
$\mathfrak{R}$-contramodule is a countably presentable (flat)
$\mathfrak{R}$-contramodule.

###### Proof.

The class of flat $\mathfrak{R}$-contramodules is closed under kernels of
surjective morphisms in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ by
Lemma 8.4(a). Now if
$0\longrightarrow\mathfrak{H}\longrightarrow\mathfrak{G}\longrightarrow\mathfrak{F}\longrightarrow
0$ is a short exact sequence of flat left $\mathfrak{R}$-contramodules, then
Lemma 8.4(b) tells that
$0\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{H}\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{G}\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}\longrightarrow
0$ is a short exact sequence of left $\mathfrak{R}/\mathfrak{I}$-modules for
any open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$. If the
$\mathfrak{R}$-contramodules $\mathfrak{F}$ and $\mathfrak{G}$ are countably
presentable, then so are the $\mathfrak{R}/\mathfrak{I}$-modules
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}$ and
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{G}$, by Lemma 9.5.
So Lemma 4.1 tells that the $\mathfrak{R}/\mathfrak{I}$-module
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{H}$ is countably
presentable, and it remains to apply Lemma 9.5 again in order to conclude that
the $\mathfrak{R}$-contramodule $\mathfrak{H}$ is countably presentable.

Alternatively, one can deduce the lemma from Corollary 9.7 essentially in the
same way as the proof of Lemma 4.1 deduces it from Lemma 1.3. Then one needs
to use the facts that the kernel of any surjective morphism of countably
presentable $\mathfrak{R}$-contramodules is a countably generated
$\mathfrak{R}$-contramodule, and any countably generated projective
$\mathfrak{R}$-contramodule is countably presentable. ∎

######  Question 11.2.

Similarly to Question 9.8, we do _not_ know, and it would be interesting to
learn, whether the assertion of Lemma 11.1 holds for left contramodules over
any complete, separated topological ring $\mathfrak{R}$ with a countable base
of neighborhoods of zero consising of open _right_ ideals.

The following proposition describes short exact sequences of flat
$\mathfrak{R}$-contramodules.

######  Proposition 11.3.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the category of short exact sequences of flat
$\mathfrak{R}$-contramodules is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of this category are precisely all the short
exact sequences of countably presentable flat $\mathfrak{R}$-contramodules.
Consequently, every short exact sequence of flat $\mathfrak{R}$-contramodules
is an $\aleph_{1}$-directed colimit of short exact sequences of countably
presentable flat $\mathfrak{R}$-contramodules.

###### Proof.

The argument is similar to the proof of Theorem 10.1 and uses Lemmas 4.3 and
8.4(b) together with Lemmas 8.6(b) and 9.5. The assertion is obtained by
applying Proposition 1.4 and Theorem 1.6 for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$.

Let
$\mathfrak{R}\supset\mathfrak{I}_{1}\supset\mathfrak{I}_{2}\supset\mathfrak{I}_{3}\supset\dotsb$
be a countable base of neighborhoods of zero in $\mathfrak{R}$ consisting of
open two-sided ideals. Let $\mathsf{K}=\mathsf{L}$ be the Cartesian product of
the categories of short exact sequences of flat left modules over the rings
$\mathfrak{R}/\mathfrak{I}_{n}$, taken over the integers $n\geq 1$. Consider
the pair of functors $F_{1}$,
$F_{2}\colon\mathsf{K}\rightrightarrows\mathsf{L}$ similar to the one in the
proof of Theorem 10.1.

Then Lemmas 8.4(b) and 8.6(b) imply that the isomorpher category $\mathsf{C}$
is equivalent to the category of short exact sequences of flat left
$\mathfrak{R}$-contramodules (one needs to use also the fact that directed
limits of countable sequences of surjective maps of short exact sequences of
abelian groups are short exact sequences of abelian groups again). The
categories $\mathsf{K}$ and $\mathsf{L}$ are $\aleph_{1}$-accessible by Lemma
4.3 and Proposition 1.4. Theorem 1.6 is applicable; it tells that the category
$\mathsf{C}$ is $\aleph_{1}$-accessible and provides a description of its full
subcategory of $\aleph_{1}$-presentable objects. To compare it with the
description asserted in the proposition, one needs to use Lemma 9.5. ∎

The assertion of Proposition 11.3 (together with Theorem 10.1) can be restated
by saying that the exact category of flat $\mathfrak{R}$-contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ is a _locally
$\aleph_{1}$-coherent exact category_ in the sense of [38, Section 1].

Let us say that an acyclic complex of flat $\mathfrak{R}$-contramodules is
_pure acyclic_ if its $\mathfrak{R}$-contramodules of cocycles are flat. For
topological rings $\mathfrak{R}$ with a countable base of neighborhoods of
zero, this is consistent with the notion of _contratensor purity_ [32, Section
3], [39, Section 13]. It is worth noticing that any bounded above acyclic
complex of flat contramodules is pure acyclic by Lemma 8.4(a).

######  Corollary 11.4.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the category of pure acyclic complexes of flat
$\mathfrak{R}$-contramodules is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of this category are precisely all the pure
acyclic complexes of countably presentable flat $\mathfrak{R}$-contramodules.
Consequently, every pure acyclic complex of flat $\mathfrak{R}$-contramodules
is an $\aleph_{1}$-directed colimit of pure acyclic complexes of countably
presentable flat $\mathfrak{R}$-contramodules.

###### Proof.

Similar to the proof of Corollary 4.5. One can either prove the assertion of
the corollary by a direct argument similar to the proof of Proposition 11.3
and using [37, Corollary 7.13] instead of Lemma 4.3, or deduce the corollary
from Proposition 11.3 using the argument with building pure acyclic complexes
by splicing short exact sequences as in the proof of Corollary 4.5. ∎

##  12\. Cotorsion Periodicity for Contramodules

For the sake of completeness of the exposition, we start with presenting a
weak version of flat/projective periodicity for contramodules. The definition
of a _periodic object_ (with respect to a class of objects in an abelian
category) was given in Section 5. The following proposition is a contramodule
generalization of [4, Theorem 2.5] and [25, Remark 2.15] (see also [8,
Proposition 7.6]).

######  Proposition 12.1.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then

(a) any projective-periodic flat $\mathfrak{R}$-contramodule is projective;

(b) in any acyclic complex of projective $\mathfrak{R}$-contramodules with
flat $\mathfrak{R}$-contramodules of cocycles, the contramodules of cocycles
are actually projective (so the complex is contractible).

###### Proof.

This is an easy corollary of the the flat/projective periodicity for modules
over a ring ([4, Theorem 2.5], [25, Remark 2.15]) together with the basic
properties of flat and projective contramodules (Lemmas 8.4(b) and 8.7).

Part (a): let
$0\longrightarrow\mathfrak{F}\longrightarrow\mathfrak{P}\longrightarrow\mathfrak{F}\longrightarrow
0$ be a short exact sequence of flat left $\mathfrak{R}$-contramodules with
projective $\mathfrak{R}$-contramodule $\mathfrak{P}$. By Lemma 8.4(b), for
any open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$, the short sequence
of left $\mathfrak{R}/\mathfrak{I}$-modules
$0\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}\longrightarrow(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}\longrightarrow
0$ is exact. Clearly, the $\mathfrak{R}/\mathfrak{I}$-module
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}$ is flat and the
$\mathfrak{R}/\mathfrak{I}$-module
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}$ is projective
(see the discussion in Sections 7–8, cf. Lemmas 8.6(b) and 8.7). By [4,
Theorem 2.5], it follows that the $\mathfrak{R}/\mathfrak{I}$-module
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{F}$ is projective
for every open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$. Using Lemma
8.7, we conclude that the $\mathfrak{R}$-contramodule $\mathfrak{F}$ is
projective.

Part (b): let $\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be
a pure acyclic complex of projective left $\mathfrak{R}$-contramodules. By
Lemma 8.4(b), for any open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$,
the complex of left $\mathfrak{R}/\mathfrak{I}$-modules
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is (pure) acyclic. Moreover, if $\mathfrak{Z}^{n}$, $n\in\mathbb{Z}$, are the
$\mathfrak{R}$-contramodules of cocycles of the complex
$\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, then
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{Z}^{n}$ are the
$\mathfrak{R}/\mathfrak{I}$-modules of cocycles of the complex
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
The complex
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is also a complex of projective left $\mathfrak{R}/\mathfrak{I}$-modules. By
[25, Theorem 8.6 and Remark 2.15] or [8, Proposition 7.6], it follows that the
complex
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible and the $\mathfrak{R}/\mathfrak{I}$-modules
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{Z}^{n}$ are
projective. Once again, we use Lemma 8.7 in order to conclude that the
$\mathfrak{R}$-contramodules $\mathfrak{Z}^{n}$ are projective and the complex
$\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is contractible.
∎

Now we pass to the cotorsion periodicity for $\mathfrak{R}$-contramodules,
which is the main topic of this section. Let us denote by
$\operatorname{Ext}^{\mathfrak{R},*}({-},{-})$ the Ext groups in the abelian
category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. A left
$\mathfrak{R}$-contramodule $\mathfrak{B}$ is said to be _cotorsion_ [28,
Section D.5], [40, Definition 7.3] if
$\operatorname{Ext}^{\mathfrak{R},1}(\mathfrak{F},\mathfrak{B})=0$ for all
flat left $\mathfrak{R}$-contramodules $\mathfrak{F}$.

Introduce the notation
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}}$ for
the full subcategory of cotorsion $\mathfrak{R}$-contramodules in the abelian
category $\mathsf{C}=\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.
Obviously, the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}}$ is closed
under extensions in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$, so it
inherits an exact category structure from the abelian exact structure of
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

Similarly, the full subcategory of flat $\mathfrak{R}$-contramodules
$\mathsf{F}=\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ is
closed under extensions in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
(assuming a countable base of neighborhoods of zero in $\mathfrak{R}$) by
Lemma 8.4(a). So the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ also inherits
an exact category structure from
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

The following corollary summarizes some of our previous results and
observations. It is a contramodule version of [44, Corollary 5.6 and Lemma
9.1] and Corollary 5.2 above.

######  Corollary 12.2.

Let $\mathfrak{R}$ be a complete, separated topological ring.

(a) Assume that $\mathfrak{R}$ has a countable base of neighborhoods of zero
consisting of open right ideals. Then the class all flat left
$\mathfrak{R}$-contramodules is resolving and closed under directed colimits
in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. Consequently, one has
$\operatorname{Ext}^{\mathfrak{R},n}(\mathfrak{F},\mathfrak{B})=0$ for all
flat left $\mathfrak{R}$-contramodules $\mathfrak{F}$, all cotorsion left
$\mathfrak{R}$-contramodules $\mathfrak{B}$, and all integers $n\geq 1$. The
directed colimit functors in the exact category of flat contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ are exact (i.
e., the directed colimits of admissible short exact sequences are are
admissible short exact sequences).

(b) Assume that $\mathfrak{R}$ has a countable base of neighborhoods of zero
consisting of open two-sided ideals. Then any flat $\mathfrak{R}$-contramodule
is a directed (in fact, $\aleph_{1}$-directed) colimit of flat
$\mathfrak{R}$-contramodules having projective dimensions not exceeding $1$ in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

###### Proof.

Part (a): the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}\subset\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
is resolving by Lemma 8.4(a) and closed under directed colimits according to
the discussion in Sections 7–8. The Ext vanishing assertion follows by [44,
Lemma 6.1]. The directed colimits in the exact category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ (though not
in the abelian category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ !)
are exact by Lemma 8.8.

In part (b), all flat $\mathfrak{R}$-contramodules are $\aleph_{1}$-directed
colimits of countably presentable flat $\mathfrak{R}$-contramodules by Theorem
10.1, and countably presentable flat $\mathfrak{R}$-contramodules have
projective dimension at most $1$ by Corollary 9.7. ∎

The following result is a contramodule generalization of the cotorsion
periodicity theorem of Bazzoni, Cortés-Izurdiaga, and Estrada [2, Theorem
1.2(2) or Proposition 4.8(2)]. It is also a contramodule analogue of [44,
Theorem 9.2] and Theorem 5.4 above.

######  Theorem 12.3.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then any cotorsion-periodic $\mathfrak{R}$-contramodule is cotorsion.

###### Proof.

This is another application of Theorem 5.3, whose assumptions are satisfied in
the situation at hand by Corollary 12.2. ∎

######  Corollary 12.4.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Let $\mathfrak{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be
an acyclic complex in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ whose
terms $\mathfrak{B}^{n}$ are cotorsion $\mathfrak{R}$-contramodules for all
$n\in\mathbb{Z}$. Then the $\mathfrak{R}$-contramodules of cocycles of the
complex $\mathfrak{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are also
cotorsion.

###### Proof.

This is a contramodule generalization of [2, Theorem 5.1(2)], and an analogue
of [44, Corollary 9.4] and Corollary 5.5 above. The infinite product functors
are exact in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$, and the class
of cotorsion $\mathfrak{R}$-contramodules is closed under (direct summands
and) infinite products. So the argument of [14, Proposition 2] is applicable,
deducing the assertion of the corollary from Theorem 12.3. ∎

The definition of a _coresolving subcategory_ is dual to that of a resolving
one. A class of objects $\mathsf{B}$ in an abelian category $\mathsf{C}$ is
said to be _cogenerating_ if every object of $\mathsf{C}$ is a subobject of an
object from $\mathsf{B}$. A class of objects $\mathsf{B}\subset\mathsf{C}$ is
said to be _coresolving_ if it is cogenerating, closed under extensions, and
closed under cokernels of monomorphisms in $\mathsf{C}$.

######  Proposition 12.5.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then the class
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}}$ of
all cotorsion left $\mathfrak{R}$-contramodules is coresolving in
$\mathsf{C}=\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

###### Proof.

It is obvious from the definition that the class of all cotorsion
$\mathfrak{R}$-contramodules is closed under extensions, and it follows from
the higher Ext vanishing assertion of Corollary 12.2(a) that
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}}$ is also closed
under cokernels of monomorphisms in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. The most nontrivial
assertion in the proposition is that the class of cotorsion
$\mathfrak{R}$-contramodules is cogenerating. This is a part of [40, Corollary
7.8]. (Notice that there are usually _not_ enough injective objects in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.) ∎

Recall the notation $\mathsf{K}(\mathsf{E})$ for the homotopy category of an
additive category $\mathsf{E}$. For an abelian (or exact) category
$\mathsf{E}$, we denote by
$\mathsf{Ac}(\mathsf{E})\subset\mathsf{K}(\mathsf{E})$ the full subcategory of
acyclic complexes. So the derived category $\mathsf{D}(\mathsf{E})$ is the
Verdier quotient category
$\mathsf{D}(\mathsf{E})=\mathsf{K}(\mathsf{E})/\mathsf{Ac}(\mathsf{E})$.

######  Proposition 12.6.

Let $\mathsf{C}$ be an abelian category with exact functors of countable
product, and let $\mathsf{B}\subset\mathsf{C}$ be a cogenerating subcategory
closed under countable products. Then for any complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in $\mathsf{C}$ there
exists a complex $B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
$\mathsf{B}$ together with a quasi-isomorphism
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of complexes in
$\mathsf{C}$. Consequently, the inclusion of additive categories
$\mathsf{B}\longrightarrow\mathsf{C}$ induces a triangulated equivalence of
Verdier quotient categories

$\frac{\mathsf{K}(\mathsf{B})}{\mathsf{K}(\mathsf{B})\cap\mathsf{Ac}(\mathsf{C})}\overset{\simeq}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\frac{\mathsf{K}(\mathsf{C})}{\mathsf{Ac}(\mathsf{C})}\,=\,\mathsf{D}(\mathsf{C}).$

###### Proof.

The proof of the first assertion is similar to the construction of
$K$-projective and $K$-injective resolutions of complexes of modules in [47,
Section 3]. Let us spell out the construction as follows. For every
$n\in\mathbb{Z}$, consider the canonical truncation $\tau_{\geq
n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the complex
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Since the complex
$\tau_{\geq n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is bounded
below and the subcategory $\mathsf{B}\subset\mathsf{C}$ is cogenerating, one
can easily find a (bounded below) complex
${}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in $\mathsf{B}$
together with a quasi-isomorphism $\tau_{\geq
n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
in $\mathsf{C}$. Moreover, one can choose the complex
${}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ so that the
morphism of complexes $\tau_{\geq
n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a termwise monomorphism. The composition
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\tau_{\geq
n}C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
provides a morphism of complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.

Now the induced morphism into the product
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}=\prod_{n\in\mathbb{Z}}{}^{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a termwise monomorphism of complexes in $\mathsf{C}$ inducing monomorphisms
on all the cohomology objects. Furthermore,
$B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}$ is a complex in
$\mathsf{B}$. Next we apply the same construction to the cokernel of the
morphism of complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}$, embedding the complex
$B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}/C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
into a complex $B^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}}$ in
$\mathsf{B}$ in such a way that the morphism of complexes
$B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}/C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}}$ is (not only termwise
monic, but also) induces monomorphisms on all the cohomology objects.
Iterating the procedure, we obtain an exact complex of complexes

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{\text{\smaller\smaller$\scriptstyle\bullet$}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muB^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muB^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muB^{2,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

such that passing to the cohomology objects produces an exact complex

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(C^{\text{\smaller\smaller$\scriptstyle\bullet$}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(B^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(B^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(B^{2,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

of graded objects in $\mathsf{C}$.

Finally, we totalize the bicomplex
$B^{{\text{\smaller\smaller$\scriptstyle\bullet$}},{\text{\smaller\smaller$\scriptstyle\bullet$}}}$
by taking infinite products along the diagonals. Put
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Tot}^{\sqcap}(B^{{\text{\smaller\smaller$\scriptstyle\bullet$}},{\text{\smaller\smaller$\scriptstyle\bullet$}}})$
to be the product totalization. Then
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex in
$\mathsf{B}$. Furthermore, the resulting morphism of complexes
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a quasi-isomorphism by
the next Lemma 12.7. This proves the first assertion of the proposition, and
the second one then follows by [44, Lemma 9.5]. An opposite version of this
argument can be found in [28, proof of Proposition A.4.3]. ∎

The next lemma is a classical result of Eilenberg and Moore [12].

######  Lemma 12.7.

Let $\mathsf{C}$ be an abelian category with exact functors of countable
product, and let

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{2,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{3,{\text{\smaller\smaller$\scriptstyle\bullet$}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be a bounded below complex of complexes in $\mathsf{C}$. Assume that, passing
to the cohomology objects, we obtain an acyclic complex

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(C^{0,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(C^{1,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(C^{2,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muH^{*}(C^{3,{\text{\smaller\smaller$\scriptstyle\bullet$}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

of graded objects in $\mathsf{C}$. Then the product totalization
$\operatorname{Tot}^{\sqcap}(C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},{\text{\smaller\smaller$\scriptstyle\bullet$}}})$
of the bicomplex
$C^{{\text{\smaller\smaller$\scriptstyle\bullet$}},{\text{\smaller\smaller$\scriptstyle\bullet$}}}$
is an acyclic complex in $\mathsf{C}$.

###### Proof.

This is a particular case of [12, Theorem 7.4] or (essentially) the dual
version of [28, Lemma A.4.4]. ∎

The following corollary is a contramodule/ind-affine ind-scheme analogue of
[44, Corollary 9.7] and Corollary 5.6 above.

######  Corollary 12.8.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the inclusion of exact/abelian categories
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
induces an equivalence of their unbounded derived categories,

$\mathsf{D}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}^{\mathsf{cot}})\,\simeq\mathsf{D}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}}).$

###### Proof.

It suffices to compare Propositions 12.5 and 12.6 with Corollary 12.4. ∎

##  13\. Exact Sequences of Flat Contramodules II

In this short section we present contramodule/ind-affine ind-scheme analogues
of the results of Section 6. We refer to Section 6 for the discussion of
_self-resolving subcategories_ in abelian/exact categories.

The following lemma is a contramodule analogue of Corollary 6.4.

######  Lemma 13.1.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then

(a) the full subcategory $\mathsf{G}$ of all countably presentable flat
$\mathfrak{R}$-contramodules is self-resolving in the abelian category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$;

(b) the homological dimension of the exact category $\mathsf{G}$ (with the
exact structure inherited from
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$) does not exceed $1$.

###### Proof.

Part (a): the class of all flat $\mathfrak{R}$-contramodules is closed under
extensions in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ by Lemma
8.4(a). The assertion that the class of all countably presentable
$\mathfrak{R}$-contramodules is closed under extensions is provable in the
same way as the similar assertion for modules over a ring. Hence the class
$\mathsf{G}$ of all countably presentable flat $\mathfrak{R}$-contramodules is
closed under extensions in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.
The class $\mathsf{G}$ is also closed under kernels of epimorphisms in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ by Lemma 11.1. Finally, the
class $\mathsf{G}$ is self-generating in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$, since every object of
$\mathsf{G}$ is a quotient object in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ of an object from
$\mathsf{G}$ (namely, of the free $\mathfrak{R}$-contramodule with a countable
set of generators) that is projective in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

Part (b): the assertion follows directly from Corollary 9.7. One can also
invoke Lemma 6.3 here to the effect that it follows from part (a) that the Ext
groups computed in $\mathsf{G}$ agree with the ones computed in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. ∎

The following theorem is a contramodule/ind-affine ind-scheme version of [13,
Theorem 2.4 (1) $\Leftrightarrow$ (3)] or [25, Theorem 8.6 (ii)
$\Leftrightarrow$ (iii)], and an analogue of Theorem 6.5 above. It should be
also compared to [43, Corollary 0.4 or Proposition 8.11]. In the latter
context, Theorem 13.2 can be interpreted as saying that (over a topological
ring $\mathfrak{R}$ with a countable topology base of two-sided ideals) _for
the exact category of flat $\mathfrak{R}$-contramodules, the derived category
coincides with the_ (_suitably defined_) _coderived category_.

######  Theorem 13.2.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open _two-sided_
ideals. Then the following classes of complexes of left
$\mathfrak{R}$-contramodules coincide:

  1. (1)

all pure acyclic complexes of flat $\mathfrak{R}$-contramodules;

  2. (2)

the closure of the class of all contractible complexes of countably
presentable flat $\mathfrak{R}$-contramodules under extensions and directed
colimits;

  3. (3)

all $\aleph_{1}$-directed colimits of totalizations of short exact sequences
of complexes of countably presentable flat $\mathfrak{R}$-contramodules.

###### Proof.

The totalization of a short exact sequence of complexes is an extension of the
two cones of identity endomorphisms of the rightmost and leftmost complexes in
the short exact sequence. Furthermore, the class of all pure acyclic complexes
of flat $\mathfrak{R}$-contramodules is closed under extensions and directed
colimits (by Lemmas 8.4(a) and 8.8), and the contractible complexes of flat
$\mathfrak{R}$-contramodules are pure acyclic. This proves the inclusions (3)
$\subset$ (2) $\subset$ (1).

Conversely, to prove that (1) $\subset$ (3), recall that all pure acyclic
complexes of flat $\mathfrak{R}$-contramodules are $\aleph_{1}$-directed
colimits of pure acyclic complexes of countably presentable flat
$\mathfrak{R}$-contramodules by Corollary 11.4. The exact category
$\mathsf{G}$ of countably presentable flat $\mathfrak{R}$-contramodules has
homological dimension $\leq 1$ by Lemma 13.1(b). Hence the pure acyclic
complexes of countably presentable flat $\mathfrak{R}$-contramodules are
absolutely acyclic by Lemma 6.1, and it remains to refer to Proposition 6.2
(for $d=1$). ∎

######  Questions 13.3.

The main results of this paper concerning contramodules, such as Theorems
10.1, 12.3, and and 13.2, as well as Propositions 10.2 and 11.3, and Corollary
11.4 assume existence of a countable topology base of two-sided ideals in the
topological ring $\mathfrak{R}$. Are any of these theorems valid for complete,
separated topological rings with a countable base of neighborhoods of zero
consisting of open _right_ ideals? It would be interesting to know; but in
fact, even Questions 9.8 and 11.2 seem to be open.

## References

  * [1] J. Adámek, J. Rosický. Locally presentable and accessible categories. London Math. Society Lecture Note Series 189, Cambridge University Press, 1994.
  * [2] S. Bazzoni, M. Cortés-Izurdiaga, S. Estrada. Periodic modules and acyclic complexes. Algebras and Represent. Theory 23, #5, p. 1861–1883, 2020. arXiv:1704.06672 [math.RA]
  * [3] A. Beilinson, V. Drinfeld. Quantization of Hitchin’s integrable system and Hecke eigensheaves. February 2000. Available from http://www.math.utexas.edu/~benzvi/Langlands.html or http://math.uchicago.edu/~drinfeld/langlands.html
  * [4] D. J. Benson, K. R. Goodearl. Periodic flat modules, and flat modules for finite groups. Pacific Journ. of Math. 196, #1, p. 45–67, 2000.
  * [5] T. Brzezinski, R. Wisbauer. Corings and comodules. London Mathematical Society Lecture Note Series, 309. Cambridge University Press, Cambridge, 2003.
  * [6] T. Bühler. Exact categories. Expositiones Math. 28, #1, p. 1–69, 2010. arXiv:0811.1480 [math.HO]
  * [7] B. Chorny, J. Rosický. Class-locally presentable and class-accessible categories. Journ. of Pure and Appl. Algebra 216, #10, p. 2113–2125, 2012. arXiv:1110.0605 [math.CT]
  * [8] L. W. Christensen, H. Holm. The direct limit closure of perfect complexes. Journ. of Pure and Appl. Algebra 219, #3, p. 449–463, 2015. arXiv:1301.0731 [math.RA]
  * [9] R. Colpi, K. R. Fuller. Tilting objects in abelian categories and quasitilted rings. Trans. Amer. Math. Soc. 359, #2, p. 741–765, 2007\.
  * [10] P. Čoupek, J. Št’ovíček. Cotilting sheaves on Noetherian schemes. Math. Zeitschrift 296, #1–2, p. 275–312, 2020. arXiv:1707.01677 [math.AG]
  * [11] A. I. Efimov, L. Positselski. Coherent analogues of matrix factorizations and relative singularity categories. Algebra and Number Theory 9, #5, p. 1159–1292, 2015\. arXiv:1102.0261 [math.CT]
  * [12] S. Eilenberg, J. C. Moore. Limits and spectral sequences. Topology 1, p. 1–23, 1962.
  * [13] E. E. Enochs, J. R. García Rozas. Flat covers of complexes. Journ. of Algebra 210, #1, p. 86–102, 1998.
  * [14] S. Estrada, X. Fu, A. Iacob. Totally acyclic complexes. Journ. of Algebra 470, p. 300–319, 2017. arXiv:1603.03850 [math.AC]
  * [15] S. Estrada, P. A. Guil Asensio, S. Odabaşi. A Lazard-like theorem for quasi-coherent sheaves. Algebras and Represent. Theory 16, #4, p. 1193–1205, 2013. arXiv:1109.0439 [math.AG]
  * [16] R. Göbel, J. Trlifaj. Approximations and endomorphism algebras of modules. Second Revised and Extended Edition. De Gruyter Expositions in Mathematics 41, De Gruyter, Berlin–Boston, 2012.
  * [17] V. E. Govorov. On flat modules (Russian). Sibir. Mat. Zh. 6, p. 300–304, 1965.
  * [18] S. Henry. When does $\operatorname{Ind}_{\kappa}(C^{I})\simeq\operatorname{Ind}_{\kappa}(C)^{I}$? Electronic preprint arXiv:2307.06664 [math.CT].
  * [19] B. Keller. Derived categories and their uses. M. Hazewinkel, Ed., Handbook of Algebra, vol. 1, Elsevier, 1996, p. 671–701.
  * [20] M. Kontsevich, A. Rosenberg. Noncommutative smooth spaces. The Gelfand Mathematical Seminars 1996–1999, p. 85–108, Birkhäuser Boston, Boston, MA, 2000. arXiv:math.AG/9812158
  * [21] M. Kontsevich, A. Rosenberg. Noncommutative spaces and flat descent. Max-Planck-Institut für Mathematik (Bonn) preprint MPIM 2004-36.
  * [22] D. Lazard. Autour de la platitude. Bull. Soc. Math. France 97, p. 81–128, 1969.
  * [23] S. Mac Lane. Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998.
  * [24] D. Murfet. Derived categories of quasi-coherent sheaves. Notes, October 2006. Available from http://www.therisingsea.org/notes
  * [25] A. Neeman. The homotopy category of flat modules, and Grothendieck duality. Inventiones Math. 174, #2, p. 255–308, 2008.
  * [26] H.-E. Porst. On corings and comodules. Archivum Mathematicum 42, #4, p. 419–425, 2006.
  * [27] L. Positselski. Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne vol. 70, Birkhäuser/Springer Basel, 2010. xxiv+349 pp. arXiv:0708.3398 [math.CT]
  * [28] L. Positselski. Contraherent cosheaves. Electronic preprint arXiv:1209.2995 [math.CT].
  * [29] L. Positselski. Contramodules. Confluentes Math. 13, #2, p. 93–182, 2021. arXiv:1503.00991 [math.CT]
  * [30] L. Positselski. Contraadjusted modules, contramodules, and reduced cotorsion modules. Moscow Math. Journ. 17, #3, p. 385–455, 2017. arXiv:1605.03934 [math.CT]
  * [31] L. Positselski. Flat ring epimorphisms of countable type. Glasgow Math. Journ. 62, #2, p. 383–439, 2020. arXiv:1808.00937 [math.RA]
  * [32] L. Positselski. Contramodules over pro-perfect topological rings. Forum Mathematicum 34, #1, p. 1–39, 2022. arXiv:1807.10671 [math.CT]
  * [33] L. Positselski. An explicit self-dual construction of complete cotorsion pairs in the relative context. Rendiconti Semin. Matem. Univ. Padova 149, p. 191–253, 2023. arXiv:2006.01778 [math.RA]
  * [34] L. Positselski. Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes: Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. Birkhäuser/Springer Nature, Cham, Switzerland, 2023. xix+216 pp. arXiv:2104.05517 [math.AG]
  * [35] L. Positselski. Exact DG-categories and fully faithful triangulated inclusion functors. Electronic preprint arXiv:2110.08237 [math.CT].
  * [36] L. Positselski. Differential graded Koszul duality: An introductory survey. Bulletin of the London Math. Society 55, #4, p. 1551–1640, 2023. arXiv:2207.07063 [math.CT]
  * [37] L. Positselski. Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT].
  * [38] L. Positselski. Locally coherent exact categories. Electronic preprint arXiv:2311.02418 [math.CT].
  * [39] L. Positselski, P. Příhoda, J. Trlifaj. Closure properties of $\varinjlim\mathcal{C}$. Journ. of Algebra 606, p. 30–103, 2022. arXiv:2110.13105 [math.RA]
  * [40] L. Positselski, J. Rosický. Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories. Journ. of Algebra 483, p. 83–128, 2017. arXiv:1512.08119 [math.CT]
  * [41] L. Positselski, J. Št’ovíček. The tilting-cotilting correspondence. Internat. Math. Research Notices 2021, #1, p. 189–274, 2021. arXiv:1710.02230 [math.CT]
  * [42] L. Positselski, J. Št’ovíček. Derived, coderived, and contraderived categories of locally presentable abelian categories. Journ. of Pure and Appl. Algebra 226, #4, article ID 106883, 2022, 39 pp. arXiv:2101.10797 [math.CT]
  * [43] L. Positselski, J. Št’ovíček. Coderived and contraderived categories of locally presentable abelian DG-categories. Electronic preprint arXiv:2210.08237 [math.CT].
  * [44] L. Positselski, J. Št’ovíček. Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG].
  * [45] G. Raptis, J. Rosický. The accessibility rank of weak equivalences. Theory and Appl. of Categories 30, no. 19, p. 687–703, 2015. arXiv:1403.3042 [math.AT]
  * [46] A.-M. Simon. Approximations of complete modules by complete big Cohen–Macaulay modules over a Cohen–Macaulay local ring. Algebras and Representation Theory 12, #2–5, p. 385–400, 2009.
  * [47] N. Spaltenstein. Resolutions of unbounded complexes. Compositio Math. 65, #2, p.121–154, 1988.
  * [48] M. Sweedler. The predual theorem to the Jacobson–Bourbaki theorem. Trans. of the Amer. Math. Soc. 213, p. 391–406, 1975.
  * [49] B. Totaro. The resolution property for schemes and stacks. Journ. für die reine und angew. Mathematik 577, p. 1–22, 2004. arXiv:math.AG/0207210
  * [50] F. Ulmer. Bialgebras in locally presentable categories. Preprint, University of Wuppertal, Summer 1977. Available from https://math.cas.cz/~positselski
  * [51] A. Yekutieli. On flatness and completion for infinitely generated modules over noetherian rings. Communicat. in Algebra 39, #11, p. 4221–4245, 2011\. arXiv:0902.4378 [math.AC]