# Flat comodules and contramodules as directed colimits, and cotorsion
periodicity

Leonid Positselski  Institute of Mathematics, Czech Academy of Sciences
Žitná 25, 115 67 Praha 1
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

This paper is a follow-up to [44]. We consider two algebraic settings of
comodules over a coring and contramodules over a topological ring with a
countable base of two-sided ideals. These correspond to two (noncommutative)
algebraic geometry settings of certain kind of stacks and ind-affine ind-
schemes. In the context of a coring $\mathcal{C}$ over a noncommutative ring
$A$, we show that all $A$-flat $\mathcal{C}$-comodules are
$\aleph_{1}$-directed colimits of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules. In the context of a complete, separated topological
ring $\mathfrak{R}$ with a countable base of neighborhoods of zero consisting
of two-sided ideals, we prove that all flat $\mathfrak{R}$-contramodules are
$\aleph_{1}$-directed colimits of countably presentable flat
$\mathfrak{R}$-contramodules. We also describe arbitrary complexes, short
exact sequences, and pure acyclic complexes of $A$-flat
$\mathcal{C}$-comodules and flat $\mathfrak{R}$-contramodules as
$\aleph_{1}$-directed colimits of similar complexes of countably presentable
objects. The arguments are based on a very general category-theoretic
technique going back to an unpublished 1977 preprint of Ulmer and rediscovered
in [37]. Applications to cotorsion periodicity and coderived categories of
flat objects in the respective settings are discussed. In particular, in any
acyclic complex of cotorsion $\mathfrak{R}$-contramodules, all the
contramodules of cocycles are cotorsion.

###### Contents

  1. 1 Preliminaries on Accessible Categories
  2. 2 Preliminaries on Corings and Comodules
  3. 3 $A$-Flat $\mathcal{C}$-Comodules as Directed Colimits
  4. 4 Exact Sequences of $A$-Flat $\mathcal{C}$-Comodules as Directed Colimits
  5. 5 Cotorsion Periodicity for Comodules
  6. 6 Exact Sequences of $A$-Flat $\mathcal{C}$-Comodules II
  7. 7 Preliminaries on Topological Rings and Contramodules
  8. 8 Separated and Flat Contramodules
  9. 9 Countably Presentable Flat Contramodules
  10. 10 Flat Contramodules as Directed Colimits
  11. 11 Exact Sequences of Flat Contramodules as Directed Colimits
  12. 12 Cotorsion Periodicity for Contramodules
  13. 13 Exact Sequences of Flat Contramodules II

## Introduction

The classical _Govorov–Lazard theorem_ [17, 22] tells that all flat modules
(over an arbitrary associative ring $R$) are directed colimits of projective
modules, and in fact, even of finitely generated free $R$-modules. In the
context of algebraic geometry, over a nonaffine scheme $X$, there are usually
_not_ enough projective objects in the abelian category of quasi-coherent
sheaves $X{\operatorname{\mathsf{--qcoh}}}$, which makes the role of flat
quasi-coherent sheaves even more important than the role of flat modules in
module theory. What should a suitable version of the Govorov–Lazard theorem
tell about flat quasi-coherent sheaves?

One could try to use locally free or locally projective quasi-coherent sheaves
in the role of projective modules. But it is still an open problem whether
there are enough locally projective quasi-coherent sheaves on $X$ under any
reasonable assumptions on a scheme $X$ [49]. On the other hand, it is known
that there are enough flat quasi-coherent sheaves on any quasi-compact semi-
separated scheme [24, Section 2.4], [11, Lemma A.1]. So describing flat quasi-
coherent sheaves is a worthwhile undertaking.

The approach in this paper follows the idea that, in many homological algebra
contexts, one can use objects of finite projective dimension in lieu of
projective ones. It is known that any countably presentable flat $R$-module
has projective dimension $\leq 1$ [16, Corollary 2.23]. Thus our suggested
answer to the question in the first paragraph is this: the Govorov–Lazard
theorem in algebraic geometry should tell that any flat sheaf is a directed
colimit of locally countably presentable flat ones. Notice, for comparison,
that any finitely presentable flat module is projective.

A partial result was obtained in the paper [15], where it was shown that,
under certain additional assumptions on a quasi-compact semi-separated scheme
$X$, any flat quasi-coherent sheaf on $X$ is a directed colimit of locally
countably generated flat quasi-coherent sheaves locally of projective
dimension $\leq 1$ [15, Theorem B or Theorem 4.9]. In full generality, the
desired assertion for quasi-compact quasi-separated schemes (or even more
generally, for countably quasi-compact, countably quasi-separated schemes) $X$
was proved in the preprint [44]: any flat quasi-coherent sheaf on $X$ is an
$\aleph_{1}$-directed colimit of locally countably presentable flat quasi-
coherent sheaves [44, Theorem 2.4 or Theorem 3.5]. Here an _$\aleph_{1}$
-directed colimit_ means the directed colimit of a diagram indexed by an
$\aleph_{1}$-directed poset, i. e., a poset in which every countable subset
has an upper bound.

The present paper aims to extend the results of [44] to two algebraic geometry
settings more general than schemes: the stacks and the ind-schemes. In fact,
we don’t assume our rings to be commutative in this paper, so it also extends
the results of [44] into certain noncommutative algebraic geometry realms.

The observation that certain noncommutative stacks $X$ can be described by
corings $\mathcal{C}$ over noncommutative rings $A$ is due to Kontsevich and
Rosenberg [20, Section 2], [21]. The quasi-coherent sheaves on $X$ are then
interpreted as left $\mathcal{C}$-comodules (which form an abelian category
whenever $\mathcal{C}$ is a flat right $A$-module). The quasi-coherent sheaf
corresponding to a $\mathcal{C}$-comodule $\mathcal{M}$ is flat if and only if
$\mathcal{M}$ is a flat $A$-module.

In commutative algebraic geometry, this description applies to the stacks $X$
admitting a flat affine surjective morphism $U\longrightarrow X$ from an
affine scheme $U$. Specifically, one has $A=\mathcal{O}(U)$ and
$\mathcal{C}=\mathcal{O}(U\times_{X}U)$. For example, if $X$ is a quasi-
compact semi-separated scheme covered by a finite collection of affine open
subschemes $U_{\alpha}\subset X$, then one can take $U$ to be the disjoint
union $U=\coprod_{\alpha}U_{\alpha}$. Then
$A=\bigoplus_{\alpha}\mathcal{O}(U_{\alpha})$ and
$\mathcal{C}=\bigoplus_{\alpha,\beta}\mathcal{O}(U_{\alpha}\cap U_{\beta})$.
The category of quasi-coherent sheaves on $X$ is equivalent to the category of
$\mathcal{C}$-comodules, and the inverse image functor
$X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
U{\operatorname{\mathsf{--qcoh}}}$ corresponds to the forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
A{\operatorname{\mathsf{--Mod}}}$. The $\mathcal{C}$-comodule structure on a
given $A$-module describes the descent/gluing datum needed to glue a quasi-
coherent sheaf on $X$ from a given collection of quasi-coherent sheaves on
$U_{\alpha}$.

Let $\mathcal{C}$ be a coring over a noncommutative ring $A$ (in the sense of
the paper [48] and the book [5]). Then we prove that any
$\mathcal{C}$-comodule that is flat as an $A$-module can be obtained as an
$\aleph_{1}$-directed colimit of $\mathcal{C}$-comodules that are flat and
countably presentable as $A$-modules (see Theorem 3.1). Similarly, any
$\mathcal{C}$-comodule is an $\aleph_{1}$-directed colimit of
$\mathcal{C}$-comodules that are countably presentable as $A$-modules (Remark
3.2). Furthermore, any complex of $A$-flat $\mathcal{C}$-comodules is an
$\aleph_{1}$-directed colimit of complexes of $A$-countably presentable
$A$-flat $\mathcal{C}$-comodules (by Proposition 3.3). We also show that any
short exact sequence of $A$-flat $\mathcal{C}$-comodules is an
$\aleph_{1}$-directed colimit of short exact sequences of $A$-countably
presentable $A$-flat $\mathcal{C}$-comodules (this is the result of our
Proposition 4.4). More generally, any $A$-pure acyclic complex of $A$-flat
$\mathcal{C}$-comodules is an $\aleph_{1}$-directed colimit of $A$-pure
acyclic complexes of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules (see Corollary 4.5).

A left $R$-module $P$ is said to be _cotorsion_ if
$\operatorname{Ext}^{1}_{R}(F,P)=0$ for all flat left $R$-modules $F$. The
_cotorsion periodicity theorem_ of Bazzoni, Cortés-Izurdiaga, and Estrada [2]
claims that, in any acyclic complex of cotorsion $R$-modules, all the modules
of cocycles are also cotorsion [2, Theorem 1.2(2), Proposition 4.8(2), or
Theorem 5.1(2)].

The following comodule version of cotorsion periodicity theorem is obtained in
this paper. Assume that $\mathcal{C}$ is a flat right $A$-module, all left
$\mathcal{C}$-comodules are quotients of $A$-flat left
$\mathcal{C}$-comodules, and all left $\mathcal{C}$-comodules having finite
projective dimension as left $A$-modules also have finite projective dimension
as $\mathcal{C}$-comodules. Let us say that a left $\mathcal{C}$-comodule
$\mathcal{B}$ is _cotorsion_ if
$\operatorname{Ext}^{1}_{\mathcal{C}}(\mathcal{F},\mathcal{B})=0$ for all
$A$-flat left $\mathcal{C}$-comodules $\mathcal{F}$. Here
$\operatorname{Ext}^{*}_{\mathcal{C}}$ denotes the $\operatorname{Ext}$ groups
in the abelian category of of left $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. Then, in any acyclic complex
of cotorsion left $\mathcal{C}$-comodules, the comodules of cocycles are also
cotorsion (see Theorem 5.4 and Corollary 5.5). As a corollary of this
periodicity theorem, we conclude that the derived category of the abelian
category of left $\mathcal{C}$-comodules is equivalent to the derived category
of the exact category of cotorsion left $\mathcal{C}$-comodules (Corollary
5.6).

Under the same assumptions as in the previous paragraph, we also obtain the
following description of $A$-pure acyclic complexes of $A$-flat left
$\mathcal{C}$-comodules. Such complexes are $\aleph_{1}$-directed colimits of
totalizations of finite acyclic complexes (of a certain fixed length) of
complexes of $A$-countably presentable $A$-flat left $\mathcal{C}$-comodules.
This result, based on [27, Lemma 2.1] and [35, proof of Proposition 8.8],
provides a comodule version of the well-known description of pure acyclic
complexes of flat $R$-modules as directed colimits of finite contractible
complexes of finitely generated projective $R$-modules [13, Theorem 2.4 (1)
$\Leftrightarrow$ (3)], [25, Theorem 8.6 (ii) $\Leftrightarrow$ (iii)]. This
is our Theorem 6.5.

The discussion of ind-schemes in this paper is mostly restricted to strict
ind-affine $\aleph_{0}$-ind-schemes, i. e., the ind-schemes represented by
countable directed diagrams of closed immersions of affine schemes. The
category of such ind-schemes $\mathfrak{X}$ is anti-equivalent to the category
of complete, separated topological commutative rings $\mathfrak{R}$ with a
countable base of neighborhoods of zero consisting of open ideals [3, Example
7.11.2(i)], [34, Example 1.6(2)]. Flat pro-quasi-coherent pro-sheaves on
$\mathfrak{X}$ (in the sense of [3, Section 7.11.3], [34, Section 3.4]) are
described by flat $\mathfrak{R}$-contramodules, while the category of all pro-
quasi-coherent pro-sheaves on $\mathfrak{X}$ is equivalent to the category of
_separated_ $\mathfrak{R}$-contramodules [34, Examples 3.8(1–2)].

As a noncommutative generalization of this class of ind-schemes, we consider
complete, separated topological associative rings $\mathfrak{R}$ with a
countable base of neighborhoods of zero consisting of open _two-sided_ ideals.
Contramodules over such topological rings $\mathfrak{R}$ were discussed in the
long preprint [28, Appendix D], while the more general case of a countable
base of _right_ ideals was studied in the paper [40].

In this context (assuming a countable topology base of two-sided ideals in
$\mathfrak{R}$), we prove that any flat $\mathfrak{R}$-contramodule is an
$\aleph_{1}$-directed colimit of countably presentable flat
$\mathfrak{R}$-contramodules (see Theorem 10.1). Furthermore, any complex of
flat $\mathfrak{R}$-contramodules is an $\aleph_{1}$-directed colimit of
complexes of countably presentable flat $\mathfrak{R}$-contramodules (by
Proposition 10.2). We also show that any 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
(Proposition 11.3), and any acyclic complex of flat
$\mathfrak{R}$-contramodules with flat $\mathfrak{R}$-contramodules of
cocycles is an $\aleph_{1}$-directed colimit of such complexes of countably
presentable flat $\mathfrak{R}$-contramodules (Corollary 11.4). It follows
that any acyclic complexes of flat $\mathfrak{R}$-contramodules with flat
$\mathfrak{R}$-contramodules of cocycles is an $\aleph_{1}$-directed colimit
of totalizations of short exact sequences of complexes of flat
$\mathfrak{R}$-contramodules (Theorem 13.2).

Then we deduce the following contramodule version of cotorsion periodicity
theorem. A left $\mathfrak{R}$-contramodule $\mathfrak{B}$ is said to be
_cotorsion_ if
$\operatorname{Ext}^{\mathfrak{R},1}(\mathfrak{F},\mathfrak{B})=0$ for all
flat left $\mathfrak{R}$-contramodules $\mathfrak{F}$. Here
$\operatorname{Ext}^{\mathfrak{R},*}$ denotes the $\operatorname{Ext}$ groups
in the abelian category of left $\mathfrak{R}$-contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. The theorem claims that, in
any acyclic complex of cotorsion $\mathfrak{R}$-contramodules, the
contramodules of cocycles are also cotorsion (see Theorem 12.3 and Corollary
12.4). As a corollary of this periodicity theorem, we deduce an equivalence
between the derived category of the abelian category of left
$\mathfrak{R}$-contramodules and the derived category of the exact category of
cotorsion left $\mathfrak{R}$-contramodules (Corollary 12.8).

Periodicity theorems can be thought of as expressing special properties of
directed colimit closures of exact categories (see the preprint [38] for a
discussion of the general concept of such directed colimit closure). In
particular, the category of flat modules is the directed colimit closure of
the category of finitely generated projective modules, which is split exact.
In this context, the flat/projective periodicity theorem [4, 25] can be
interpreted as saying that _for the exact category of flat modules, the
contraderived category coincides with the derived category_ , while the
cotorsion periodicity theorem [2] is closely related to the assertion that
_for the exact category of flat modules, the coderived category coincides with
the derived category_.

We refer the reader to the survey paper [36, Section 7] for a discussion of
the history and philosophy of the coderived and contraderived categories (see
also [42, Remark 9.2]). More advanced discussions of the coderived and
contraderived categories in the context relevant to the present paper can be
found in the papers [42, 43] (see [35] for a different point of view). In this
paper we observe that, under suitable assumptions, the results describing the
pure acyclic complexes of flat objects as the directed colimits of pure
acyclic complexes of countably presentable flat objects can be interpreted as
implying that _for the exact category of such flat objects, the coderived
category coincides with the derived category_. The description of the
coderived category obtained in [43, Corollary 0.4 or Proposition 8.11] is used
as the reference point for the analogy or comparison here, in connection with
our Theorems 6.5 and 13.2.

Let us say a few words about the proofs. The proofs of the main results of
this paper are based on very general category-theoretic observations
concerning preservation of $\kappa$-accessible categories and
$\kappa$-presentable objects (for a regular cardinal $\kappa$) by category-
theoretic constructions such as the pseudopullback [7, Proposition 3.1], [45,
Pseudopullback Theorem 2.2], the equifier [37, Section 3], and the inserter
[37, Section 4]. These results, going back to an unpublished 1977 preprint of
Ulmer [50] and rediscovered in [37], depend on the assumption of existence of
a smaller infinite cardinal $\lambda<\kappa$ such that the $\kappa$-accessible
categories involved have colimits of $\lambda$-indexed chains and the functors
involved preserve such colimits. In the situation at hand, we take
$\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$. The key observation is that the
classes of $A$-flat $\mathcal{C}$-comodules and flat
$\mathfrak{R}$-contramodules, as well as various classes of complexes of flat
comodules and flat contramodules, are preserved by all directed colimits. This
allows to prove that the respective categories are $\aleph_{1}$-accessible,
and describe their full subcategories of $\aleph_{1}$-presentable objects.

### Acknowledgement

I am grateful to Jan Št’ovíček for helpful discussions. The author is
supported by the GAČR project 23-05148S and the Czech Academy of Sciences (RVO
67985840).

##  1\. Preliminaries on Accessible Categories

Let $\kappa$ be a regular cardinal. We refer to [1, Definition 1.4, Theorem
1.5, Corollary 1.5, Definition 1.13(1), and Remark 1.21] for the discussion of
_$\kappa$ -directed posets_ vs.  _$\kappa$ -filtered small categories_, and
accordingly, $\kappa$-directed vs. $\kappa$-filtered colimits.

Let $\mathsf{C}$ be a category with $\kappa$-directed (equivalently,
$\kappa$-filtered) colimits. An object $S\in\mathsf{C}$ is said to be
_$\kappa$ -presentable_ if the functor
$\operatorname{Hom}_{\mathsf{C}}(S,{-})\colon\mathsf{C}\longrightarrow\mathsf{Sets}$
preserves $\kappa$-directed colimits (see [1, Definition 1.13(2)]). We will
denote the class of all $\kappa$-presentable objects of $\mathsf{C}$ by
$\mathsf{C}_{<\kappa}\subset\mathsf{C}$. The $\aleph_{0}$-presentable objects
are called finitely presentable (see [1, Definition 1.1]); and the
$\aleph_{1}$-presentable objects can be similarly called _countably
presentable_.

This category-theoretic terminology is consistent with the module-theoretic
one. Given an associative ring $A$, an $A$-module $M$ is $\kappa$-presentable
as an object of the category of left $A$-modules
$A{\operatorname{\mathsf{--Mod}}}$ (in the sense of the definition above) if
and only if it is the cokernel of a morphism of free $A$-modules with less
than $\kappa$ generators.

Given an additive category $\mathsf{E}$, we denote by
$\mathsf{Com}(\mathsf{E})$ the category of (unbounded) cochain complexes in
$\mathsf{E}$. Clearly, if $\kappa$-directed colimits exist in $\mathsf{E}$,
then they also exist in $\mathsf{Com}(\mathsf{E})$. The following lemma is
standard.

######  Lemma 1.1.

(a) Let $\mathsf{E}$ be an additive category with directed colimits. Then any
_bounded_ complex of finitely presentable objects of $\mathsf{E}$ is a
finitely presentable object in $\mathsf{Com}(\mathsf{E})$.

(b) Let $\kappa$ be an _uncountable_ regular cardinal and $\mathsf{E}$ be an
additive category with $\kappa$-directed colimits. Then any complex of
$\kappa$-presentable objects of $\mathsf{E}$ is a $\kappa$-presentable object
in $\mathsf{Com}(\mathsf{E})$.

###### Proof.

Part (a) holds, because directed colimits commute with finite limits in the
category of abelian groups. Part (b) is true due to the fact that
$\aleph_{1}$-directed colimits commute with countable limits of abelian
groups. ∎

A category $\mathsf{C}$ with $\kappa$-directed colimits is said to be
_$\kappa$ -accessible_ if there is a _set_ of $\kappa$-presentable objects
$\mathsf{S}\subset\mathsf{C}$ such that every object of $\mathsf{C}$ is a
$\kappa$-directed colimit of objects from $\mathsf{S}$ (see [1, Definition
2.1]). If this is the case, then the $\kappa$-presentable objects of
$\mathsf{C}$ are precisely all the retracts of objects from $\mathsf{S}$. A
$\kappa$-accessible category $\mathsf{C}$ is said to be _locally
$\kappa$-presentable_ if all colimits exist in $\mathsf{C}$ (see [1,
Definition 1.17 and Theorem 1.20]). The $\aleph_{0}$-accessible categories are
called _finitely accessible_ [1, Remark 2.2(1)], and the locally
$\aleph_{0}$-presentable categories are called _locally finitely presentable_
[1, Definition 1.9 and Theorem 1.11].

We denote by $A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}\subset
A{\operatorname{\mathsf{--Mod}}}$ the full subcategory of flat left
$A$-modules.

######  Lemma 1.2.

For any ring $A$ and any regular cardinal $\kappa$, the category of flat left
$A$-modules $A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ is
$\kappa$-accessible. The $\kappa$-presentable objects of
$A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ are precisely all the
$\kappa$-presentable flat $A$-modules (i. e., the flat $A$-modules that are
$\kappa$-presentable in $A{\operatorname{\mathsf{--Mod}}}$).

###### Proof.

The Govorov–Lazard description of flat $A$-modules as the directed colimits of
finitely generated projective $A$-modules [17], [22], [16, Corollary 2.22]
means that the category $A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ is
finitely accessible, and the finitely generated projective $A$-modules are its
finitely presentable objects. By [1, Theorem 2.11 and Example 2.13(1)], it
follows that the category $A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$
is $\kappa$-presentable for all regular cardinals $\kappa$; and the argument
in [1, proof of Theorem 2.11 (iv) $\Rightarrow$ (i)] implies the desired
description of $\kappa$-presentable objects. The point is that all flat
$A$-modules that are $\kappa$-presentable in
$A{\operatorname{\mathsf{--Mod}}}$ are also $\kappa$-presentable in
$A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$, since the full subcategory
$A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ is closed under directed
colimits in $A{\operatorname{\mathsf{--Mod}}}$. On the other hand, any
directed colimit of $\kappa$-presentable flat $A$-modules indexed by a
directed poset of the cardinality smaller than $\kappa$ is a
$\kappa$-presentable flat $A$-module again. ∎

The following lemma illustrates the significance of countably presentable (i.
e., $\aleph_{1}$-presentable) flat modules.

######  Lemma 1.3.

Any countably presentable flat module over an associative ring $A$ is a
countable directed colimit of finitely generated free $A$-modules.
Consequently, the projective dimension of any countably presentable flat
module is less than or equal to $1$.

###### Proof.

See [16, Corollary 2.23]. ∎

In the rest of this section, we discuss several category-theoretic
constructions which will be used in this paper: the product, the
pseudopullback, the isomorpher, the inserter, the equifier, and the diagram
category. We recall results from the papers [50, 7, 45, 37] concerning
$\kappa$-accessibility of the categories produced by such constructions and
the descriptions of $\kappa$-presentable objects in these categories.

In almost all the contexts, we will consider a regular cardinal $\kappa$ and a
smaller infinite cardinal $\lambda<\kappa$ (so $\kappa$ has to be
uncountable). In the applications in the main body of the paper, we will use
$\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$.

A _$\lambda$ -indexed chain_ (of objects and morphisms) in a category
$\mathsf{C}$ is a functor $\lambda\longrightarrow\mathsf{C}$, where $\lambda$
is considered as an ordinal, and this ordinal, viewed as an ordered set, is
considered as a small category. In other words, a $\lambda$-indexed chain is a
directed diagram $(C_{i}\to C_{j})_{0\leq i<j<\lambda}$ in $\mathsf{C}$.

######  Proposition 1.4.

Let $\kappa$ be a regular cardinal and $\Xi$ be a set of the cardinality
smaller than $\kappa$. Let $(\mathsf{K}_{\xi})_{\xi\in\Xi}$ be a family of
$\kappa$-accessible categories indexed by the set $\Xi$. Then the Cartesian
product $\mathsf{K}=\prod_{\xi\in\Xi}\mathsf{K}_{\xi}$ is also a
$\kappa$-accessible category. The $\kappa$-presentable objects of $\mathsf{K}$
are precisely all the collections of objects
$(S_{\xi}\in\mathsf{K}_{\xi})_{\xi\in\Xi}$ with
$S_{\xi}\in(\mathsf{K}_{\xi})_{<\kappa}$ for all $\xi\in\Xi$.

###### Proof.

This is a corrected version of [1, proof of Proposition 2.67]. See [37,
Proposition 2.1] for the details. ∎

Let $\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ be three categories,
and let $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ and
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be two functors. The
_pseudopullback_ $\mathsf{C}$ of the pair of functors $F_{1}$, $F_{2}$ is
defined as the category of triples $(K_{1},K_{2},\theta)$, where
$K_{1}\in\mathsf{K}_{1}$ and $K_{2}\in\mathsf{K}_{2}$ are two objects, and
$\theta\colon F_{1}(K_{1})\simeq F_{2}(K_{2})$ is an isomorphism in
$\mathsf{L}$.

######  Theorem 1.5.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ be three
$\kappa$-accessible categories where all $\lambda$-indexed chains have
colimits. Assume that two functors
$F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ and
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ preserve
$\kappa$-directed colimits and colimits of $\lambda$-indexed chains. Assume
further that the functors $F_{1}$ and $F_{2}$ take $\kappa$-presentable
objects to $\kappa$-presentable objects. Then the pseudopullback category
$\mathsf{C}$ is $\kappa$-accessible. The $\kappa$-presentable objects of
$\mathsf{C}$ are precisely all the triples $(S_{1},S_{2},\theta)\in\mathsf{C}$
with $S_{1}\in(\mathsf{K}_{1})_{<\kappa}$ and
$S_{2}\in(\mathsf{K}_{2})_{<\kappa}$.

###### Proof.

This is [45, Pseudopullback Theorem 2.2], based on the argument in [7, proof
of Proposition 3.1]. The assertion essentially goes back to [50, Remark
3.2(I), Theorem 3.8, Corollary 3.9, and Remark 3.11(II)]. See also [37,
Corollary 5.1]. ∎

Let $\mathsf{K}$ and $\mathsf{L}$ be two categories, and let $F_{1}$,
$F_{2}\colon\mathsf{K}\rightrightarrows\mathsf{L}$ be two parallel functors.
The _isomorpher_ $\mathsf{C}$ of the pair of functors $F_{1}$, $F_{2}$ is
defined as the category of pairs $(K,\theta)$, where $K\in\mathsf{K}$ is an
object and $\theta\colon F_{1}(K)\simeq F_{2}(K)$ is an isomorphism in
$\mathsf{L}$.

######  Theorem 1.6.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}$ and $\mathsf{L}$ be two $\kappa$-accessible
categories where all $\lambda$-indexed chains have colimits. Assume that two
parallel functors $F_{1}$, $F_{2}\colon\mathsf{K}\rightrightarrows\mathsf{L}$
preserve $\kappa$-directed colimits and colimits of $\lambda$-indexed chains.
Assume further that the functors $F_{1}$ and $F_{2}$ take $\kappa$-presentable
objects to $\kappa$-presentable objects. Then the isomorpher category
$\mathsf{C}$ is $\kappa$-accessible. The $\kappa$-presentable objects of
$\mathsf{C}$ are precisely all the pairs $(S,\theta)\in\mathsf{C}$ with
$S\in\mathsf{K}_{<\kappa}$.

###### Proof.

This is an equivalent reformulation of Theorem 1.5; see [37, Remark 5.2]. The
assertion can be also viewed as a particular case of [50, Remark 3.2(I),
Theorem 3.8, Corollary 3.9, and Remark 3.11(II)]. ∎

Let $\mathsf{K}$ and $\mathsf{L}$ be two categories, and $F$,
$G\colon\mathsf{K}\rightrightarrows\mathsf{L}$ be two parallel functors. The
_inserter_ $\mathsf{D}$ of the pair of functors $F$, $G$ is defined [1,
Section 2.71] as the category of pairs $(K,\phi)$, where $K\in\mathsf{K}$ is
an object and $\phi\colon F(K)\longrightarrow G(K)$ is a morphism in
$\mathsf{L}$.

######  Theorem 1.7.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}$ and $\mathsf{L}$ be two $\kappa$-accessible
categories where all $\lambda$-indexed chains have colimits. Assume that two
parallel functors $F$, $G\colon\mathsf{K}\rightrightarrows\mathsf{L}$ preserve
$\kappa$-directed colimits and colimits of $\lambda$-indexed chains. Assume
further that the functor $F$ takes $\kappa$-presentable objects to
$\kappa$-presentable objects. Then the inserter category $\mathsf{D}$ is
$\kappa$-accessible. The $\kappa$-presentable objects of $\mathsf{D}$ are
precisely all the pairs $(S,\theta)\in\mathsf{D}$ with
$S\in\mathsf{K}_{<\kappa}$.

###### Proof.

This is [50, Theorem 3.8, Corollary 3.9, and Remark 3.11(II)] or [37, Theorem
4.1]. ∎

Let $\mathsf{D}$ and $\mathsf{M}$ be two categories, $F$,
$G\colon\mathsf{D}\rightrightarrows\mathsf{M}$ be two parallel functors, and
$\phi$, $\psi\colon F\rightrightarrows G$ be two parallel natural
transformations. The _equifier_ $\mathsf{E}$ of the pair of natural
transformations $\phi$, $\psi$ is defined [1, Lemma 2.76] as the full
subcategory in $\mathsf{D}$ consisting of all objects $E\in\mathsf{D}$ for
which the two morphisms $\phi_{E}\colon F(E)\longrightarrow G(E)$ and
$\psi_{E}\colon F(E)\longrightarrow G(E)$ are equal to each other, i. e.,
$\phi_{E}=\psi_{E}$.

######  Theorem 1.8.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{D}$ and $\mathsf{M}$ be two $\kappa$-accessible
categories where all $\lambda$-indexed chains have colimits. Assume that two
parallel functors $F$, $G\colon\mathsf{D}\rightrightarrows\mathsf{M}$ preserve
$\kappa$-directed colimits and colimits of $\lambda$-indexed chains. Assume
further that the functor $F$ takes $\kappa$-presentable objects to
$\kappa$-presentable objects. Let $\phi$, $\psi\colon F\rightrightarrows G$ be
two parallel natural transformations. Then the equifier category $\mathsf{E}$
is $\kappa$-accessible. The $\kappa$-presentable objects of $\mathsf{E}$ are
precisely all the objects of $\mathsf{E}$ that are $\kappa$-presentable as
objects of $\mathsf{D}$.

###### Proof.

This is [50, Theorem 3.8, Corollary 3.9, and Remark 3.11(II)] or [37, Theorem
3.1]. ∎

Let $k$ be a commutative ring. The notion of a _$\kappa$ -presented $k$-linear
category_ spelled out in [37, Section 6] represents an intuitively clear idea
of a quiver with less than $\kappa$-vertices, less than $\kappa$ arrows, and
less than $\kappa$ relations between $k$-linear combinations of various
iterated compositions of the arrows. We refer to [37] for the details. Given a
small $k$-linear category $D$ and a $k$-linear category $\mathsf{K}$, we
denote by $\mathsf{Fun}_{k}(D,\mathsf{K})$ the category of $k$-linear functors
$D\longrightarrow\mathsf{K}$.

######  Theorem 1.9.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{C}$ be a $\kappa$-accessible $k$-linear category where
all $\lambda$-indexed chains have colimits, and let $D$ be a
$\kappa$-presented $k$-linear category. Then the diagram category
$\mathsf{Fun}_{k}(D,\mathsf{C})$ is $\kappa$-accessible. The
$\kappa$-presentable objects of $\mathsf{Fun}_{k}(D,\mathsf{C})$ are precisely
all the $k$-linear functors $D\longrightarrow\mathsf{C}_{<\kappa}$.

###### Proof.

This is [37, Theorem 6.2]. ∎

##  2\. Preliminaries on Corings and Comodules

The following definitions seem to go back to Sweedler’s paper [48]. A detailed
exposition can be found in the book [5]. The book [27] and the survey paper
[29, Sections 2.5 and 3.4] can be used as further references.

A _coring_ over an associative ring $A$ is a comonoid object in the monoidal
category of $A$-$A$-bimodules with respect to the tensor product over $A$. In
other words, a coring $\mathcal{C}$ over $A$ is an $A$-$A$-bimodule endowed
with $A$-$A$-bimodule maps of _comultiplication_
$\mu\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}$ and
_counit_ $\epsilon\colon\mathcal{C}\longrightarrow A$ satisfying the usual
coassociativity and counitality axioms. Specifically, the two compositions
$(\mu\otimes{\mathrm{id}}_{\mathcal{C}})\circ\mu$ and
$({\mathrm{id}}_{\mathcal{C}}\otimes\mu)\circ\mu$ must be equal to each other,

$\mathcal{C}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}\rightrightarrows\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}\mathcal{C},$

and both the compositions
$(\epsilon\otimes{\mathrm{id}}_{\mathcal{C}})\circ\mu$ and
$({\mathrm{id}}_{\mathcal{C}}\otimes\epsilon)\circ\mu$ must be equal to the
identity map ${\mathrm{id}}_{\mathcal{C}}$,

$\mathcal{C}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}\rightrightarrows\mathcal{C}.$

The category of left $A$-modules $A{\operatorname{\mathsf{--Mod}}}$ is a left
module category (with respect to the tensor product operation) over the
monoidal category of $A$-$A$-bimodules
$A{\operatorname{\mathsf{--Bimod--}}}A$. A _left $\mathcal{C}$-comodule_ is a
comodule object in the module category $A{\operatorname{\mathsf{--Mod}}}$ over
the comonoid object $\mathcal{C}$ in the monoidal category
$A{\operatorname{\mathsf{--Bimod--}}}A$. In other words, a left comodule
$\mathcal{M}$ over $\mathcal{C}$ is a left $A$-module endowed with a left
$A$-module map of _left coaction_
$\nu\colon\mathcal{M}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{M}$
satisfying the usual coassociativity and counitality axioms together with the
comultiplication map $\mu$ and the counit map $\epsilon$ of the coring
$\mathcal{C}$. Specifically, the two compositions
$(\mu\otimes{\mathrm{id}}_{\mathcal{M}})\circ\nu$ and
$({\mathrm{id}}_{\mathcal{C}}\otimes\nu)\circ\nu$ must be equal to each other,

$\mathcal{M}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{M}\rightrightarrows\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}\mathcal{M},$

and the composition $(\epsilon\otimes{\mathrm{id}}_{\mathcal{M}})\circ\nu$
must be equal to the identity map ${\mathrm{id}}_{\mathcal{M}}$,

$\mathcal{M}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{M}\longrightarrow\mathcal{M}.$

We denote the additive category of left $\mathcal{C}$-comodules by
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, and use the notation
$\operatorname{Hom}_{\mathcal{C}}({-},{-})$ for the groups of morphisms in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. A $\mathcal{C}$-comodule is
said to be _$A$ -flat_ if it is flat as an $A$-module. The full subcategory of
$A$-flat left $\mathcal{C}$-comodules is denoted by
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}\subset\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

######  Lemma 2.1.

Let $\mathcal{C}$ be a coring over a ring $A$.

(a) All colimits exist in the additive category of left
$\mathcal{C}$-comodules. The forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
A{\operatorname{\mathsf{--Mod}}}$ preserves colimits (in particular,
coproducts).

(b) If $\mathcal{C}$ is a flat right $A$-module, then the category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is a Grothendieck abelian
category, and the forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
A{\operatorname{\mathsf{--Mod}}}$ is exact.

(c) Conversely, if the category $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$
is abelian _and_ the forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
A{\operatorname{\mathsf{--Mod}}}$ is exact, then $\mathcal{C}$ is a flat right
$A$-module.

(d) There exists a coring $\mathcal{C}$ such that the category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is _not_ abelian.

###### Proof.

Part (a) follows from the fact that the tensor product functor
$\mathcal{C}\otimes_{A}{-}$ preserves colimits in
$A{\operatorname{\mathsf{--Mod}}}$. Parts (b–c) are covered by [5, Sections
18.6, 18.14 and 18.16]; see also [29, Proposition 2.12(a)]. A counterexample
for part (d) can be found in [28, Example B.1.1]. ∎

When $\mathcal{C}$ is not a flat right $A$-module, exact sequences of
arbitrary left $\mathcal{C}$-comodules are not well-behaved. But there is
still a well-behaved notion of a _short exact sequence of $A$-flat left
$\mathcal{C}$-comodules_, defined as a short sequence of $A$-flat left
$\mathcal{C}$-comodules that is exact in $A{\operatorname{\mathsf{--Mod}}}$.
We refer to the survey [6] for background material on exact categories in the
sense of Quillen.

######  Lemma 2.2.

The additive category of $A$-flat $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
endowed with the class of short exact sequences defined above is an exact
category.

###### Proof.

Essentially, it suffices to check that the kernel of any surjective morphism
of $A$-flat $\mathcal{C}$-comodules exists in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
and agrees with the kernel of the same morphism computed in the category of
flat $A$-modules. For this purpose, one needs to use the fact that the
functors $\mathcal{C}\otimes_{A}{-}$ and
$\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}{-}$ take short exact sequences
of flat $A$-modules to short exact sequences of $A$-modules. ∎

More generally, an exact sequence of left $A$-modules is called _pure_ if it
stays exact after the functor $N\otimes_{A}{-}$ is applied to it, for every
right $A$-module $N$. An _$A$ -pure short exact sequence of
$\mathcal{C}$-comodules_ is a short sequence of $\mathcal{C}$-comodules that
is pure exact as a short sequence of $A$-modules. The category of left
$\mathcal{C}$-comodules $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ with
$A$-pure short exact sequences is also an exact category.

A left $\mathcal{C}$-comodule is said to be _$A$ -countably presentable_ if it
is countably presentable as an $A$-module. Similarly, a $\mathcal{C}$-comodule
is said to be _$A$ -finitely presentable_ if it is finitely presentable as an
$A$-module.

######  Lemma 2.3.

Let $\mathcal{C}$ be a coring over a ring $A$. Then

(a) all $A$-finitely presentable $\mathcal{C}$-comodules are finitely
presentable as objects of the additive category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$;

(b) all _bounded_ complexes of $A$-finitely presentable
$\mathcal{C}$-comodules are finitely presentable as objects of the additive
category of complexes of $\mathcal{C}$-comodules;

(c) all $A$-countably presentable $\mathcal{C}$-comodules are countably
presentable (i. e., $\aleph_{1}$-presentable) as objects of the additive
category $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$;

(d) all complexes of $A$-countably presentable $\mathcal{C}$-comodules are
countably presentable as objects of the additive category of complexes of
$\mathcal{C}$-comodules.

###### Proof.

Part (b) follows from part (a) according to Lemma 1.1(a); and part (d) follows
from part (c) in view of Lemma 1.1(b). Parts (a) and (c) follow from the facts
that, for any two left $\mathcal{C}$-comodules $\mathcal{L}$ and
$\mathcal{M}$, the abelian group
$\operatorname{Hom}_{\mathcal{C}}(\mathcal{L},\mathcal{M})$ can be computed as
the kernel of (the difference of) a natural pair of maps

$\operatorname{Hom}_{A}(\mathcal{L},\mathcal{M})\,\rightrightarrows\,\operatorname{Hom}_{A}(\mathcal{L},\>\mathcal{C}\otimes_{A}\mathcal{M}),$

and the functor $\mathcal{C}\otimes_{A}{-}$ preserves colimits. ∎

For the converse assertions to Lemma 2.3(c–d), see Remarks 3.2 and 3.4 below.
It will be also explained there that the additive categories of left
$\mathcal{C}$-comodules $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ and
complexes of left $\mathcal{C}$-comodules
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}})$ are locally
$\aleph_{1}$-presentable.

######  Lemma 2.4.

Let $\mathcal{C}$ be a coring over a ring $A$. Assume that $\mathcal{C}$ is a
flat left and right $A$-module, and the ring $A$ has finite weak global
dimension (or in other words, finite $\operatorname{Tor}$-dimension). Then
every $\mathcal{C}$-comodule is a quotient comodule of an $A$-flat
$\mathcal{C}$-comodule.

###### Proof.

This is [27, Lemma 1.1.3]. For a discussion of this construction in a module-
theoretic context, see [33, Section 2.2]. ∎

######  Example 2.5.

This example is due to Kontsevich and Rosenberg [20, Section 2], [21]. Let $X$
be a quasi-compact semi-separated scheme, and let
$X=\bigcup_{\alpha}U_{\alpha}$ be a finite covering of $X$ by affine open
subschemes. Let $U$ denote the disjoint union $U=\coprod_{\alpha}U_{\alpha}$;
so $U$ is an affine scheme and there is a natural surjective flat affine
morphism of schemes $U\longrightarrow X$. Let $A=\mathcal{O}(U)$ and
$\mathcal{C}=\mathcal{O}(U\times_{X}U)$ denote the rings of functions on the
affine schemes $U$ and $U\times_{X}U$. Then the two projections $p_{1}$,
$p_{2}\colon U\times_{X}U\rightrightarrows U$ induce two ring homomorphisms
$A\rightrightarrows\mathcal{C}$, endowing $\mathcal{C}$ with two $A$-module
structures (which we view as the “left” and the “right” one).

The diagonal morphism $\delta\colon U\longrightarrow U\times_{X}U$ induces a
ring homomorphism $\delta^{\vee}\colon\mathcal{C}\longrightarrow A$.
Furthermore, we have
$\mathcal{C}\otimes_{A}\mathcal{C}=\mathcal{O}((U\times_{X}U)\times_{U}(U\times_{X}U))=\mathcal{O}(U\times_{X}U\times_{X}U)$.
The projection $q\colon U\times_{X}U\times_{X}U\longrightarrow U\times_{X}U$
dropping the second (inner) coordinate induces a ring homomorphism
$q^{\vee}\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}$.
The maps $\epsilon=\delta^{\vee}\colon\mathcal{C}\longrightarrow A$ and
$\mu=q^{\vee}\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}$
that we have constructed endow the $A$-$A$-bimodule $\mathcal{C}$ with a
coring structure (over the commutative ring $A$).

For any quasi-coherent sheaf $N$ on $X$, the inverse image of $N$ with respect
to the morphism $U\longrightarrow X$ produces a quasi-coherent sheaf $M$ on
$U$, that is an $A$-module $\mathcal{M}$. The inverse images of $M$ with
respect to the two projections $U\times_{X}U\rightrightarrows U$ are naturally
isomorphic as quasi-coherent sheaves on $U\times_{X}U$, that is
$p_{1}^{*}M\simeq p_{2}^{*}M$. The corresponding bimodules of sections are
$\mathcal{C}\otimes_{A}\mathcal{M}$ and $\mathcal{M}\otimes_{A}\mathcal{C}$;
so we have a natural isomorphism of $A$-$A$-bimodules
$\mathcal{M}\otimes_{A}\mathcal{C}\simeq\mathcal{C}\otimes_{A}\mathcal{M}$.
Precomposing this isomorphism with the map
$\mathcal{M}\longrightarrow\mathcal{M}\otimes_{A}\mathcal{C}$ induced by the
“left” map $A\longrightarrow\mathcal{C}$ (induced by $p_{1}$), we obtain a map
$\nu\colon\mathcal{M}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{M}$,
endowing $\mathcal{M}$ with a left $\mathcal{C}$-comodule structure. The
assignment $N\longmapsto\mathcal{M}$ is an equivalence of abelian categories
$X{\operatorname{\mathsf{--qcoh}}}\simeq\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

######  Remark 2.6.

A quasi-coherent sheaf $F$ on a scheme $X$ is said to be _flat_ if the tensor
product functor $F\otimes_{\mathcal{O}_{X}}{-}\,\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is exact, or equivalently, the
$\mathcal{O}(U)$-module $F(U)$ is flat for every affine open subscheme
$U\subset X$. It suffices to check the latter condition for the affine open
subschemes $U=U_{\alpha}$ from a given affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ of the scheme $X$.

Now let $X$ be a quasi-compact semi-separated scheme. Then it follows that, in
the context of Example 2.5, a quasi-coherent sheaf on $X$ is flat if and only
if the corresponding $\mathcal{C}$-comodule is $A$-flat.

By [24, Section 2.4] or [11, Lemma A.1], any quasi-coherent sheaf on a quasi-
compact semi-separated scheme $X$ is a quotient sheaf of a flat quasi-coherent
sheaf. Therefore, the corings $\mathcal{C}$ appearing in Example 2.5 also have
the property that any $\mathcal{C}$-comodule is a quotient comodule of an
$A$-flat $\mathcal{C}$-comodule.

##  3\. $A$-Flat $\mathcal{C}$-Comodules as Directed Colimits

The following theorem describing $A$-flat $\mathcal{C}$-comodules is the first
main result of this paper. In view of Example 2.5 and Remark 2.6, it is a
generalization of the assertion of [44, Theorem 2.4] for quasi-compact semi-
separated schemes.

######  Theorem 3.1.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Then the category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
of $A$-flat left $\mathcal{C}$-comodules is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
are precisely all the $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules. Consequently, every $A$-flat $\mathcal{C}$-comodule
is an $\aleph_{1}$-directed colimit of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules.

###### Proof.

All $A$-countably presentable $\mathcal{C}$-comodules are
$\aleph_{1}$-presentable as objects of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ by Lemma 2.3(c). Since the full
subcategory
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
is closed under directed colimits in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, it follows that all
$A$-countably presentable $A$-flat $\mathcal{C}$-comodules are
$\aleph_{1}$-presentable in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$.

The full assertion of the theorem is obtained by applying Theorems 1.7 and 1.8
for $\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$. One starts with the category
$\mathsf{K}=A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$, the category
$\mathsf{L}=A{\operatorname{\mathsf{--Mod}}}$, and the pair of parallel
functors $F$, $G\colon\mathsf{K}\rightrightarrows\mathsf{L}$ given by the
rules $F(M)=M$ and $G(M)=\mathcal{C}\otimes_{A}M$ for all $M\in
A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$. Then the inserter category
$\mathsf{D}$ is the category of all “noncoassociative, noncounital” $A$-flat
left $\mathcal{C}$-comodules, i. e., flat left $A$-modules $\mathcal{D}$
endowed with a left $A$-module morphism
$\nu\colon\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{D}$.

The category $\mathsf{K}=A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ is
$\aleph_{1}$-accessible by Lemma 1.2; and the abelian category
$A{\operatorname{\mathsf{--Mod}}}$ is locally finitely presentable, hence
locally $\kappa$-presentable for every regular cardinal $\kappa$ [1, Remark
1.20]. Theorem 1.7 is applicable (notice that the functor $F$ takes
$\aleph_{1}$-presentable objects to $\aleph_{1}$-presentable objects, as it
must; while the functor $G$ does not, and it does not have to). The theorem
tells that the category $\mathsf{D}$ is $\aleph_{1}$-accessible, and provides
a description of the $\aleph_{1}$-presentable objects of $\mathsf{D}$.

Now let $\mathsf{M}$ be the Cartesian product of categories
$\mathsf{M}=A{\operatorname{\mathsf{--Mod}}}\times
A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$. The functor
$F\colon\mathsf{D}\longrightarrow\mathsf{M}$ assigns to a noncoassociative,
noncounital $A$-flat $\mathcal{C}$-comodule $\mathcal{D}$ the pair of
$A$-modules $(\mathcal{D},\mathcal{D})$. The functor
$G\colon\mathsf{D}\longrightarrow\mathsf{M}$ assigns to $\mathcal{D}$ the pair
of $A$-modules
$(\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}\mathcal{D},\>\mathcal{D})$. The
natural transformation $\phi\colon F\longrightarrow G$ assigns to
$\mathcal{D}$ the pair of compositions
$(\mu\otimes{\mathrm{id}}_{\mathcal{D}})\circ\nu\colon\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}\mathcal{D}$
and
$(\epsilon\otimes{\mathrm{id}}_{D})\circ\nu\colon\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{D}\longrightarrow\mathcal{D}$.
The natural transformation $\psi\colon F\longrightarrow G$ assigns to
$\mathcal{D}$ the pair of maps consisting of the composition
$({\mathrm{id}}_{\mathcal{C}}\otimes_{\nu})\circ\nu\colon\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{D}\longrightarrow\mathcal{C}\otimes_{A}\mathcal{C}\otimes_{A}\mathcal{D}$
and the identity map
${\mathrm{id}}_{\mathcal{D}}\colon\mathcal{D}\longrightarrow\mathcal{D}$. Then
the equifier category $\mathsf{E}$ is the category of coassociative, counital
$A$-flat left $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$.

Proposition 1.4 tells that the category $\mathsf{M}$ is
$\aleph_{1}$-accessible, and describes its full subcategory of
$\aleph_{1}$-presentable objects. Theorem 1.8 is applicable (once again, the
functor $F$ takes $\aleph_{1}$-presentable objects to $\aleph_{1}$-presentable
objects, while the functor $G$ does not). The theorem tells that the category
$\mathsf{E}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
is $\aleph_{1}$-accessible, and provides the desired description of the
$\aleph_{1}$-presentable objects of $\mathsf{E}$. ∎

######  Remark 3.2.

Replacing the original category
$\mathsf{K}=A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ by
$\mathsf{K}=A{\operatorname{\mathsf{--Mod}}}$ (and the category
$\mathsf{M}=A{\operatorname{\mathsf{--Mod}}}\times
A{\operatorname{\mathsf{--Mod}}}_{\mathsf{flat}}$ by
$\mathsf{M}=A{\operatorname{\mathsf{--Mod}}}\times
A{\operatorname{\mathsf{--Mod}}}$) in the proof of Theorem 3.1, one obtains a
proof of the assertion that the category of all comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is $\aleph_{1}$-accessible for
any coring $\mathcal{C}$. In fact, all colimits exist in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ by Lemma 2.1(a); so the
additive category $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is even
locally $\aleph_{1}$-presentable. Furthermore, Theorems 1.7 and 1.8 tell that
the $\aleph_{1}$-presentable objects of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ are precisely all the
$A$-countably presentable $\mathcal{C}$-comodules. So any
$\mathcal{C}$-comodule is an $\aleph_{1}$-directed colimit of $A$-countably
presentable $\mathcal{C}$-comodules. Thus we obtain a much more precise
version of [26, Theorem 10].

Let us also provide a description of arbitrary complexes of $A$-flat
$\mathcal{C}$-comodules as directed colimits.

######  Proposition 3.3.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Then the category
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}})$
of complexes of $A$-flat left $\mathcal{C}$-comodules is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}})$
are precisely all the complexes of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules. Consequently, every complex of $A$-flat
$\mathcal{C}$-comodules is an $\aleph_{1}$-directed colimit of complexes of
$A$-countably presentable $A$-flat $\mathcal{C}$-comodules.

###### Proof.

All complexes of $A$-countably presentable $\mathcal{C}$-comodules are
$\aleph_{1}$-presentable as objects of
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}})$ by Lemma 2.3(d).
Since the full subcategory
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}})$
is closed under directed colimits in
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}})$, it follows that
all complexes of $A$-countably presentable $A$-flat $\mathcal{C}$-comodules
are $\aleph_{1}$-presentable in
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}})$.

The full assertion of the proposition is obtained by combining the results of
Theorems 3.1 and 1.9. All one needs to do is to produce a suitable
$\aleph_{1}$-presented $\mathbb{Z}$-linear category $D$ describing complexes
in additive categories. This simple construction can be found in [37, proof of
Corollary 7.4]. ∎

######  Remark 3.4.

Similarly to the proof of Proposition 3.3, it follows from Remark 3.2 that the
category $\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}})$ of
arbitrary complexes of $\mathcal{C}$-comodules is locally
$\aleph_{1}$-presentable. Furthermore, the $\aleph_{1}$-presentable objects of
$\mathsf{Com}(\mathcal{C}{\operatorname{\mathsf{--Comod}}})$ are precisely all
the complexes of $A$-countably presentable $\mathcal{C}$-comodules. So any
complex of $\mathcal{C}$-comodules in an $\aleph_{1}$-directed colimit of
complexes of $A$-countably presentable $\mathcal{C}$-comodules. Alternatively,
as the categories involved are locally presentable rather than only
accessible, it suffices to use a suitable additive version of [18, Theorem
1.2] instead of Theorem 1.9 in this proof.

##  4\. Exact Sequences of $A$-Flat $\mathcal{C}$-Comodules as Directed
Colimits

We start with three module-theoretic lemmas.

######  Lemma 4.1.

Let $A$ be an associative ring. Then the kernel of any surjective morphism
from a countably presentable flat $A$-module to a countably presentable flat
$A$-module is a countably presentable (flat) $A$-module.

###### Proof.

Notice that the kernel of a surjective morphism from a countably presentable
module to a countably presentable module need not be countably presentable in
general (unless the ring is countably coherent). The lemma claims that for
_flat_ modules it is.

Let $F$ be a countably presentable flat $A$-module. By Lemma 1.3, there exists
a countable directed diagram of finitely generated projective $A$-modules
$P_{0}\longrightarrow P_{1}\longrightarrow P_{2}\longrightarrow\dotsb$ such
that $F=\varinjlim_{n\geq 0}P_{n}$. Hence we have a short exact sequence of
$A$-modules
$0\longrightarrow\bigoplus_{n=0}^{\infty}P_{n}\longrightarrow\bigoplus_{n=0}^{\infty}P_{n}\longrightarrow
F\longrightarrow 0$. Put
$P^{\prime}=P^{\prime\prime}=\bigoplus_{n=0}^{\infty}P_{n}$.

Now let $0\longrightarrow H\longrightarrow G\longrightarrow F\longrightarrow
0$ be a short exact sequence of $A$-modules. Since the $A$-module $P^{\prime}$
is projective, the $A$-module morphism $P^{\prime}\longrightarrow F$ lifts
over the surjective $A$-module morphism $G\longrightarrow F$. We obtain a
commutative diagram of a morphism of short exact sequences acting by the
identity on the rightmost terms

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Consequently, there is a short exact sequence of $A$-modules $0\longrightarrow
P^{\prime\prime}\longrightarrow H\oplus P^{\prime}\longrightarrow
G\longrightarrow 0$. Now we notice that the $A$-module $P^{\prime\prime}$ is
countably presentable. If the $A$-module $G$ is countably presentable as well,
then it follows that the $A$-module $H\oplus P^{\prime}$ is countably
presentable (as the class of countably presentable $A$-modules is closed under
extensions). Consequently, the $A$-module $H$ is countably presentable as a
direct summand of a countably presentable $A$-module. ∎

For a generalization of Lemma 4.1 to arbitrary regular cardinals $\kappa$, see
[37, Corollary 7.11] or [38, Corollary 4.7].

######  Lemma 4.2.

Let $A$ be an associative ring. Then the abelian category of three-term
complexes of left $A$-modules $C^{1}\longrightarrow C^{2}\longrightarrow
C^{3}$ is locally finitely presentable, and consequently, locally
$\aleph_{1}$-presentable. The $\aleph_{1}$-presentable objects of this
category are precisely all the three-term complexes of countably presentable
$A$-modules.

###### Proof.

The assertion holds for any regular cardinal $\kappa$ in lieu of $\aleph_{1}$.
See [38, Proposition 1.1(c)] for a much more general result. ∎

######  Lemma 4.3.

For any ring $A$, the category of short exact sequences of flat $A$-modules is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of this category
are precisely all the short exact sequences of countably presentable flat
$A$-modules.

###### Proof.

Once again, the assertion holds for any regular cardinal $\kappa$. See [37,
Corollary 7.12] or [38, Propositions 1.1(c) and 4.6]. ∎

Recall from the discussion in Section 2 that the category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ need not be abelian for a
coring $\mathcal{C}$ over an associative ring $A$ in general; but one can
still speak about _short exact sequences of $A$-flat $\mathcal{C}$-comodules_.
The following proposition describes such short exact sequences.

######  Proposition 4.4.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Then the category
of short exact sequences of $A$-flat $\mathcal{C}$-comodules is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of this category
are precisely all the short exact sequences of $A$-countably presentable
$A$-flat $\mathcal{C}$-comodules. Consequently, every short exact sequence of
$A$-flat $\mathcal{C}$-comodules is an $\aleph_{1}$-directed colimit of short
exact sequences of $A$-countably presentable $A$-flat $\mathcal{C}$-comodules.

###### Proof.

Similar to the proof of Theorem 3.1, with the following changes. Use the
category of short exact sequences of flat $A$-modules (from Lemma 4.3) in the
role of $\mathsf{K}$, the category of three-term complexes of $A$-modules
(from Lemma 4.2) in the role of $\mathsf{L}$, and put
$\mathsf{M}=\mathsf{L}\times\mathsf{K}$. The functors and natural
transformations are given by the same formulas as in the proof of Theorem 3.1,
applied to short exact sequences or three-term complexes instead of
(co)modules. ∎

The assertion of Proposition 4.4 (together with Theorem 3.1) can be restated
by saying that the exact category of $A$-flat $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
from Lemma 2.2 is a _locally $\aleph_{1}$-coherent exact category_ in the
sense of [38, Section 1].

We refer to [6, Section 10] for the definition of an acyclic complex in an
exact category. Acyclic complexes in the exact category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
are called _$A$ -pure acyclic complexes of $A$-flat $\mathcal{C}$-comodules_.
Equivalently, the $A$-pure acyclic complexes of $A$-flat
$\mathcal{C}$-comodules are the complexes of $\mathcal{C}$-comodules whose
underlying complexes of $A$-modules are acyclic with flat $A$-modules of
cocycles.

######  Corollary 4.5.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Then the category
of $A$-pure acyclic complexes of $A$-flat $\mathcal{C}$-comodules is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of this category
are precisely all the $A$-pure acyclic complexes of $A$-countably presentable
$A$-flat $\mathcal{C}$-comodules. Consequently, every $A$-pure acyclic complex
of $A$-flat $\mathcal{C}$-comodules is an $\aleph_{1}$-directed colimit of
$A$-pure acyclic complexes of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules.

###### Proof.

This is provable similarly to Proposition 4.4 (using [37, Corollary 7.13]
instead of Lemma 4.3, and the obvious version of Lemma 4.2 for unbounded
complexes). This is akin to the argument used in [44, proof of Theorem 4.2].
But we prefer to spell out an argument deducing the corollary from the
proposition.

The argument is similar to [37, proof of Corollary 7.13]. An $A$-pure acyclic
complex of $A$-flat $\mathcal{C}$-comodules
$\mathcal{F}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is the same thing
as a collection of short exact sequences of $A$-flat $\mathcal{C}$-comodules
$0\longrightarrow\mathcal{G}^{n}\longrightarrow\mathcal{F}^{n}\longrightarrow\mathcal{H}^{n}\longrightarrow
0$ together with a collection of isomorphisms of $\mathcal{C}$-comodules
$\mathcal{H}^{n}\simeq\mathcal{G}^{n+1}$, $n\in\mathbb{Z}$. Therefore, the
category of $A$-pure acyclic complexes of $A$-flat $\mathcal{C}$-comodules can
be constructed from the category of short exact sequences of $A$-flat
$\mathcal{C}$-comodules using Cartesian products (as in Proposition 1.4) and
the isomorpher construction (as in Theorem 1.6).

Specifically, take $\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$. Denote by
$\mathsf{K}$ the Cartesian product of copies of the category of short exact
sequences of $A$-flat $\mathcal{C}$-comodules (from Proposition 4.4), indexed
by all the integers $n\in\mathbb{Z}$. Denote by $\mathsf{L}$ the Cartesian
product of copies of the category of $A$-flat $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$,
also indexed by $n\in\mathbb{Z}$. Let
$F_{1}\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking a
collection of short exact sequences
$(0\to\mathcal{G}^{n}\to\mathcal{F}^{n}\to\mathcal{H}^{n}\to
0)_{n\in\mathbb{Z}}$ to the collection of flat comodules
$(\mathcal{H}^{n})_{n\in\mathbb{Z}}$, and let
$F_{2}\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking the
same collection of short exact sequences to the collection of flat comodules
$(\mathcal{G}^{n+1})_{n\in\mathbb{Z}}$. Then the isomorpher category
$\mathsf{C}$ is the category of $A$-pure acyclic complexes of $A$-flat
$\mathcal{C}$-comodules.

Theorem 3.1 and Proposition 4.4 together with Proposition 1.4 tell that the
categories $\mathsf{K}$ and $\mathsf{L}$ are $\aleph_{1}$-accessible, and
describe their full subcategories of $\aleph_{1}$-presentable objects. In view
of these descriptions, Theorem 1.6 is applicable, telling that the isomorpher
category $\mathsf{C}$ is $\aleph_{1}$-accessible, and describing its full
subcategory of $\aleph_{1}$-presentable objects. ∎

##  5\. Cotorsion Periodicity for Comodules

In this section we deduce certain conditional results, based on the techniques
of [44, Section 8] and generalizing some results from [44, Section 9].

Let $\mathcal{C}$ be a coring over a ring $A$ such that $\mathcal{C}$ is a
flat right $A$-module (so the category of left $\mathcal{C}$-comodules is
abelian by Lemma 2.1(b)). Consider the following condition on the coring
$\mathcal{C}$:

  * ($*$)

Every left $\mathcal{C}$-comodule is a quotient comodule of an $A$-flat left
$\mathcal{C}$-comodule.

Two classes of corings $\mathcal{C}$ known to satisfy ($*$) are described in
Remark 2.6 and Lemma 2.4 (geometrically, these correspond to quasi-compact
semi-separated schemes and some kind of smooth stacks). We are not aware of
any example of a coring $\mathcal{C}$ over $A$ that is flat as a (left and)
right $A$-module but does not satisfy ($*$).

We will also need a second condition:

  * (${*}{*}$)

Let $\mathcal{M}$ be a left $\mathcal{C}$-comodule such that the underlying
$A$-module of $\mathcal{M}$ has finite projective dimension as an object of
the abelian category $A{\operatorname{\mathsf{--Mod}}}$. Then $\mathcal{M}$
has a finite projective dimension as an object of the abelian category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

######  Remark 5.1.

The corings $\mathcal{C}$ corresponding to quasi-compact semi-separated
schemes $X$ as per Example 2.5 satisfy (${*}{*}$). This is the result of [44,
Theorem 5.3].

On the other hand, when $A=k$ is a field and $\mathcal{C}$ is a coalgebra over
$k$, the condition (${*}{*}$) asks the comodule category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ to have finite homological
dimension. This, of course, does _not_ hold in general. For example, if
$\mathcal{C}$ is a finite-dimensional coalgebra over $k$, then the category of
$\mathcal{C}$-comodules is equivalent to the category of modules over the
$k$-vector space dual finite-dimensional algebra,
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\simeq\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$.
Taking $\mathcal{C}^{*}$ to be a finite-dimensional $k$-algebra of infinite
global dimension, one obtains an example when (${*}{*}$) does not hold.

Informally speaking, (${*}{*}$) means that “the coring $\mathcal{C}$ has
finite homological dimension relative to the ring $A$”. In the context of
algebraic geometry, one can expect this to be satisfied for many stacks in
characteristic zero, such as the quotient stacks $X/G$ of algebraic varieties
$X$ by actions of algebraic groups $G$ (because the homological dimension of
the category of algebraic group representations does not exceed the dimension
of the group, in characteristic zero). On the other hand, in prime
characteristic $p$, the quotient of a point by the action of a finite group of
the order divisible by $p$ does _not_ satisfy (${*}{*}$), as the previous
paragraph explains.

See Remarks 5.7 and 6.6 below for a further discussion.

We denote by $\operatorname{Ext}_{\mathcal{C}}^{*}({-},{-})$ the Ext groups in
the abelian category $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. A left
$\mathcal{C}$-comodule $\mathcal{B}$ is said to be _cotorsion_ [28, Section
B.2]) if $\operatorname{Ext}_{\mathcal{C}}^{1}(\mathcal{F},\mathcal{B})=0$ for
all $A$-flat left $\mathcal{C}$-comodules $\mathcal{F}$. Let
$\mathsf{B}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}^{\mathsf{cot}}$
denote the full subcategory of cotorsion $\mathcal{C}$-comodules in the
abelian category $\mathsf{C}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.
Obviously, the full subcategory
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}^{\mathsf{cot}}$ is closed under
extensions in $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, so it inherits
an exact category structure from the abelian exact structure of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

A class of objects $\mathsf{F}$ in an abelian category $\mathsf{C}$ is said to
be _resolving_ if $\mathsf{F}$ is closed under extensions and kernels of
epimorphisms in $\mathsf{C}$, and every object of $\mathsf{C}$ is a quotient
object of an object from $\mathsf{F}$.

The following corollary is our version of [44, Corollary 5.6 and Lemma 9.1].
Part (a) was already mentioned in the proof of Theorem 3.1.

######  Corollary 5.2.

Let $\mathcal{C}$ be a coring over a ring $A$.

(a) The class of all $A$-flat left $\mathcal{C}$-comodules is closed under
directed colimits in the additive category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

(b) Assume that the condition ($*$) above on the coring $\mathcal{C}$ is
satisfied. Then the class of all $A$-flat left $\mathcal{C}$-comodules is
resolving in the abelian category
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. Consequently, one has
$\operatorname{Ext}_{\mathcal{C}}^{n}(\mathcal{F},\mathcal{B})=0$ for all
$A$-flat left $\mathcal{C}$-comodules $\mathcal{F}$, all cotorsion left
$\mathcal{C}$-comodules $\mathcal{B}$, and all integers $n\geq 1$.

(c) Assume that the condition (${*}{*}$) above on the coring $\mathcal{C}$ is
satisfied. Then any $A$-flat left $\mathcal{C}$-comodule is an
$\aleph_{1}$-directed colimit of $A$-flat $\mathcal{C}$-comodules having
finite projective dimensions as objects of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

###### Proof.

Part (a) follows from Lemma 2.1(a), since the class of all flat modules is
closed under directed colimits in $A{\operatorname{\mathsf{--Mod}}}$. In the
context of part (b), condition ($*$) presumes that $\mathcal{C}$ is a flat
right $A$-module. Then the class of all $A$-flat left $\mathcal{C}$-comodules
is closed under extensions and kernels of epimorphisms in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ by Lemma 2.1(b), since the
class of all flat modules is closed under extensions and kernels of
epimorphisms in $A{\operatorname{\mathsf{--Mod}}}$. Every left
$\mathcal{C}$-comodule is an epimorphic image of an $A$-flat left
$\mathcal{C}$-comodule by the condition ($*$). The final assertion of (b) is
provided by [44, Lemma 6.1].

In part (c), all countably presentable flat $A$-modules have projective
dimensions $\leq 1$ in $A{\operatorname{\mathsf{--Mod}}}$ by Lemma 1.3. Under
condition (${*}{*}$), it follows that all $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules have finite projective dimensions in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. So it remains to refer to
Theorem 3.1. ∎

Asume that $\mathcal{C}$ is a flat right $A$-module. Let
$\mathsf{F}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}_{A{\operatorname{\mathsf{-flat}}}}$
denote the full subcategory of all $A$-flat $\mathcal{C}$-comodules in the
abelian category $\mathsf{C}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.
Since the full subcategory $\mathsf{F}$ is closed under extensions in
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, it inherits an exact category
structure from the abelian exact structure of
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. Since $\mathsf{F}$ is also
closed under directed colimits in $\mathsf{C}$, and the directed colimit
functors are exact in
$\mathsf{C}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, it follows that
$\mathsf{F}$ is an _exact category with exact directed colimits_ in the sense
of [44, Section 7], i. e., all directed colimits exist and directed colimits
of admissible short exact sequences are admissible short exact sequences in
$\mathsf{F}$.

Let $\mathsf{C}$ be an abelian category and $\mathsf{B}\subset\mathsf{C}$ be a
class of objects. An object $M\in\mathsf{C}$ is called _$\mathsf{B}$
-periodic_ if there exists a short exact sequence $0\longrightarrow
M\longrightarrow B\longrightarrow M\longrightarrow 0$ in $\mathsf{C}$ with $M$
as both the rightmost and the leftmost term, and a middle object
$B\in\mathsf{B}$.

The following theorem summarizes the technology of proving cotorsion
periodicity theorems going back to [2] and developed in its present form in
[44].

######  Theorem 5.3.

Let $\mathsf{C}$ be an abelian category with coproducts and
$\mathsf{F}\subset\mathsf{C}$ be a resolving subcategory closed under directed
colimits. Assume that the directed colimit functors in $\mathsf{F}$ take
admissible short exact sequences to admissible short exact sequences (in the
exact category structure on $\mathsf{F}$ inherited from the abelian exact
structure of $\mathsf{C}$, i. e., the short sequences in $\mathsf{F}$ that are
exact in $\mathsf{C}$). Assume further that the category $\mathsf{F}$
coincides with the directed colimit closure of its full subcategory consisting
of all the objects of $\mathsf{F}$ that have finite projective dimensions in
$\mathsf{C}$. Let $\mathsf{B}=\mathsf{F}^{\perp_{1}}\subset\mathsf{C}$ be the
full subcategory of all objects $B\in\mathsf{C}$ such that
$\operatorname{Ext}^{1}_{\mathsf{C}}(F,B)=0$ for all $F\in\mathsf{F}$. Then
any $\mathsf{B}$-periodic object of $\mathsf{C}$ belongs to $\mathsf{B}$.

###### Proof.

This is a particular case of [44, Corollary 8.2]. ∎

Now we can prove our cotorsion periodicity theorem for comodules.

######  Theorem 5.4.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Assume that both
the conditions ($*$) and (${*}{*}$) on the coring $\mathcal{C}$ are satisfied.
Then any cotorsion-periodic left $\mathcal{C}$-comodule is cotorsion.

###### Proof.

This is our generalization of [44, Theorem 9.2]. The assumptions of Theorem
5.3 are satisfied by Corollary 5.2 (cf. the discussion above). ∎

######  Corollary 5.5.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Assume that both
the conditions ($*$) and (${*}{*}$) on the coring $\mathcal{C}$ are satisfied.
Let $\mathcal{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic
complex in $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ whose terms
$\mathcal{B}^{n}$ are cotorsion left $\mathcal{C}$-comodules for all
$n\in\mathbb{Z}$. Then the $\mathcal{C}$-comodules of cocycles of the complex
$\mathcal{B}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are also
cotorsion.

###### Proof.

This is our generalization of [44, Corollary 9.4]. To deduce it from Theorem
5.4, we apply [44, Proposition 8.4]. It remains to check the assumptions of
that proposition from [44].

Indeed, the abelian category $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is
Grothendieck by Lemma 2.1(b), so it has infinite products. The full
subcategory $\mathsf{B}$ of cotorsion left $\mathcal{C}$-comodules is closed
under infinite products in the abelian category
$\mathsf{C}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ by [9, Corollary
8.3] or [10, Corollary A.2]. It is also obvious that $\mathsf{B}$ is closed
under direct summands in $\mathsf{C}$.

Finally, any left $\mathcal{C}$-comodule is a quotient comodule of $A$-flat
$\mathcal{C}$-comodule by the condition ($*$), and any $A$-flat
$\mathcal{C}$-comodule is a quotient comodule of a coproduct of $A$-countably
presentable $A$-flat left $\mathcal{C}$-comodules by Theorem 3.1. The
underlying $A$-modules of the $\mathcal{C}$-comodules of the latter kind have
projective dimension $\leq 1$ by Lemma 1.3. Such $\mathcal{C}$-comodules have
finite projective dimensions in $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$
by the condition (${*}{*}$). Therefore, every left $\mathcal{C}$-comodule is a
quotient comodule of an $A$-flat left $\mathcal{C}$-comodule having finite
projective dimension in $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$. It
remains to recall that
$\operatorname{Ext}_{\mathcal{C}}^{n}(\mathcal{F},\mathcal{B})=0$ for any
$A$-flat left $\mathcal{C}$-comodule $\mathcal{F}$, any left
$\mathcal{C}$-comodule $\mathcal{B}\in\mathsf{B}$, and all integers $n\geq 1$
by Corollary 5.2(b). ∎

######  Corollary 5.6.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Assume that both
the conditions ($*$) and (${*}{*}$) on the coring $\mathcal{C}$ are satisfied.
Then the inclusion of exact/abelian categories
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}^{\mathsf{cot}}\longrightarrow\mathcal{C}{\operatorname{\mathsf{--Comod}}}$
induces an equivalence of their unbounded derived categories,

$\mathsf{D}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}^{\mathsf{cot}})\,\simeq\,\mathsf{D}(\mathcal{C}{\operatorname{\mathsf{--Comod}}}).$

###### Proof.

This is our generalization of [44, Corollary 9.7]. To deduce it from Corollary
5.5, it suffices to refer to [44, Proposition 9.6]. ∎

######  Remark 5.7.

The assumption of condition (${*}{*}$) _cannot_ be dropped in any one of the
assertions of Theorem 5.4, Corollary 5.5, or Corollary 5.6. It suffices to
choose a field $A=k$ and a finite-dimensional coalgebra $\mathcal{C}$ over $k$
whose dual algebra $\mathcal{C}^{*}$ is a Frobenius algebra of infinite global
dimension (e. g., the algebra of dual numbers
$\mathcal{C}^{*}=k[\epsilon]/(\epsilon^{2})$). Then
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}=\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$
(cf. Remark 5.1), all $\mathcal{C}$-comodules are $A$-flat, and the cotorsion
$\mathcal{C}$-comodules are the injective $\mathcal{C}$-comodules, or
equivalently the injective $\mathcal{C}^{*}$-modules. Pick any
$\mathcal{C}^{*}$-module of infinite projective/injective dimension; then its
two-sided projective-injective resolution is an unbounded acyclic complex of
injective $\mathcal{C}$-comodules refuting the conclusions of both Corollaries
5.5 and 5.6. Then it is clear from [8, proof of Proposition 7.6] or [14,
Proposition 2] that the conclusion of Theorem 5.4 does not hold for the
coalgebra/coring $\mathcal{C}$, either.

##  6\. Exact Sequences of $A$-Flat $\mathcal{C}$-Comodules II

In this section we also deduce some results conditional upon the assumptions
($*$) and (${*}{*}$) from Section 5 on a coring $\mathcal{C}$.

We start with an abstract category-theoretic discussion of absolutely acyclic
complexes in exact categories of finite homological dimension. Given an
additive category $\mathsf{E}$, we denote by $\mathsf{K}(\mathsf{E})$ the
triangulated homotopy category of (unbounded cochain complexes in)
$\mathsf{E}$.

Let $\mathsf{E}$ be an exact category. Any (termwise admissible) short exact
sequence of complexes in $\mathsf{E}$ can be viewed as a bicomplex with three
rows, so one can construct its total complex. A complex in $\mathsf{E}$ is
said to be _absolutely acyclic_ [27, Sections 2 and 4], [28, Section A.1],
[35, Section 5.1] if it belongs to the minimal thick subcategory of the
homotopy category $\mathsf{K}(\mathsf{E})$ containing the totalizations of
short exact sequences in $\mathsf{E}$. In other words, a complex in
$\mathsf{E}$ is absolutely acyclic if and only if it can be constructed as a
homotopy direct summand of a complex obtained from the totalizations of short
exact sequences of complexes in $\mathsf{E}$ by passing to cones of closed
morphisms of complexes a finite number of times.

Equivalently, the full subcategory of absolutely acyclic complexes in
$\mathsf{Com}(\mathsf{E})$ can be constructed as the closure of the full
subcategory of contractible complexes under extensions (in the termwise exact
structure on $\mathsf{Com}(\mathsf{E})$) and direct summands. This is
essentially shown in [43, Proposition 8.10].

Denote by $\operatorname{Ext}_{\mathsf{E}}^{*}({-},{-})$ the Yoneda Ext
functor in an exact category $\mathsf{E}$. The exact category $\mathsf{E}$ is
said to have _homological dimension $\leq d$_ (where $d\geq-1$ is an integer)
if $\operatorname{Ext}_{\mathsf{E}}^{n}(X,Y)=0$ for all $X$, $Y\in\mathsf{E}$
and $n>d$.

######  Lemma 6.1.

In an exact category of finite homological dimension, all acyclic complexes
are absolutely acyclic.

###### Proof.

This is [27, Lemma 2.1]. ∎

######  Proposition 6.2.

Let $\mathsf{E}$ be an exact category of homological dimension $\leq d$, where
$d$ is a finite integer. Then any absolutely acyclic complex in $\mathsf{E}$
is a direct summand of the totalization of a $(d+\nobreak 2)$-term exact
complex $0\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{d}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{d-1}\longrightarrow\dotsb\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{0}\longrightarrow
E^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{-1}\longrightarrow 0$ of
complexes in $\mathsf{E}$.

###### Proof.

This result is a particular case of a similar assertion for _exact DG-
categories_ , which is essentially proved in [35, proof of Proposition 8.8].
More precisely, as a particular case of [35, proof of Proposition 8.8]
(corresponding to [35, Examples 4.39 and 6.1(4)]) one obtains the claim that
any absolutely acyclic complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in $\mathsf{E}$ is a
_homotopy direct summand_ of the totalization
$T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of a $(d+\nobreak 2)$-term
exact complex of complexes in $\mathsf{E}$. Here a “homotopy direct summand”
means a direct summand in the homotopy category $\mathsf{K}(\mathsf{E})$. It
remains to show that a homotopy direct summand can be turned into a strict
direct summand.

Now we have a pair of closed morphisms of complexes $i\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
T^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and $p\colon
T^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ such that the composition
$pi\colon A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to the identity
morphism ${\mathrm{id}}\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. It follows that the
difference $pi-{\mathrm{id}}\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ factorizes as
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, where
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is the cone of identity
morphism ${\mathrm{id}}\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is the natural closed
morphism, while
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is some arbitrary closed
morphism. Hence the identity morphism ${\mathrm{id}}\colon
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ factorizes through the
direct sum $T^{\text{\smaller\smaller$\scriptstyle\bullet$}}\oplus
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as a morphism in the
category of complexes in $\mathsf{E}$. Finally, we point out that
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is the totalization of a
$2$-term exact complex $0\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ of
complexes in $\mathsf{E}$. This takes care of all cases with $d\geq 0$, while
in the trivial case of $d=-1$ we have $\mathsf{E}=0$ and there is nothing to
prove. ∎

Let $\mathsf{F}$ be an exact category. A class of objects
$\mathsf{G}\subset\mathsf{F}$ is said to be _self-generating_ [44, Section 6]
if for any admissible epimorphism $F\longrightarrow H$ in $\mathsf{F}$ with
$H\in\mathsf{G}$ there exists a morphism $G\longrightarrow F$ in $\mathsf{F}$
with $G\in\mathsf{G}$ such that the composition $G\longrightarrow
F\longrightarrow H$ is an admissible epimorphism in $\mathsf{F}$. A class of
objects $\mathsf{G}$ in $\mathsf{F}$ is said to be _self-resolving_ [35,
Section 7.1], [44, Section 8] if it is self-generating, closed under
extensions, and closed under kernels of admissible epimorphisms.

Clearly, any self-resolving full subcategory $\mathsf{G}$ in an exact category
$\mathsf{F}$ interits an exact category structure from $\mathsf{F}$ (since
$\mathsf{G}$ is closed under extensions in $\mathsf{F}$). The following lemma
is straightforward and well-known.

######  Lemma 6.3.

Let $\mathsf{F}$ be an exact category and $\mathsf{G}\subset\mathsf{F}$ be a
self-resolving full subcategory. Then the inclusion of exact categories
$\mathsf{G}\longrightarrow\mathsf{F}$ induces isomorphisms on the
$\operatorname{Ext}$ groups,

$\operatorname{Ext}_{\mathsf{G}}^{n}(X,Y)\simeq\operatorname{Ext}_{\mathsf{F}}^{n}(X,Y)\qquad\text{for
all\/ $X$, $Y\in\mathsf{G}$ and\/ $n\geq 0$}.$

###### Proof.

See [19, Section 12] or [28, Proposition A.2.1]. ∎

Now we return to comodules over a coring $\mathcal{C}$ over an associative
ring $A$.

######  Corollary 6.4.

Let $\mathcal{C}$ be a coring over an associative ring $A$ such that
$\mathcal{C}$ is a flat right $A$-module.

(a) The full subcategory $\mathsf{G}$ of all $A$-countably presentable
$A$-flat left $\mathcal{C}$-comodules is self-resolving in the exact category
$\mathsf{F}$ of all $A$-flat left $\mathcal{C}$-comodules.

(b) If the conditions ($*$) and (${*}{*}$) on the coring $\mathcal{C}$ are
satisfied, then the exact category $\mathsf{G}$ of $A$-countably presentable
$A$-flat left $\mathcal{C}$-comodules has finite homological dimension.

###### Proof.

In part (a), the assertion that $\mathsf{G}$ is self-generating in
$\mathsf{F}$ is an easy corollary of Theorem 3.1. Let $\mathcal{H}$ be an
$A$-countably presentable $A$-flat left $\mathcal{C}$-comodule, $\mathcal{F}$
be an $A$-flat $\mathcal{C}$-comodule, and
$\mathcal{F}\longrightarrow\mathcal{H}$ be a surjective $\mathcal{C}$-comodule
morphism. By Theorem 3.1, there exists a directed (actually, an
$\aleph_{1}$-directed) poset $\Xi$ and a $\Xi$-indexed diagram of
$A$-countably presentable $A$-flat $\mathcal{C}$-comodules
$(\mathcal{G}_{\xi})_{\xi\in\Xi}$ for which
$\mathcal{F}=\varinjlim_{\xi\in\Xi}\mathcal{G}_{\xi}$. Then it is clear that
there exists an index $\xi_{0}\in\Xi$ such that the composition
$\mathcal{G}_{\xi_{0}}\longrightarrow\mathcal{F}\longrightarrow\mathcal{H}$ is
surjective. The full subcategory $\mathsf{G}$ is closed under kernels of
surjective homomorphisms in $\mathsf{F}$ by Lemma 4.1, and the assertion that
$\mathsf{G}$ is closed under extensions in $\mathsf{F}$ is provable similarly
to Corollary 5.2(b).

In part (b), the full subcategory $\mathsf{G}$ is self-resolving in
$\mathsf{F}$ by part (a), and the full subcategory $\mathsf{F}$ is resolving
in the abelian category
$\mathsf{C}=\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ by Corollary 5.2(b).
Consequently, $\mathsf{G}$ is self-resolving in $\mathsf{C}$. The underlying
$A$-modules of all the $\mathcal{C}$-comodules from $\mathsf{G}$ have
projective dimensions at most $1$ by Lemma 1.3, hence the objects of
$\mathsf{G}$ has finite projective dimensions in $\mathsf{C}$ by condition
(${*}{*}$). Moreover, the latter projective dimensions must be uniformly
bounded by some fixed integer $d$, since the class of objects $\mathsf{G}$ is
closed under countable coproducts in $\mathsf{C}$. Finally, Lemma 6.3 tells
that the Ext groups computed in $\mathsf{G}$ and in $\mathsf{C}$ agree. Thus
the homological dimension of the exact category $\mathsf{G}$ cannot exceed
$d$. ∎

The following theorem is a comodule version of [13, Theorem 2.4 (1)
$\Leftrightarrow$ (3)] or [25, Theorem 8.6 (ii) $\Leftrightarrow$ (iii)]. It
should be also compared to [43, Corollary 0.4 or Proposition 8.11]. In the
latter context, one can say that Theorem 6.5 means that (under the conditions
($*$) and (${*}{*}$)) _for the exact category of $A$-flat
$\mathcal{C}$-comodules, the derived category coincides with the_ (_suitably
defined_) _coderived category_.

######  Theorem 6.5.

Let $\mathcal{C}$ be a coring over an associative ring $A$. Assume that both
the conditions ($*$) and (${*}{*}$) from Section 5 on the coring $\mathcal{C}$
are satisfied. Then there exists a finite integer $d\geq 0$ such that the
following classes of complexes of left $\mathcal{C}$-comodules coincide:

  1. (1)

all $A$-pure acyclic complexes of $A$-flat $\mathcal{C}$-comodules;

  2. (2)

the closure of the class of all contractible complexes of $A$-countably
presentable $A$-flat $\mathcal{C}$-comodules under extensions and directed
colimits;

  3. (3)

all $\aleph_{1}$-directed colimits of totalizations of $(d+\nobreak 2)$-term
exact complexes of complexes of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules.

###### Proof.

One can easily see that the totalization of a $(d+\nobreak 2)$-term exact
complex of complexes is a finitely iterated extension of $d$ cones of identity
endomorphisms of complexes. Furthermore, $\aleph_{1}$-directed colimits are a
subclass of directed colimits, the class of all $A$-pure acyclic complexes of
$A$-flat $\mathcal{C}$-comodules is closed under extensions and directed
colimits, and the contractible complexes of $A$-flat $\mathcal{C}$-comodules
are $A$-pure acyclic. This proves the inclusions (3) $\subset$ (2) $\subset$
(1).

Conversely, to prove that (1) $\subset$ (3), recall that all $A$-pure acyclic
complexes of $A$-flat $\mathcal{C}$-comodules are $\aleph_{1}$-directed
colimits of $A$-pure acyclic complexes of $A$-countably presentable $A$-flat
$\mathcal{C}$-comodules by Corollary 4.5. The exact category $\mathsf{G}$ of
$A$-countably presentable $A$-flat $\mathcal{C}$-comodules has finite
homological dimension by Corollary 6.4(b). Hence the $A$-pure acyclic
complexes of $A$-countably presentable $A$-flat $\mathcal{C}$-comodules are
absolutely acyclic by Lemma 6.1, and it remains to refer to Proposition 6.2. ∎

######  Remark 6.6.

Without the assumption (${*}{*}$), the conclusion of Theorem 6.5 certainly
does _not_ hold (cf. Remarks 5.1 and 5.7). In fact, for a coalgebra
$\mathcal{C}$ over a field $A=k$, if the global dimension of $\mathcal{C}$ is
infinite, then the coderived category of $\mathcal{C}$-comodules is usually
quite different from their derived category [36, Section 7].

##  7\. Preliminaries on Topological Rings and Contramodules

The material of this section goes back to [27, Remark A.3], [28, Appendix D],
[29, Section 2.1], and [40, Section 5]. For more recent references, see [32,
Section 2] or [31, Section 2]. The exposition in [41, Sections 6–7] may be the
most accessible one for a beginner. In this paper, we are mostly interested in
contramodules over complete, separated topological associative rings with a
countable base of neighborhoods of zero consisting of open two-sided ideals.

Let $\mathfrak{A}$ be a topological abelian group where open subgroups form a
base of neighborhoods of zero. Consider the natural map to the directed limit
$\lambda_{\mathfrak{A}}\colon\mathfrak{A}\longrightarrow\varprojlim_{\mathfrak{U}\subset\mathfrak{A}}\mathfrak{A}/\mathfrak{U}$,
where $\mathfrak{U}$ ranges over the open subgroups of $\mathfrak{A}$. The
topological group $\mathfrak{A}$ is said to be _separated_ if the map
$\lambda_{\mathfrak{A}}$ is injective, and _complete_ if
$\lambda_{\mathfrak{A}}$ is surjective.

Let $\mathfrak{R}$ be a complete, separated topological ring with a base of
neighborhoods of zero consisting of open right ideals. A right
$\mathfrak{R}$-module $\mathcal{N}$ is said to be _discrete_ if the action map
$\mathcal{N}\times\mathfrak{R}\longrightarrow\mathcal{N}$ is continuous in the
given topology of $\mathfrak{R}$ and the discrete topology on $\mathcal{N}$.
Equivalently, a right $\mathfrak{R}$-module $\mathcal{N}$ is discrete if, for
every element $x\in\mathcal{N}$, the annihilator of $x$ in $\mathfrak{R}$ is
an open right ideal. The full subcategory of discrete right
$\mathfrak{R}$-modules
${\operatorname{\mathsf{Discr--}}}\mathfrak{R}\subset{\operatorname{\mathsf{Mod--}}}\mathfrak{R}$
is a Grothendieck abelian category. The inclusion functor
${\operatorname{\mathsf{Discr--}}}\mathfrak{R}\longrightarrow{\operatorname{\mathsf{Mod--}}}\mathfrak{R}$
is exact and preserves all colimits.

Given an abelian group $A$ and a set $X$, we use $A[X]=A^{(X)}$ as a notation
for the direct sum of $X$ copies of $A$. The elements of $A[X]$ are
interpreted as finitely supported formal linear combinations $\sum_{x\in
X}a_{x}x$ of elements of $X$ with the coefficients $a_{x}\in A$ (where
$a_{x}=0$ for all but a finite subset of indices $x\in X$).

Let $\mathfrak{A}$ be a complete, separated topological abelian group with a
base of neighborhoods of zero consisting of open subgroups. Given a set $X$,
we denote by $\mathfrak{A}[[X]]$ the directed limit of abelian groups
$\varprojlim_{\mathfrak{U}\subset\mathfrak{A}}(\mathfrak{A}/\mathfrak{U})[X]$
(where $\mathfrak{U}$ ranges over the open subgroups of $\mathfrak{A}$). The
elements of $\mathfrak{A}[[X]]$ are interpreted as infinite formal linear
combinations $\sum_{x\in X}a_{x}x$ with the families of coefficients
$a_{x}\in\mathfrak{A}$ _converging to zero in the topology of $\mathfrak{A}$_.
The latter condition means that, for every open subgroup
$\mathfrak{U}\subset\mathfrak{A}$, one has $a_{x}\in\mathfrak{U}$ for all but
a finite subset of indices $x\in X$.

Given a map of sets $X\longrightarrow Y$, the induced (pushforward) map
$\mathfrak{A}[[f]]\colon\mathfrak{A}[[X]]\longrightarrow\mathfrak{A}[[Y]]$ is
defined by the rule $\sum_{x\in X}a_{x}x\longmapsto\sum_{y\in Y}b_{y}y$, where
$b_{y}=\sum_{x\in X}^{f(x)=y}a_{x}$ for every $y\in Y$. Here the latter
infinite sum (defining the element $b_{y}\in\mathfrak{A}$) is _not_ formal;
rather, it is understood as the limit of finite partial sums in the topology
of $\mathfrak{A}$. Due to this construction, the assigment
$X\longmapsto\mathfrak{A}[[X]]$ is a covariant functor
$\mathsf{Sets}\longrightarrow\mathsf{Ab}$ from the category of sets to the
category of abelian groups. We will mostly view it as an endofunctor on the
category of sets,
$\mathfrak{A}[[{-}]]\colon\mathsf{Sets}\longrightarrow\mathsf{Sets}$, ignoring
the abelian group structures on the sets $\mathfrak{A}[[X]]$.

Let $\mathfrak{R}$ be a complete, separated topological ring where open right
ideals form a base of neighborhoods of zero. Then the functor
$\mathfrak{R}[[{-}]]\colon\mathsf{Sets}\longrightarrow\mathsf{Sets}$ acquires
a natural structure of a _monad_ on the category of sets. We refer to [23,
Chapter VI] (see also [41, Section 6]) for the background discussion of monads
and algebras over monads. Specifically, for any set $X$, the monad unit
$\epsilon_{X}\colon X\longrightarrow\mathfrak{R}[[X]]$ is the obvious natural
map taking every element $x\in X$ to the formal linear combination $\sum_{y\in
X}r_{y}y$, where $r_{x}=1$ and $r_{y}=0$ for all $y\neq x$.

The monad multiplication
$\phi_{X}\colon\mathfrak{R}[[\mathfrak{R}[[X]]]]\longrightarrow\mathfrak{R}[[X]]$
is the natural “opening of parentheses” map producing a formal linear
combination from a formal linear combination of formal linear combinations.
This construction involves taking products of pairs of elements in
$\mathfrak{R}$ and infinite sums of zero-convergent families of elements. The
conditions that $\mathfrak{R}$ is complete, separated, and open right ideals
form a base of neighborhoods of zero guarantee the convergence.

_Left $\mathfrak{R}$-contramodules_ are defined as algebras over the monad
$(\mathfrak{R}[[{-}]],\epsilon,\phi)$ on the category of sets. We usually
prefer to speak of _modules_ (rather than algebras) _over monads_ in this
context, as contramodules are more similar to modules over rings than to
algebras. So, specifically, a left $\mathfrak{R}$-contramodule $\mathfrak{P}$
is a set endowed with a _left contraaction_ map
$\pi\colon\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}$ satisfying
the following _contraassociativity_ and _contraunitality_ axioms. The two
compositions $\pi\circ\phi_{\mathfrak{P}}$ and $\pi\circ\mathfrak{R}[[\pi]]$
must be equal to each other,

$\mathfrak{R}[[\mathfrak{R}[[\mathfrak{P}]]]]\rightrightarrows\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P},$

and the composition $\pi\circ\epsilon_{\mathfrak{P}}$ must be ideal to the
identity map ${\mathrm{id}}_{\mathfrak{P}}$,

$\mathfrak{P}\longrightarrow\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}.$

We denote the category of left $\mathfrak{R}$-contramodules by
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

Restricting the contraaction map
$\pi\colon\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}$ to the
subset of finitely supported formal linear combinations
$\mathfrak{R}[\mathfrak{P}]\subset\mathfrak{R}[[\mathfrak{P}]]$, one
constructs the underlying left $\mathfrak{R}$-module structure on a left
$\mathfrak{R}$-contramodule $\mathfrak{P}$ [41, Sections 6.1–6.2]. Hence the
forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Mod}}}$.
The category of left $\mathfrak{R}$-contramodules is abelian. All set-indexed
products and coproducts (hence also limits and colimits) exist in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. The forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Mod}}}$
is exact, faithful, and preserves infinite products. We use the notation
$\operatorname{Hom}^{\mathfrak{R}}({-},{-})$ for the groups of morphisms in
the abelian category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

For any set $X$, the set/abelian group $\mathfrak{R}[[X]]$ has a natural left
$\mathfrak{R}$-contramodule structure with the contraaction map
$\pi=\phi_{X}\colon\mathfrak{R}[[\mathfrak{R}[[X]]]]\longrightarrow\mathfrak{R}[[X]]$.
The $\mathfrak{R}$-contramodules $\mathfrak{R}[[X]]$ are called the _free_
left $\mathfrak{R}$-contramodules. For any left $\mathfrak{R}$-contramodule
$\mathfrak{P}$, the $\mathfrak{R}$-contramodule morphisms
$\mathfrak{R}[[X]]\longrightarrow\mathfrak{P}$ correspond bijectively to maps
of sets $X\longrightarrow\mathfrak{P}$; so there is a natural isomorphism of
abelian groups

$\operatorname{Hom}^{\mathfrak{R}}(\mathfrak{R}[[X]],\mathfrak{P})\simeq\operatorname{Hom}_{\mathsf{Sets}}(X,\mathfrak{P}).$

There are enough projective objects in the abelian category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$; the projective
$\mathfrak{R}$-contramodules are precisely the direct summands of the free
ones.

Given a left $\mathfrak{R}$-contramodule $\mathfrak{P}$ and a closed subgroup
$\mathfrak{A}\subset\mathfrak{R}$, we denote by
$\mathfrak{A}\rightthreetimes\mathfrak{P}\subset\mathfrak{P}$ the image of the
composition
$\mathfrak{A}[[\mathfrak{P}]]\longrightarrow\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}$
of the natural inclusion
$\mathfrak{A}[[\mathfrak{P}]]\hookrightarrow\mathfrak{R}[[\mathfrak{P}]]$ with
the contraaction map
$\pi\colon\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}$. So
$\mathfrak{A}\rightthreetimes\mathfrak{P}$ is a subgroup in $\mathfrak{P}$
(since $\pi$ is a left $\mathfrak{R}$-contramodule morphism, hence in
particular an abelian group homomorphism, and $\mathfrak{A}[[\mathfrak{P}]]$
is subgroup in $\mathfrak{R}[[\mathfrak{P}]]$). If
$\mathfrak{J}\subset\mathfrak{R}$ is a closed left ideal, then
$\mathfrak{J}\rightthreetimes\mathfrak{P}$ is an
$\mathfrak{R}$-subcontramodule in $\mathfrak{P}$ (since
$\mathfrak{J}[[\mathfrak{P}]]$ is an $\mathfrak{R}$-subcontramodule in the
free contramodule $\mathfrak{R}[[\mathfrak{P}]]$).

Let $\mathcal{N}$ be a discrete right $\mathfrak{R}$-module and $\mathfrak{P}$
be a left $\mathfrak{R}$-contramodule. Then the _contratensor product_
$\mathcal{N}\odot_{\mathfrak{R}}\mathfrak{P}$ is the abelian group constructed
as the cokernel of (the difference of) the natural pair of maps

$\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{R}[[\mathfrak{P}]]\,\rightrightarrows\,\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{P}.$

Here the first map
$\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{P}$
is induced by the contraaction map
$\pi\colon\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathfrak{P}$, while the
second map
$\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{R}[[\mathfrak{P}]]\longrightarrow\mathcal{N}\otimes_{\mathbb{Z}}\mathfrak{P}$
is defined by the rule

$y\otimes\sum\nolimits_{p\in\mathfrak{P}}r_{p}p\longmapsto\sum\nolimits_{p\in\mathfrak{P}}yr_{p}\otimes
p$

for all $y\in\mathcal{N}$ and
$\sum_{p\in\mathfrak{P}}r_{p}p\in\mathfrak{R}[[\mathfrak{P}]]$. Here the sum
in the right-hand side is well-defined, because one has $yr_{p}=0$ in
$\mathcal{N}$ for all but a finite subset of indices $p\in\mathfrak{P}$.

The contratensor product functor
$\odot_{\mathfrak{R}}\colon{\operatorname{\mathsf{Discr--}}}\mathfrak{R}\times\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{Ab}$
preserves all colimits in both the arguments. For any discrete right
$\mathfrak{R}$-module $\mathcal{N}$ and any set $X$, one has a natural
isomorphism of abelian groups

$\mathcal{N}\odot_{\mathfrak{R}}\mathfrak{R}[[X]]\simeq\mathcal{N}[X]=\mathcal{N}^{(X)}.$

For any closed right ideal $\mathfrak{J}\subset\mathfrak{R}$, one has

$\mathfrak{J}\rightthreetimes(\mathfrak{R}[[X]])=\mathfrak{J}[[X]]\subset\mathfrak{R}[[X]].$

For any open right ideal $\mathfrak{I}\subset\mathfrak{R}$ and any left
$\mathfrak{R}$-contamodule $\mathfrak{P}$, one has

$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{P}\simeq\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P}).$

In particular,
$(\mathfrak{R}/\mathfrak{I})\odot_{\mathfrak{R}}\mathfrak{R}[[X]]\simeq\mathfrak{R}[[X]]/(\mathfrak{I}\rightthreetimes\mathfrak{R}[[X]])=\mathfrak{R}[[X]]/\mathfrak{I}[[X]]\simeq(\mathfrak{R}/\mathfrak{I})[X]$.

For any open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$ and any left
$\mathfrak{R}$-contramodule $\mathfrak{P}$, the quotient group
$\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$ has a natural module
structure over the quotient ring $\mathfrak{R}/\mathfrak{I}$. The subgroup
$\mathfrak{I}\rightthreetimes\mathfrak{P}$ is an
$\mathfrak{R}$-subcontramodule in $\mathfrak{P}$, as mentioned above; and the
quotient $\mathfrak{R}$-contramodule structure on
$\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$ comes from the
$\mathfrak{R}/\mathfrak{I}$-module structure.

##  8\. Separated and Flat Contramodules

In this section we discuss contramodules over a topological ring with a
_countable_ base of neighborhoods of zero. There are no new results in this
section, which is based on [28, Section D.1] and [40, Section 6]. A brief
survey is available in [29, Section 3.3]. A more recent exposition can be
found in [31, Sections 4–5]. A connection with algebraic geometry is discussed
in [34, Chapter 3].

Given a left $\mathfrak{R}$-contramodule $\mathfrak{P}$, consider the natural
map to the directed limit
$\lambda_{\mathfrak{R},\mathfrak{P}}\colon\mathfrak{P}\longrightarrow\varprojlim_{\mathfrak{I}\subset\mathfrak{R}}\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$,
where $\mathfrak{I}$ ranges over the open right ideals of $\mathfrak{R}$. A
left $\mathfrak{R}$-contramodule $\mathfrak{P}$ is said to be _separated_ if
the map $\lambda_{\mathfrak{R},\mathfrak{P}}$ is injective, and _complete_ if
$\lambda_{\mathfrak{R},\mathfrak{P}}$ is surjective. We denote full
subcategory of separated $\mathfrak{R}$-contamodules by
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}\subset\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

######  Lemma 8.1.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then all left $\mathfrak{R}$-contramodules are complete (but they _need not_
be separated).

###### Proof.

This goes back, at least, to [27, Lemma A.2.3 and Remark A.3]. Counterexamples
of nonseparated contramodules appeared in [27, Section A.1] (see also [46,
Example 2.5], [51, Example 3.20], [29, Section 1.5], [30, Example 2.7(1)], and
a further discussion in Remark 8.5). The completeness assertion for
contramodules over topological rings with a countable topology base consisting
of two-sided ideals can be found in [28, Lemma D.1.1], and the general case of
a countable topology base of right ideals is covered by [40, Lemma 6.3(b)].
See also [29, Lemmas 3.6 and 3.9]. ∎

######  Lemma 8.2.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then, for any nonzero left $\mathfrak{R}$-contramodule $\mathfrak{P}$, there
exists an open right ideal $\mathfrak{I}\subset\mathfrak{R}$ such that
$\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})\neq 0$.

###### Proof.

This assertion belongs to a class of results known as “contramodule Nakayama
lemmas” and going back, at least, to [27, Lemma A.2.1]. The assertion for
contramodules over topological rings with a countable topology base of two-
sided ideals can be found in [28, Lemma D.1.2], and the general case of a
countable base of right ideals is covered by [40, Lemma 6.14]. See also [29,
Lemmas 2.1(b) and 3.22]. ∎

A left $\mathfrak{R}$-contramodule $\mathfrak{F}$ is called _flat_ if the
contratensor product functor
${-}\odot_{\mathfrak{R}}\nobreak\mathfrak{F}\colon\allowbreak{\operatorname{\mathsf{Discr--}}}\mathfrak{R}\longrightarrow\mathsf{Ab}$
is exact on the abelian category of discrete right $\mathfrak{R}$-modules. All
projective $\mathfrak{R}$-contramodules are separated and flat. All directed
colimits and coproducts of flat $\mathfrak{R}$-contramodules (computed in the
category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$) are flat. We denote
the full subcategory of flat $\mathfrak{R}$-contramodules by
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}\subset\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

When $\mathfrak{R}$ has a base of neighborhoods of zero consisting of open
two-sided ideals $\mathfrak{I}$, the discrete right $\mathfrak{R}$-modules are
the directed unions of modules over the quotient rings
$\mathfrak{R}/\mathfrak{I}$. For any right $\mathfrak{R}/\mathfrak{I}$-module
$N$ and any left $\mathfrak{R}$-contramodule $\mathfrak{P}$, one has

$N\odot_{\mathfrak{R}}\mathfrak{P}\simeq
N\otimes_{\mathfrak{R}/\mathfrak{I}}\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P}).$

Hence, in this case, a left $\mathfrak{R}$-contramodule $\mathfrak{F}$ is flat
if and only if the left $\mathfrak{R}/\mathfrak{I}$-module
$\mathfrak{F}/(\mathfrak{I}\rightthreetimes\mathfrak{F})$ is flat for every
open two-sided ideal $\mathfrak{I}\subset\mathfrak{R}$.

######  Lemma 8.3.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then all _flat_ left $\mathfrak{R}$-contramodules are separated.

###### Proof.

The case of topological rings with a countable topology base of two-sided
ideals can be found in [28, Lemma D.1.7] (cf. the definition of a “flat
contramodule” in [28, the paragraph before Lemma D.1.4], which is stated
differently than our definition above). The general case of a countable base
of right ideals is covered by [40, Corollary 6.15]. See also [29, Propositions
3.7(a) and 3.10(a)]. ∎

######  Lemma 8.4.

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

(a) the class of all flat left $\mathfrak{R}$-contramodules is closed under
extensions and kernels of epimorphisms in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$;

(b) for any discrete right $\mathfrak{R}$-module $\mathcal{N}$, the
contratensor product functor
$\mathcal{N}\odot_{\mathfrak{R}}\nobreak{-}\,\colon\allowbreak\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{Ab}$
takes short exact sequences of flat left $\mathfrak{R}$-contramodules to short
exact sequences of abelian groups.

###### Proof.

The case of topological rings with a countable topology base of two-sided
ideals can be found in [28, Lemmas D.1.4 and D.1.5]. The general case of a
countable base of right ideals is covered by [40, Lemma 6.7 or 6.10,
Corollaries 6.8, 6.13 and 6.15], with a summary in [40, Corollary 7.1(a–b)].
See also [29, Propositions 3.7(b) and 3.10(b)]. ∎

######  Remark 8.5.

Lemma 8.4 tells that the class of flat $\mathfrak{R}$-contramodules is well-
behaved (under the assumption of a countable topology base in $\mathfrak{R}$).
But the class of separated $\mathfrak{R}$-contramodules is _not_ well-behaved.

In suffices to consider the case of the ring of $p$-adic integers
$\mathfrak{R}=\mathbb{Z}_{p}$ (in the $p$-adic topology), or the ring of
formal Taylor power series $\mathfrak{R}=k[[t]]$ in one variable $t$ over a
field $k$ (in the $t$-adic topology). In both cases, the forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Mod}}}$
is fully faithful; in fact, even the forgetful functors
$\mathbb{Z}_{p}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{Ab}$
and $k[[t]]{\operatorname{\mathsf{--Contra}}}\longrightarrow
k[t]{\operatorname{\mathsf{--Mod}}}$ are fully faithful. The essential images
of these forgetful functors are described in [29, Section 2.2] and the
references therein.

With these descriptions in mind, there is a now-classical counterexample
showing that the the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ is not closed
under extensions in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ [46,
Example 2.5] and the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ is _not_
abelian [30, Example 2.7(1)]. Furthermore, the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ is _not_
closed under directed colimits, and in fact, _not_ even closed under
coproducts in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ [29, Section
1.5]. (See also [34, Example 3.1] and Example 8.9 below.)

However, separated contramodules are sometimes easier to work with than
arbitrary ones, as the following lemma illustrates.

######  Lemma 8.6.

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 $\mathfrak{I}_{n}\subset\mathfrak{R}$, $n\geq 1$.

(a) The full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ of separated
$\mathfrak{R}$-contramodules $\mathfrak{M}$ in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ is equivalent to the category
formed by the following sets of data:

  1. (1)

for every $n\geq 1$, a left $\mathfrak{R}/\mathfrak{I}_{n}$-module $M_{n}$ is
given; and

  2. (2)

for every $n\geq 1$, an isomorphism of left
$\mathfrak{R}/\mathfrak{I}_{n}$-modules

$M_{n}\,\simeq\,\mathfrak{R}/\mathfrak{I}_{n}\otimes_{\mathfrak{R}/\mathfrak{I}_{n+1}}M_{n+1}$

is given.

(b) The full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ of flat
$\mathfrak{R}$-contramodules $\mathfrak{F}$ in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ is equivalent to the category
formed by the following sets of data:

  1. (1)

for every $n\geq 1$, a flat left $\mathfrak{R}/\mathfrak{I}_{n}$-module
$F_{n}$ is given; and

  2. (2)

for every $n\geq 1$, an isomorphism of left
$\mathfrak{R}/\mathfrak{I}_{n}$-modules

$F_{n}\,\simeq\,\mathfrak{R}/\mathfrak{I}_{n}\otimes_{\mathfrak{R}/\mathfrak{I}_{n+1}}F_{n+1}$

is given.

Here, in both parts (a) and (b), morphisms between the described sets of data
are defined in the obvious way as collections module morphisms forming
commutative diagrams with the reduction isomorphisms.

###### Proof.

In part (a), the equivalence of categories assigns to a separated left
$\mathfrak{R}$-contramodule $\mathfrak{M}$ the collection of modules
$M_{n}=(\mathfrak{R}/\mathfrak{I}_{n})\odot_{\mathfrak{R}}\mathfrak{M}=\mathfrak{M}/(\mathfrak{I}_{n}\rightthreetimes\mathfrak{M})$,
endowed with the obvious reduction isomorphisms. The inverse functor assigns
to a collection of modules $M_{n}$ endowed with reduction isomorphisms the
separated $\mathfrak{R}$-contramodule $\mathfrak{M}=\varprojlim_{n\geq
0}M_{n}$. Here the $\mathfrak{R}/\mathfrak{I}_{n}$-modules $M_{n}$ are viewed
as $\mathfrak{R}$-contramodules via the natural surjective ring homomorphisms
$\mathfrak{R}\longrightarrow\mathfrak{R}/\mathfrak{I}_{n}$, and the directed
limit can be computed in the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ (as the forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Mod}}}$
preserves all limits). The assertion that these rules indeed define mutually
inverse equivalences of categories is [28, Lemma D.1.3] (see [40, Corollary
6.4] for a generalization to topological rings with a countable base of right
ideals). Lemma 8.1 needs to be used here.

The equivalence of categories in part (b) is obtained by restricting the
equivalence of part (a) to the respective full subcategories on both sides.
Lemma 8.3 plays an important role here. ∎

######  Lemma 8.7.

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
a left $\mathfrak{R}$-contramodule $\mathfrak{P}$ is projective if and only if
(it is separated and flat and) the corresponding reduction datum in Lemma 8.6
consists of projective modules, that is, if and only if the
$\mathfrak{R}/\mathfrak{I}_{n}$-module
$\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$ is projective for
every $n\geq 1$.

###### Proof.

This is [28, Corollary D.1.10(a)]; see also [29, Proposition 3.8]. ∎

The category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ is better
behaved than $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$
in that $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ is abelian (cf.
Remark 8.5). Still, the directed colimit functors are _not_ exact in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$, as one can see, e. g., from
[40, Examples 4.4]. The next lemma tells that the directed colimit functors
_are_ exact in the exact category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{flat}}$ (assuming a
countable base of neighborhoods of zero in $\mathfrak{R}$).

######  Lemma 8.8.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then the directed colimit of any directed diagram of short exact sequences of
_flat_ left $\mathfrak{R}$-contramodules (computed in the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$) is a short exact sequence of
(flat) left $\mathfrak{R}$-contramodules.

###### Proof.

In the case of a topological ring with a countable topology base of two-sided
ideals, the assertion is easily deduced from the fact that the class of all
flat left $\mathfrak{R}$-contramodules is closed under directed colimits in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ and the description of flat
left $\mathfrak{R}$-contramodules in Lemma 8.6(b) together with Lemma 8.4(b).
The general case of a countable base of right ideals is covered by [40, Lemma
6.16] (see also [40, Proposition 6.17, Remark 6.18, and Corollary 7.1(c)] and
[29, Proposition 3.10(c)]). ∎

######  Example 8.9.

In the context of algebraic geometry, a _strict ind-affine $\aleph_{0}$-ind-
scheme_ $\mathfrak{X}$ is an ind-object in the category of affine schemes
representable by a countable directed diagram of closed immersions of affine
schemes $X_{1}\longrightarrow X_{2}\longrightarrow
X_{2}\longrightarrow\dotsb$. So one has $X_{n}=\operatorname{Spec}R_{n}$ for
some commutative rings $R_{n}$, and the scheme morphisms $X_{n}\longrightarrow
X_{n+1}$ correspond to surjective ring homomorphisms $R_{n+1}\longrightarrow
R_{n}$.

The category of strict ind-affine $\aleph_{0}$-ind-schemes is equivalent to
the category of complete, separated topological commutative rings having a
countable base of neighborhoods of zero consisting of two-sided ideals. To an
ind-scheme
$\mathfrak{X}=\mathop{\text{``$\varinjlim$''\\!\\!}}\nolimits_{n\geq 1}X_{n}$
as above, the topological ring $\mathfrak{R}=\varprojlim_{n\geq 0}R_{n}$ (with
the topology of projective limit of discrete rings $R_{n}$) is assigned [3,
Example 7.11.2(i)], [34, Example 1.6(2)].

In this context, the sets of data from Lemma 8.6(a) are called
“$\mathcal{O}^{p}$-modules on $\mathfrak{X}$” in [3, Section 7.11.3] or _pro-
quasi-coherent pro-sheaves on $\mathfrak{X}$_ in [34, Section 3.1]. The sets
of data from Lemma 8.6(b) are called “flat $\mathcal{O}^{p}$-modules on
$\mathfrak{X}$” in [3, Section 7.11.3] or _flat pro-quasi-coherent pro-sheaves
on $\mathfrak{X}$_ in [34, Section 3.4]. So the category of pro-quasi-coherent
pro-sheaves on $\mathfrak{X}$ is equivalent to the category of separated
$\mathfrak{R}$-contramodules, while the category of flat pro-quasi-coherent
pro-sheaves on $\mathfrak{X}$ is equivalent to the category of flat
$\mathfrak{R}$-contramodules. We refer to [3, Remark 7.11.3(ii)] and [34,
Examples 3.1 and 3.8] for further discussion. For a discussion of non-ind-
affine $\aleph_{0}$-ind-schemes with further details, see Remark 10.3 below.

##  9\. Countably Presentable Flat Contramodules

We refer to Section 1 above for the definitions of _$\kappa$ -presentable
objects_, _locally $\kappa$-presentable categories_, and _$\kappa$ -accessible
categories_, with the references to [1].

######  Proposition 9.1.

Let $\kappa$ be an uncountable regular cardinal and $\mathfrak{R}$ be a
complete, separated topological ring with a base of neighborhoods of zero of
the cardinality less than $\kappa$, consisting of open right ideals.

(a) The category of left $\mathfrak{R}$-contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ is locally
$\kappa$-presentable. A left $\mathfrak{R}$-contramodule $\mathfrak{P}$ is
$\kappa$-presentable as an object of the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ if and only if it is the
cokernel of a morphism of free left $\mathfrak{R}$-contramodules
$\mathfrak{R}[[Y]]\longrightarrow\mathfrak{R}[[X]]$ with the sets $X$ and $Y$
having cardinalities less than $\kappa$.

(b) The category of complexes of left $\mathfrak{R}$-contramodules
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}})$ is locally
$\kappa$-presentable. A complex of left $\mathfrak{R}$-contramodules
$\mathfrak{P}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
$\kappa$-presentable as an object of the category
$\mathsf{Com}(\mathfrak{R}{\operatorname{\mathsf{--Contra}}})$ if and only if
all its terms $\mathfrak{P}^{n}$, $n\in\mathbb{Z}$, are $\kappa$-presentable
$\mathfrak{R}$-contramodules.

###### Proof.

Part (a): the “if” assertion is easily provable using the fact that the
forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Mod}}}$
preserves $\kappa$-directed colimits. The latter follows from the observation
that any zero-convergent family of elements in $\mathfrak{R}$ has cardinality
less than $\kappa$. To prove the “only if” assertion, one then needs to check
that all left $\mathfrak{R}$-contramodules are $\kappa$-directed colimits of
the cokernels of morphisms of free contramodules with less than $\kappa$
generators, and the class of all such cokernels is closed under direct
summands. The first assertion of part (a) follows immediately (cf. the
discussion in [40, Sections 1.1 and 5] and [41, Sections 6.2 and 6.4]).

Part (b): the “if” assertion follows from the “if” assertion of part (a) by
Lemma 1.1(b). To deduce the “only if”, one can check directly that all
complexes of left $\mathfrak{R}$-contramodules are $\kappa$-directed colimits
of complexes whose terms are the cokernels of morphisms of free contramodules
with less than $\kappa$-generators. The first assertion is then also clear
(cf. [42, second paragraph of the proof of Lemma 6.3]). Alternatively, one can
refer to Theorem 1.9 or a suitable additive version of [18, Theorem 1.2],
similarly to the argument in Proposition 3.3 and Remark 3.4. ∎

Now let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals.
Then the fully faithful inclusion functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
has a left adjoint functor
$\Lambda_{\mathfrak{R}}\colon\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$
assigning to every left $\mathfrak{R}$-contramodule $\mathfrak{P}$ its maximal
separated quotient $\mathfrak{R}$-contramodule

$\Lambda_{\mathfrak{R}}(\mathfrak{P})=\varprojlim\nolimits_{\mathfrak{I}\subset\mathfrak{R}}\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P}).$

It is explained in [40, Lemma 6.2] or [31, Proposition 4.2(b)] how to endow
the directed limit
$\varprojlim_{\mathfrak{I}\subset\mathfrak{R}}\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$
computed in the category of abelian groups with a left
$\mathfrak{R}$-contramodule structure. The natural morphism
$\lambda_{\mathfrak{R},\mathfrak{P}}\colon\mathfrak{P}\longrightarrow\varprojlim_{\mathfrak{I}\subset\mathfrak{R}}\mathfrak{P}/(\mathfrak{I}\rightthreetimes\mathfrak{P})$
is the adjunction unit.

######  Lemma 9.2.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals. Let
$\mathfrak{P}$ be a left $\mathfrak{R}$-contramodule such that the separated
left $\mathfrak{R}$-contramodule $\Lambda_{\mathfrak{R}}(\mathfrak{P})$ is
flat. Then the $\mathfrak{R}$-contramodule $\mathfrak{P}$ is flat and
separated, and the natural morphism
$\lambda_{\mathfrak{R},\mathfrak{P}}\colon\mathfrak{P}\longrightarrow\Lambda_{\mathfrak{R}}(\mathfrak{P})$
is an isomorphism.

###### Proof.

This is an intermediate step in the proofs of [28, Corollary D.1.7] and [40,
Corollary 6.15] (which are restated above as Lemma 8.3). The argument is based
on the contramodule Nakayama lemma (Lemma 8.2 above). Alternatively, the same
proof can be phrased as follows. One observes that, for any left
$\mathfrak{R}$-contramodule $\mathfrak{P}$, the adjunction morphism
$\lambda_{\mathfrak{R},\mathfrak{P}}$ induces an isomorphism of the
contratensor product functors
${-}\odot_{\mathfrak{R}}\mathfrak{P}\simeq{-}\odot_{\mathfrak{R}}\Lambda_{\mathfrak{R}}(\mathfrak{P})$
(essentially, the nonseparated part of $\mathfrak{P}$ is killed by
contratensor products). This shows that flatness of
$\Lambda_{\mathfrak{R}}(\mathfrak{P})$ implies flatness of $\mathfrak{P}$.
Then the separatedness and completeness of $\mathfrak{P}$ follow from Lemmas
8.1 and 8.3. ∎

######  Proposition 9.3.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals, and
let $\kappa$ be an uncountable regular cardinal. Then the category of
separated left $\mathfrak{R}$-contramodules
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ is locally
$\kappa$-presentable. A separated left $\mathfrak{R}$-contramodule
$\mathfrak{M}$ is $\kappa$-presentable as an object of the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ if and only if
it has the form $\mathfrak{M}=\Lambda_{\mathfrak{R}}(\mathfrak{P})$, where
$\mathfrak{P}$ is a $\kappa$-presentable object of the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$.

###### Proof.

One needs to observe that the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}\subset\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
is closed under $\aleph_{1}$-directed colimits. Indeed, for any closed
subgroup $\mathfrak{A}\subset\mathfrak{R}$, the functor
$\mathfrak{P}\longmapsto\mathfrak{A}\rightthreetimes\mathfrak{P}$ commutes
with $\aleph_{1}$-directed colimits (since the functor
$X\longmapsto\mathfrak{A}[[X]]\colon\mathsf{Sets}\longrightarrow\mathsf{Sets}$
does), and countable directed limits also commute with $\aleph_{1}$-directed
colimits. Consequently, the inclusion functor
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$
preserves $\kappa$-directed colimits, and it follows that the left adjoint
functor
$\Lambda_{\mathfrak{R}}\colon\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$
takes $\kappa$-presentable objects to $\kappa$-presentable objects. This
proves the “if” assertion.

Furthermore, the functor $\Lambda_{\mathfrak{R}}$ being a left adjoint,
preserves all colimits. Hence all colimits exist in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ and can be
constructed by applying $\Lambda_{\mathfrak{R}}$ to the colimits in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. Since all objects of
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ are $\kappa$-directed
colimits of $\kappa$-presentable objects by Proposition 9.1(a), so are all the
objects of $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$.
Thus the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ is locally
$\kappa$-presentable.

To deduce the assertion “only if”, we notice that all the objects of
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ are
$\kappa$-directed colimits of separated contramodules of the desired form
$\mathfrak{M}=\Lambda_{\mathfrak{R}}(\mathfrak{P})$ with
$\mathfrak{P}\in\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{<\kappa}$, as
we have just shown. It remains to check that the class of all separated
contramodules of the desired form
$\mathfrak{M}=\Lambda_{\mathfrak{R}}(\mathfrak{P})$ is closed under direct
summands. Let us prove this class is closed under cokernels in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$.

Suppose given a morphism $g\colon\mathfrak{M}\longrightarrow\mathfrak{N}$,
where $\mathfrak{M}=\Lambda_{\mathfrak{R}}(\mathfrak{P})$,
$\mathfrak{N}=\Lambda_{\mathfrak{R}}(\mathfrak{Q})$, and $\mathfrak{P}$,
$\mathfrak{Q}\in\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{<\kappa}$.
Then the natural (adjunction) morphisms
$\mathfrak{P}\longrightarrow\mathfrak{M}$ and
$\mathfrak{Q}\longrightarrow\mathfrak{N}$ are surjective. Let
$\mathfrak{F}\longrightarrow\mathfrak{P}$ be a surjective morphism onto
$\mathfrak{P}$ from a free left $\mathfrak{R}$-contramodule with less than
$\kappa$ generators. Then the composition
$\mathfrak{F}\longrightarrow\mathfrak{P}\longrightarrow\mathfrak{M}\longrightarrow\mathfrak{N}$
can be lifted to a morphism $f\colon\mathfrak{F}\longrightarrow\mathfrak{Q}$.
The cokernel of the morphism $g$ in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ can be
obtained by applying $\Lambda_{\mathfrak{R}}$ to the cokernel of the morphism
$f$ in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. The latter cokernel
is $\kappa$-presentable in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. ∎

######  Lemma 9.4.

Let $\mathfrak{R}$ be a complete, separated topological ring with a
_countable_ base of neighborhoods of zero consisting of open right ideals, and
let $\kappa$ be an uncountable regular cardinal. Then a _flat_ left
$\mathfrak{R}$-contramodule is $\kappa$-presentable as an object of the
abelian category $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$ if and only
if it is $\kappa$-presentable as an object of the category
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$.

###### Proof.

The “only if” assertion holds, since the functor
$\Lambda_{\mathfrak{R}}\colon\mathfrak{R}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$
takes $\kappa$-presentable objects to $\kappa$-presentable objects, as
explained in the proof of Proposition 9.3, and flat contramodules are
separated by Lemma 8.3.

The proof of “if” is based on Lemma 9.2 and the “only if” assertion of
Proposition 9.3. If a left $\mathfrak{R}$-contramodule $\mathfrak{F}$ is
$\kappa$-presentable in
$\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$, then
Proposition 9.3 tells that
$\mathfrak{F}=\Lambda_{\mathfrak{R}}(\mathfrak{P})$, where $\mathfrak{P}$ is
$\kappa$-presentable in $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}$. Now
if $\mathfrak{F}$ is flat, then $\mathfrak{P}\simeq\mathfrak{F}$ by Lemma 9.2.
∎

######  Lemma 9.5.

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 $\mathfrak{I}_{n}\subset\mathfrak{R}$, $n\geq 1$. Let
$\kappa$ be an uncountable regular cardinal. Then a separated
$\mathfrak{R}$-contramodule $\mathfrak{M}$ is $\kappa$-presentable as an
object of $\mathfrak{R}{\operatorname{\mathsf{--Contra}}}_{\mathsf{sep}}$ if
and only if the $\mathfrak{R}/\mathfrak{I}_{n}$-module
$M_{n}=\mathfrak{M}/(\mathfrak{I}_{n}\rightthreetimes\mathfrak{M})$ is
$\kappa$-presentable for every $n\geq 1$. Consequently, a flat