# Homological full-and-faithfulness
of comodule inclusion
and contramodule forgetful functors

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

In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let
$\Upsilon\colon C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ be the natural functor of inclusion of
the category of $C$-comodules into the category of $C^{*}$-modules, and let
$\Theta\colon C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ be the natural forgetful functor. We
prove that the functor $\Upsilon$ induces a fully faithful triangulated
functor on bounded (below) derived categories if and only if the functor
$\Theta$ induces a fully faithful triangulated functor on bounded (above)
derived categories, and if and only if the $k$-vector space
$\operatorname{Ext}_{C}^{n}(k,k)$ is finite-dimensional for all $n\geq 0$. We
call such coalgebras “weakly finitely Koszul”.

###### Contents

  1. 1 Preliminaries on Coalgebras, Comodules, and Contramodules
  2. 2 Conilpotent Coalgebras and Minimal Resolutions
  3. 3 The Left and Right Comodule and Contramodule Ext Comparison
  4. 4 Weakly Finite Koszulity Implies Comodule Ext Isomorphism
  5. 5 Comodule Ext Isomorphism Implies Weakly Finite Koszulity
  6. 6 Weakly Finite Koszulity Implies Contramodule Ext Isomorphism
  7. 7 Contramodule Ext Isomorphism Implies Weakly Finite Koszulity
  8. 8 Half-Bounded Derived Full-and-Faithfulness
  9. 9 Co-Noetherian and Cocoherent Conilpotent Coalgebras
  10. 10 Cocommutative Conilpotent Coalgebras

## Introduction

In this paper we work with coassociative, counital coalgebras over a field
$k$. For any such coalgebra $C$, the dual $k$-vector space $C^{*}$ is
naturally an associative, unital algebra over $k$. One has to choose between
two opposite ways of defining the multiplication on $C^{*}$. We prefer the
notation in which any left $C$-comodule becomes a left $C^{*}$-module, and any
right $C$-comodule becomes a right $C^{*}$-module.

It is known, at least, since 1960s that the resulting exact functor
$\Upsilon\colon C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is fully faithful [41, Section 2.1].
Following the terminology of the book [41], $C^{*}$-modules coming from
$C$-comodules are often called “rational” in the literature. The essential
image of the functor $\Upsilon$ is a _hereditary pretorsion class_ in
$C^{*}{\operatorname{\mathsf{--Mod}}}$: this means that the full subcategory
$\Upsilon(C{\operatorname{\mathsf{--Comod}}})$ is closed under subobjects,
quotient objects, and infinite coproducts in
$C^{*}{\operatorname{\mathsf{--Mod}}}$. However, this full subcategory need
_not_ be closed under extensions. In other words,
$\Upsilon(C{\operatorname{\mathsf{--Comod}}})$ is _not_ always a torsion class
or a Serre subcategory in $C^{*}{\operatorname{\mathsf{--Mod}}}$. A module
extension of two comodules need not be a comodule.

When is the essential image of $\Upsilon$ closed under extensions in
$C^{*}{\operatorname{\mathsf{--Mod}}}$ ? There is a vast body of literature on
this topic, including the papers [38, 40, 15, 4, 3, 43, 10]. In this paper we
discuss further questions going in this direction, under an additional
assumption. The assumption is that the coalgebra $C$ is _conilpotent_. The
conilpotent coalgebras were called “pointed irreducible” in the terminology of
[41]. For a conilpotent coalgebra $C$, the full subcategory
$C{\operatorname{\mathsf{--Comod}}}$ is closed under extensions in
$C^{*}{\operatorname{\mathsf{--Mod}}}$ if and only if the coalgebra $C$ is
_finitely cogenerated_ [38, Corollary 2.4 and Section 2.5], [40, Theorem 4.6],
[15, Corollary 21], [4, Theorem 2.8], [3, Proposition 3.13 and Corollary
3.14], [10, Lemma 1.2 and Theorem 4.8]. If $C$ is not finitely cogenerated,
then there is a two-dimensional $C^{*}$-module which is not a $C$-comodule,
but an extension of two one-dimensional $C$-comodules.

Let $\mathsf{A}$ and $\mathsf{B}$ be two abelian categories, and
$\Phi\colon\mathsf{B}\longrightarrow\mathsf{A}$ be a fully faithful exact
functor. Then the essential image of $\Phi$ is a full subcategory closed under
kernels and cokernels in $\mathsf{A}$. The full subcategory $\Phi(\mathsf{B})$
is closed under extensions in $\mathsf{A}$ if and only if the functor $\Phi$
induces an isomorphism

(1) $\Phi\colon\operatorname{Ext}^{1}_{\mathsf{B}}(X,Y)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{1}_{\mathsf{A}}(\Phi(X),\Phi(Y))$

for all objects $X$, $Y\in\mathsf{B}$. Generally speaking, for a fully
faithful exact functor $\Phi$, the map (1) is a monomorphism, but not
necessarily an isomorphism.

Thus, the following question is a natural extension of the question about
extension closedness of $\Upsilon(C{\operatorname{\mathsf{--Comod}}})$ in
$C^{*}{\operatorname{\mathsf{--Mod}}}$. Put
$\mathsf{B}=C{\operatorname{\mathsf{--Comod}}}$,
$\mathsf{A}=C^{*}{\operatorname{\mathsf{--Mod}}}$, and $\Phi=\Upsilon$.
Consider the induced maps on the Ext spaces

(2) $\Phi\colon\operatorname{Ext}^{i}_{\mathsf{B}}(X,Y)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{i}_{\mathsf{A}}(\Phi(X),\Phi(Y)).$

When is the map (2) in isomorphism for all $X$, $Y\in\mathsf{B}$ ? Generally
speaking, for an exact functor of abelian categories
$\Phi\colon\mathsf{B}\longrightarrow\mathsf{A}$, all one can say is that the
map (2) is a monomorphism for $i=n+1$ and all $X$, $Y\in\mathsf{B}$ whenever
it is an isomorphism for $i=n$ and all $X$, $Y\in\mathsf{B}$.

Let $C$ be a conilpotent coalgebra over a field $k$. Then the one-dimensional
$k$-vector space $k$ has a unique left $C$-comodule structure (and a unique
right $C$-comodule structure) provided by the unique coaugmentation of $C$. In
this context, we show that the maps (2) are isomorphisms for $\Phi=\Upsilon$
and all $1\leq i\leq n$ if and only if the $k$-vector spaces
$\operatorname{Ext}^{i}_{C}(k,k)$ (computed in the abelian category of left or
right $C$-comodules) are finite-dimensional for all $1\leq i\leq n$. In
particular, $\operatorname{Ext}^{1}_{C}(k,k)$ is the vector space of
cogenerators of a conilpotent coalgebra $C$; so $C$ is finitely cogenerated if
and only if $\operatorname{Ext}^{1}_{C}(k,k)$ is finite-dimensional.

Alongside with the abelian categories of left and right _comodules_ over a
coalgebra $C$, there are much less familiar, but no less natural abelian
categories of left and right _$C$ -contramodules_ [22]. Endowing the dual
vector space $C^{*}$ to a coalgebra $C$ with the natural algebra structure in
which any left $C$-comodule is a left $C^{*}$-module and any right
$C$-comodule is a right $C^{*}$-module, one also obtains a natural left
$C^{*}$-module structure on any left $C$-contramodule. So there is an exact
forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$.

The functor $\Theta$ is _not_ in general fully faithful. It was shown in the
paper [28, Theorem 2.1] that that the functor $\Theta$ is fully faithful for
any finitely cogenerated conilpotent coalgebra $C$. In this paper we
demonstrate a counterexample proving the (much easier) converse implication:
if a conilpotent coalgebra $C$ is _not_ finitely cogenerated, then the functor
$\Theta$ is _not_ fully faithful.

More generally, the maps (2) are isomorphisms for $\Phi=\Theta$ and all $0\leq
i\leq n-1$ (for separated contramodules $Y$; and also for $0\leq i\leq n-2$
and arbitrary $Y$) if and only if the $k$-vector spaces
$\operatorname{Ext}^{i}_{C}(k,k)$ are finite-dimensional for all $1\leq i\leq
n$. Summarizing the assertions for comodules and contramodules, let us state
the following theorem.

######  Theorem 0.1.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 1$ be an
integer. Then the following five conditions are equivalent:

  1. (i)

the map

$\operatorname{Ext}^{i}_{C}(L,M)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{i}_{C^{*}}(L,M)$

induced by the inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is an isomorphism for all left
$C$-comodules $L$ and $M$, and all $1\leq i\leq n$;

  2. (ii)

the map

$\operatorname{Ext}^{i}_{C^{\mathrm{op}}}(L,M)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{i}_{C^{*}{}^{\mathrm{op}}}(L,M)$

induced by the inclusion functor
${\operatorname{\mathsf{Comod--}}}C\longrightarrow{\operatorname{\mathsf{Mod--}}}C^{*}$
is an isomorphism for all right $C$-comodules $L$ and $M$, and all $1\leq
i\leq n$;

  3. (iii)

the map

(3) $\operatorname{Ext}^{C,i}(P,Q)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{i}_{C^{*}}(P,Q)$

induced by the forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is an isomorphism for all left
$C$-contramodules $P$, all _separated_ left $C$-contramodules $Q$, and all
$0\leq i\leq n-1$;

  4. (iv)

the $k$-vector space $\operatorname{Ext}^{i}_{C}(k,k)$ is finite-dimensional
for all $1\leq i\leq n$;

  5. (v)

the $k$-vector space $\operatorname{Ext}^{C,i}(k,k)$ is finite-dimensional for
all $1\leq i\leq n$.

If any one of the equivalent conditions (i–v) holds, then the map (3) is an
isomorphism for _all_ left $C$-contramodules $P$ and $Q$ and all $0\leq i\leq
n-2$.

Notice a curious cohomological dimension shift in comparison between the
assertions about the comodule and contramodule inclusion/forgetful functors.
Let $n\geq 1$ be the minimal integer for which the vector space
$\operatorname{Ext}^{n}_{C}(k,k)$ is infinite-dimensional. Then the map
$\operatorname{Ext}^{n}_{C}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(k,k)$
induced by the functor $\Upsilon$ is injective, but not surjective. In fact,
the dimension cardinality of the vector space
$\operatorname{Ext}^{n}_{C^{*}}(k,k)$ is larger than that of
$\operatorname{Ext}^{n}_{C}(k,k)$ in this case:
$\operatorname{Ext}^{n}_{C^{*}}(k,k)$ is as large as the double dual vector
space $\operatorname{Ext}^{n}_{C}(k,k)^{**}$ to
$\operatorname{Ext}^{n}_{C}(k,k)$.

At the same time, denoting by $T$ an infinite-dimensional $k$-vector space
endowed with the trivial $C$-contramodule structure, the map
$\operatorname{Ext}^{C,n}(T,k)\longrightarrow\operatorname{Ext}_{C^{*}}^{n}(T,k)$
induced by the functor $\Theta$ is _not injective_ (for the integer $n$ as in
the previous paragraph). Consequently, there exists a projective
$C$-contramodule $P$ such that the map
$\operatorname{Ext}^{C,n-1}(P,k)\longrightarrow\operatorname{Ext}_{C^{*}}^{n-1}(P,k)$
is injective, but not surjective. For $n>1$ this, of course, means that
$\operatorname{Ext}_{C^{*}}^{n-1}(P,k)\neq 0$, while
$\operatorname{Ext}^{C,n-1}(P,k)=0$ as it should be. Notice that both the
contramodules $P$ and $k$ are separated.

Let us emphasize that _we do not know_ whether the equivalent conditions (i–v)
of Theorem 0.1 imply bijectivity of the maps (3) for arbitrary (nonseparated)
contramodules $Q$ and $i=n-1$. This remains an open question.

Returning to the discussion of a fully faithful exact functor
$\Phi\colon\mathsf{B}\longrightarrow\mathsf{A}$, it is clear that the maps of
Ext groups (2) induced by $\Phi$ are isomorphisms if and only if the induced
triangulated functor between the bounded derived categories
$\Phi^{\mathsf{b}}\colon\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
is fully faithful. Now assume that there are enough injective objects in the
abelian category $\mathsf{A}$ and the functor
$\Phi\colon\mathsf{B}\longrightarrow\mathsf{A}$ has a right adjoint. Then the
functor $\Phi^{\mathsf{b}}$ is fully faithful if and only if the similar
functor between the bounded below derived categories
$\Phi^{+}\colon\mathsf{D}^{+}(\mathsf{B})\longrightarrow\mathsf{D}^{+}(\mathsf{A})$
is fully faithful.

Dually, assume that there are enough projective objects in $\mathsf{A}$ and
the functor $\Phi\colon\mathsf{B}\longrightarrow\mathsf{A}$ has a left
adjoint. Then the functor $\Phi^{\mathsf{b}}$ is fully faithful if and only if
the similar functor between the bounded above derived categories
$\Phi^{-}\colon\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
is fully faithful [29, Proposition 6.5].

In the situation at hand, there are enough projective and injective objects in
the categories of modules of associative rings. The comodule inclusion functor
$\Upsilon\colon C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ has a right adjoint functor
$\Gamma\colon C^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
C{\operatorname{\mathsf{--Comod}}}$, while the contramodule forgetful functor
$\Theta\colon C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ has a left adjoint functor $\Delta\colon
C^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
C{\operatorname{\mathsf{--Contra}}}$. Consequently, Theorem 0.1 implies the
following result about full-and-faithfulness of induced triangulated functors.

######  Theorem 0.2.

For any conilpotent coalgebra $C$ over a field $k$, the following eight
conditions are equivalent:

  1. (i)

the triangulated functor
$\Upsilon^{\mathsf{b}}\colon\mathsf{D}^{\mathsf{b}}(C{\operatorname{\mathsf{--Comod}}})\longrightarrow\mathsf{D}^{\mathsf{b}}(C^{*}{\operatorname{\mathsf{--Mod}}})$
induced by the comodule inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is fully faithful;

  2. (ii)

the triangulated functor
$\Upsilon^{+}\colon\mathsf{D}^{+}(C{\operatorname{\mathsf{--Comod}}})\longrightarrow\mathsf{D}^{+}(C^{*}{\operatorname{\mathsf{--Mod}}})$
induced by the comodule inclusion functor $\Upsilon$ is fully faithful;

  3. (iii)

the triangulated functor
$\mathsf{D}^{\mathsf{b}}({\operatorname{\mathsf{Comod--}}}C)\longrightarrow\mathsf{D}^{\mathsf{b}}({\operatorname{\mathsf{Mod--}}}C^{*})$
induced by the comodule inclusion functor
${\operatorname{\mathsf{Comod--}}}C\longrightarrow{\operatorname{\mathsf{Mod--}}}C^{*}$
is fully faithful;

  4. (iv)

the triangulated functor
$\mathsf{D}^{+}({\operatorname{\mathsf{Comod--}}}C)\longrightarrow\mathsf{D}^{+}({\operatorname{\mathsf{Mod--}}}C^{*})$
induced by the comodule inclusion functor is fully faithful;

  5. (v)

the triangulated functor
$\Theta^{\mathsf{b}}\colon\mathsf{D}^{\mathsf{b}}(C{\operatorname{\mathsf{--Contra}}})\longrightarrow\mathsf{D}^{\mathsf{b}}(C^{*}{\operatorname{\mathsf{--Mod}}})$
induced by the contramodule forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is fully faithful;

  6. (vi)

the triangulated functor
$\Theta^{-}\colon\mathsf{D}^{-}(C{\operatorname{\mathsf{--Contra}}})\longrightarrow\mathsf{D}^{-}(C^{*}{\operatorname{\mathsf{--Mod}}})$
induced by the contramodule forgetful functor $\Theta$ is fully faithful;

  7. (vii)

the $k$-vector space $\operatorname{Ext}^{n}_{C}(k,k)$ is finite-dimensional
for all $n\geq 0$;

  8. (viii)

the $k$-vector space $\operatorname{Ext}^{C,n}(k,k)$ is finite-dimensional for
all $n\geq 0$.

Let us discuss the condition of finite-dimensionality of the Ext spaces
$\operatorname{Ext}_{C}^{i}(k,k)$ for all $i\geq 0$ in some detail. One can
consider the special case when the coalgebra $C$ is positively graded with
finite-dimensional components; so $C$ is the graded dual coalgebra to a
positively graded algebra $A=\bigoplus_{m=0}^{\infty}A_{m}$ with $\dim
A_{m}<\infty$ for all $m\geq 0$ and $A_{0}=k$. Then one has
$\operatorname{Ext}_{C}^{i}(k,k)\simeq\bigoplus_{j=i}^{\infty}\operatorname{Ext}_{A}^{i,j}(k,k)$
and
$\operatorname{Ext}_{A}^{i}(k,k)\simeq\prod_{j=i}^{\infty}\operatorname{Ext}_{A}^{i,j}(k,k)$,
where $i$ is the usual cohomological grading on the Ext spaces, while the
_internal grading_ $j$ is induced by the grading on $A$ (cf. [17, Section 1 of
Chapter 1], [33, Section 2.1], and [19, Section 2]).

Assume further that $A$ is multiplicatively generated by $A_{1}$ with
relations of degree $2$; so $A$ is a quadratic algebra with finite-dimensional
components over $k$. Then the vector spaces $\operatorname{Ext}_{C}^{i}(k,k)$
or $\operatorname{Ext}_{A}^{i}(k,k)$ still need _not_ be finite-dimensional;
in fact, $\operatorname{Ext}_{C}^{3}(k,k)$ or (equivalently)
$\operatorname{Ext}_{A}^{3}(k,k)$ can be infinite-dimensional already [1,
Theorem 7.6], [7], [17, Section 6 of Chapter 6].

Nevertheless, for a (_homogeneous_) _Koszul_ $k$-algebra $A$ with finite-
dimensional grading components [37, 17] and the graded dual coalgebra $C$, the
vector spaces $\operatorname{Ext}_{C}^{i}(k,k)$ and
$\operatorname{Ext}_{A}^{i}(k,k)$ are, of course, finite-dimensional for all
$i\geq 0$. More generally, for a finitely cogenerated conilpotent coalgebra
$C$, all the vector spaces $\operatorname{Ext}_{C}^{i}(k,k)$ are finite-
dimensional whenever the Ext-algebra $\operatorname{Ext}_{C}^{*}(k,k)$ is
generated by $\operatorname{Ext}_{C}^{1}(k,k)$. This includes the important
particular case when the Ext-algebra $\operatorname{Ext}_{C}^{*}(k,k)$ is
Koszul (but the coalgebra $C$ need not be graded) described in [36, Main
Theorem] and [24, Sections 5–6]. Here we keep assuming that the coalgebra $C$
is finitely cogenerated.

With these important special cases in mind, we call a conilpotent coalgebra
$C$ _weakly finitely Koszul_ if the vector space
$\operatorname{Ext}_{C}^{n}(k,k)$ is finite-dimensional for every $n\geq 0$.
This terminology goes back to [33, Sections 5.4 and 5.7].

Let us emphasize that _we do not know_ whether the induced triangulated
functors between the unbounded derived categories

$\Upsilon^{\varnothing}\colon\mathsf{D}(C{\operatorname{\mathsf{--Comod}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}(C^{*}{\operatorname{\mathsf{--Mod}}})\quad\text{and}\quad\Theta^{\varnothing}\colon\mathsf{D}(C{\operatorname{\mathsf{--Contra}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}(C^{*}{\operatorname{\mathsf{--Mod}}})$

are fully faithful under the equivalent conditions of Theorem 0.2 (or even for
a graded coalgebra $C$ graded dual to a Koszul algebra $A$ with finite-
dimensional components) in general. However, the triangulated functors
$\Upsilon^{\varnothing}$ and $\Theta^{\varnothing}$ are known to be fully
faithful for any finitely cogenerated conilpotent _cocommutative_ coalgebra
$C$. This is a particular case of [23, Theorems 1.3 and 2.9] (applied to the
complete Noetherian commutative local $k$-algebra $R=C^{*}$ with its maximal
ideal $I=\mathfrak{m}$).

In this connection, it should first of all be mentioned that all finitely
cogenerated conilpotent cocommutative coalgebras $C$ are weakly finitely
Koszul since they are Artinian, and consequently, co-Noetherian [27, Section
2]. Quite generally, any left or right co-Noetherian conilpotent coalgebra is
weakly finitely Koszul (and so is any finitely cogenerated left or right
cocoherent coalgebra). But this does not seem to be enough. The proofs of [23,
Theorems 1.3 and 2.9] are essentially based on the observation that _the right
adjoint functor $\Gamma$ to $\Upsilon$ and the left adjoint functor $\Delta$
to $\Theta$ have finite homological dimensions_ (cf. [26, Theorem 6.4] and
[29, Proposition 6.5]). We cannot think of any general noncommutative versions
of these properties, proved in [18] and [23] using commutative Koszul
complexes.

### Acknowledgement

I am grateful to Victor Roca i Lucio and Teresa Conde for stimulating
correspondence. I also wish to thank Jan Št’ovíček for helpful conversations.
The author is supported by the GAČR project 23-05148S and the Czech Academy of
Sciences (RVO 67985840).

##  1\. Preliminaries on Coalgebras, Comodules, and Contramodules

Unless otherwise mentioned, all _coalgebras_ , _comodules_ , and
_contramodules_ in this paper are presumed to be coassociaitve and counital;
all _coalgebra homomorphisms_ are presumed to preserve the counit. Dually, all
_rings_ , _algebras_ , and _modules_ are presumed to be associative and
unital.

A (coassociative, counital) _coalgebra_ $C$ over a field $k$ is a $k$-vector
space endowed with $k$-linear maps of _comultiplication_ $\mu\colon
C\longrightarrow C\otimes_{k}C$ and _counit_ $\epsilon\colon C\longrightarrow
k$ satisfying the usual coassociativity and counitality axioms (which can be
obtained by writing down the definition of an associative, unital $k$-algebra
in the tensor notation and inverting the arrows). We suggest the books [41,
16] as the standard reference sources on coalgebras over a field. The present
author’s surveys [22, Section 1], [34, Section 3] can be used as additional
reference sources.

Let $C$ be a coalgebra over $k$. A _right $C$-comodule_ $N$ is a $k$-vector
space endowed with a $k$-linear map of _right coaction_ $\nu\colon
N\longrightarrow N\otimes_{k}C$ satisfying the usual coassociativity and
counitality axioms (which can be obtained by inverting the arrows in the
definition of module over an associative, unital algebra). Similarly, a _left
$C$-comodule_ $M$ is a $k$-vector space endowed with a $k$-linear map of _left
coaction_ $\nu\colon M\longrightarrow C\otimes_{k}M$ satisfying the
coassociativity and counitality axioms.

A _left $C$-contramodule_ (see [5, Section III.5], [20, Sections 0.2.4 and
3.1.1, and Appendix A], [22, Sections 1.1–1.6], and [34, Section 8]) is a
$k$-vector space $P$ endowed with a $k$-linear map of _left contraaction_
$\pi\colon\operatorname{Hom}_{k}(C,P)\longrightarrow P$ satisfying the
following _contraassociativity_ and _contraunitality_ axioms. Firstly, the two
maps $\operatorname{Hom}(C,\pi)$ and
$\operatorname{Hom}(\mu,P)\colon\operatorname{Hom}_{k}(C\otimes_{k}C,\>P)\simeq\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(C,P))\longrightarrow\operatorname{Hom}_{k}(C,P)$
must have equal compositions with the contraaction map
$\pi\colon\operatorname{Hom}_{k}(C,P)\longrightarrow P$,

$\operatorname{Hom}_{k}(C\otimes_{k}C,\>P)\simeq\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(C,P))\,\rightrightarrows\,\operatorname{Hom}_{k}(C,P)\longrightarrow
P.$

Secondly, the composition of the map $\operatorname{Hom}(\epsilon,P)\colon
P\longrightarrow\operatorname{Hom}_{k}(C,P)$ with the map
$\pi\colon\operatorname{Hom}_{k}(C,P)\longrightarrow P$ must be equal to the
identity endomorphism of $P$,

$P\longrightarrow\operatorname{Hom}_{k}(C,P)\longrightarrow P.$

Here the $k$-vector space isomorphism
$\operatorname{Hom}_{k}(C\otimes_{k}C,\>P)\simeq\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(C,P))$
is obtained as a particular case of the adjunction isomorphism
$\operatorname{Hom}_{k}(U\otimes_{k}V,\>W)\simeq\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,W))$,
which holds for any vector spaces $U$, $V$, and $W$.

The definition of a _right $C$-contramodule_ is similar, with the only
difference that the isomorphism
$\operatorname{Hom}_{k}(C\otimes_{k}C,\>P)\simeq\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(C,P))$
arising as a particular case of the identification
$\operatorname{Hom}_{k}(V\otimes_{k}U,\>W)\simeq\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,W))$
is used.

For any right $C$-comodule $N$ and any $k$-vector space $V$, the vector space
$\operatorname{Hom}_{k}(N,V)$ has a natural left $C$-contramodule structure.
The left contraaction map

$\pi\colon\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(N,V))\simeq\operatorname{Hom}_{k}(N\otimes_{k}C,\>V)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(N,V)$

is constructed by applying the functor $\operatorname{Hom}_{k}({-},V)$ to the
right coaction map $\nu\colon N\longrightarrow N\otimes_{k}C$, i. e.,
$\pi=\operatorname{Hom}_{k}(\nu,V)$.

Let $C^{\mathrm{op}}$ denote the opposite coalgebra to $C$ (i. e., the same
vector space endowed with the same counit and the left-right opposite
comultiplication to that in $C$). Similarly, given a ring $A$, we denote by
$A^{\mathrm{op}}$ the opposite ring. Then right $C$-comodules are the same
things as left $C^{\mathrm{op}}$-comodules, and vice versa.

We will use the notation $\operatorname{Hom}_{C}({-},{-})$ for the vector
spaces of morphisms in the category of left $C$-comodules
$C{\operatorname{\mathsf{--Comod}}}$ and the notation
$\operatorname{Hom}_{C^{\mathrm{op}}}({-},{-})$ for the vector spaces of
morphisms in the category of right $C$-comodules
${\operatorname{\mathsf{Comod--}}}C$. The vector spaces of morphisms in the
left and right contramodule categories $C{\operatorname{\mathsf{--Contra}}}$
and ${\operatorname{\mathsf{Contra--}}}C$ will be denoted by
$\operatorname{Hom}^{C}({-},{-})$ and
$\operatorname{Hom}^{C^{\mathrm{op}}}({-},{-})$. Similarly, the Yoneda Ext
spaces are denoted by $\operatorname{Ext}_{C}^{*}({-},{-})$ in
$C{\operatorname{\mathsf{--Comod}}}$, by
$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}({-},{-})$ in
${\operatorname{\mathsf{Comod--}}}C$, and by
$\operatorname{Ext}^{C,*}({-},{-})$ in $C{\operatorname{\mathsf{--Contra}}}$.

The category of left $C$-comodules $C{\operatorname{\mathsf{--Comod}}}$ is a
locally finite Grothendieck abelian category. Any $C$-comodule is the union of
its finite-dimensional subcomodules [41, Propositions 2.1.1–2.1.2 and
Corollary 2.1.4], [34, Lemma 3.1(b)]. The forgetful functor
$C{\operatorname{\mathsf{--Comod}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ from the category of $C$-comodules to the
category of $k$-vector spaces is exact and preserves the infinite coproducts
(but _not_ the infinite products). So the coproduct (and more generally,
filtered direct limit) functors are exact in
$C{\operatorname{\mathsf{--Comod}}}$, while the functors of infinite product
are usually _not_ exact.

Left $C$-comodules of the form $C\otimes_{k}V$ and right $C$-comodules of the
form $V\otimes_{k}C$, where $V$ ranges over the $k$-vector spaces, are called
the _cofree_ $C$-comodules. For any left $C$-comodule $L$, the vector space of
$C$-comodule morphisms $L\longrightarrow C\otimes_{k}V$ is naturally
isomorphic to the vector space of $k$-linear maps $L\longrightarrow V$,

$\operatorname{Hom}_{C}(L,\>C\otimes_{k}V)\simeq\operatorname{Hom}_{k}(L,V).$

Hence the cofree comodules are injective (as objects of
$C{\operatorname{\mathsf{--Comod}}}$ or ${\operatorname{\mathsf{Comod--}}}C$).
A $C$-comodule is injective if and only if it is a direct summand of a cofree
one.

The category of left $C$-contramodules $C{\operatorname{\mathsf{--Contra}}}$
is a locally presentable abelian category with enough projective objects. The
forgetful functor $C{\operatorname{\mathsf{--Contra}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ is exact and preserves the infinite
products (but _not_ the infinite coproducts). Hence the functors of infinite
product are exact in $C{\operatorname{\mathsf{--Contra}}}$, while the
coproduct functors are usually _not_ exact.

Left $C$-contramodules of the form $\operatorname{Hom}_{k}(C,V)$, where $V\in
k{\operatorname{\mathsf{--Vect}}}$, are called the _free_ $C$-contramodules.
For any left $C$-contramodule $Q$, the vector space of $C$-contramodule
morphisms $\operatorname{Hom}_{k}(C,V)\longrightarrow Q$ is naturally
isomorphic to the vector space of $k$-linear maps $V\longrightarrow Q$,

$\operatorname{Hom}^{C}(\operatorname{Hom}_{k}(C,V),Q)\simeq\operatorname{Hom}_{k}(V,Q).$

Hence the free contramodules are projective (as objects of
$C{\operatorname{\mathsf{--Contra}}}$). A $C$-contramodule is projective if
and only if it is a direct summand of a free one.

Let us introduce a simplified version of the _Sweedler notation_ [41, Section
1.2], [16, Notation 1.4.2] for the comultiplication in $C$. Given an element
$c\in C$, we write

$\mu(c)=c_{(1)}\otimes c_{(2)}\in C\otimes_{k}C.$

Following the convention in [20, 22, 34] (which is opposite to the convention
in [41, 16]), we define the associative algebra structure on the dual vector
space $C^{*}=\operatorname{Hom}_{k}(C,k)$ to a coalgebra $C$ by the formula

$(fg)(c)=f(c_{(2)})g(c_{(1)})\qquad\text{for all $f$, $g\in C^{*}$ and $c\in
C$}.$

The counit on $C$ induces a unit in $C^{*}$ in the obvious way.

Then, for any left $C$-comodule $M$, the composition

$C^{*}\otimes_{k}M\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{*}\otimes_{k}C\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM$

of the map induced by the coaction map $\nu\colon M\longrightarrow
C\otimes_{k}M$ and the map induced by the pairing map
$C^{*}\otimes_{k}C\longrightarrow k$ endows $M$ with a left $C^{*}$-module
structure. Similarly, for any right $C$-comodule $N$, the composition

$N\otimes_{k}C^{*}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muN\otimes_{k}C\otimes_{k}C^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN$

endows $N$ with a right $C^{*}$-module structure. Finally, for any left
$C$-contramodule $P$, the composition

$C^{*}\otimes_{k}P\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(C,P)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP$

of the natural embedding of vector spaces
$C^{*}\otimes_{k}P\rightarrowtail\operatorname{Hom}_{k}(C,P)$ and the
contraaction map $\pi\colon\operatorname{Hom}_{k}(C,P)\longrightarrow P$
endows $P$ with a left $C^{*}$-module structure.

We have constructed the _comodule inclusion functors_

$\displaystyle\Upsilon\colon C{\operatorname{\mathsf{--Comod}}}$
$\displaystyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{*}{\operatorname{\mathsf{--Mod}}},$
$\displaystyle{\operatorname{\mathsf{Comod--}}}C$ $\displaystyle\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\operatorname{\mathsf{Mod--}}}C^{*}$

and the _contramodule forgetful functor_

$\Theta\colon C{\operatorname{\mathsf{--Contra}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC^{*}{\operatorname{\mathsf{--Mod}}}.$

The comodule inclusion functors (for a coalgebra $C$ over a field $k$) are
always fully faithful (see [41, Propositions 2.1.1–2.1.2 and Theorem 2.1.3(e)]
for a discussion). The contramodule forgetful functor is _not_ fully faithful
in general, as we will see in Example 7.2 below.

##  2\. Conilpotent Coalgebras and Minimal Resolutions

What we call _conilpotent_ coalgebras (in the terminology going back to [36,
Section 3.1], [19, Section 4.1]) were called “pointed irreducible” coalgebras
in [41, Section 8.0]. We refer to [34, Sections 3.3–3.4] for an introductory
discussion.

Let $D$ be a coalgebra without counit over a field $k$. For every $n\geq 1$,
there is the uniquely defined _iterated comultiplication map_ $\mu^{(n)}\colon
D\longrightarrow D^{\otimes n+1}$. The coalgebra $D$ is said to be
_conilpotent_ if for every $d\in D$ there exists $n\geq 1$ such that
$\mu^{(n)}(d)=0$ in $D^{\otimes n+1}$. Clearly, one then also has
$\mu^{(m)}(d)=0$ for all $m\geq n$.

A _coaugmentation_ $\gamma$ of a coalgebra $C$ is a homomorphism of (counital)
coalgebras $\gamma\colon k\longrightarrow C$. So the composition
$\epsilon\gamma\colon k\longrightarrow C\longrightarrow k$ must be the
identity map.

Given a coaugmented coalgebra $(C,\gamma)$, the cokernel $D=C/\gamma(k)$ of
the map $\gamma$ has a unique coalgebra structure for which the natural
surjection $C\longrightarrow D$ is a homomorphism of noncounital coalgebras. A
coaugmented coalgebra $C$ us called _conilpotent_ if the noncounital coalgebra
$D$ is conilpotent (in the sense of the definition above).

Obviously, no nonzero noncounital coalgebra homomorphisms $k\longrightarrow D$
exist for a conilpotent noncounital coalgebra $D$. Consequently, a conilpotent
coaugmented coalgebra $(C,\gamma)$ admits no other coaugmentation but
$\gamma$.

The following result is a version of Nakayama lemma for conilpotent
noncounital coalgebras.

######  Lemma 2.1.

(a) Let $D$ be a conilpotent noncounital coalgebra and $M\neq 0$ be a
noncounital left $D$-comodule. Then the coaction map $M\longrightarrow
D\otimes_{k}M$ is _not_ injective.

(b) Let $D$ be a conilpotent noncounital coalgebra and $P\neq 0$ be a
noncounital left $D$-contramodule. Then the contraaction map
$\operatorname{Hom}_{k}(D,P)\longrightarrow P$ is _not_ surjective.

###### Proof.

Part (a): for the sake of contraction, assume that the coaction map $\nu\colon
M\longrightarrow D\otimes_{k}M$ is injective. Then the iterated coaction map
$\nu^{(n)}\colon M\longrightarrow D^{\otimes n}\otimes_{k}M$ is also injective
for every $n\geq 1$.

Pick a nonzero element $x\in M$, and write $\nu(x)=\sum_{i=1}^{r}d_{r}\otimes
y_{r}$ for some $d_{i}\in D$ and $y_{i}\in M$. Choose $n\geq 1$ such that
$\mu^{(n)}(d_{i})=0$ in $D^{\otimes n+1}$ for every $1\leq i\leq r$. Then
$\nu^{(n+1)}(x)=\sum_{i=1}^{r}\mu^{(n)}(d_{i})\otimes y_{i}=0$ in $D^{\otimes
n+1}\otimes_{k}M$, a contradiction.

The proof of part (b) is a bit more involved; it can be found in [20, Lemma
A.2.1]. For a discussion of generalizations and other versions of the comodule
and contramodule Nakayama lemmas, see [22, Lemma 2.1]. ∎

Let $E$ be a subcoalgebra in a coalgebra $C$. Then in any left $C$-comodule
$M$ there exists a unique maximal subcomodule whose $C$-comodule structure
arises from an $E$-comodule structure. We denote this subcomodule, which can
be computed as the kernel of the composition of maps $M\longrightarrow
C\otimes_{k}M\longrightarrow C/E\otimes_{k}M$, by ${}_{E}M\subset M$. The
similar subcomodule of a right $C$-comodule $N$ will be denoted by
$N_{E}\subset N$.

Dually, any left $C$-contramodule $P$ admits a unique maximal quotient
contramodule whose $C$-contramodule structure arises from an $E$-contramodule
structure. We denote this quotient contramodule, which can be computed as the
cokernel of the composition of maps
$\operatorname{Hom}_{k}(C/E,P)\longrightarrow\operatorname{Hom}_{k}(C,P)\longrightarrow
P$, by ${}^{E}\\!P\twoheadleftarrow P$. A further discussion of the functors
$M\longmapsto{}_{E}M$ and $P\longmapsto{}^{E}\\!P$ can be found in [27,
Section 2] or [34, Section 8.4].

For any right $C$-comodule $N$, any subcoalgebra $E\subset C$, and any
$k$-vector space $V$, there is a natural isomorphism of left $E$-contramodules

(4)
${}^{E}\\!\operatorname{Hom}_{k}(N,V)\simeq\operatorname{Hom}_{k}(N_{E},V),$

where the left contramodule structure on the space of linear maps from a right
comodule to a vector space is constructed as explained in Section 1. The
natural isomorphism (4) follows immediately from the constructions of the
functors $M\longmapsto{}_{E}M$ and $P\longmapsto{}^{E}\\!P$ above.

In this section, we will be interested in the particular case when
$(C,\gamma)$ is a coaugmented (eventually, conilpotent) coalgebra and
$E=\gamma(k)\subset C$. In this case, we put ${}_{\gamma}M={}_{E}M$ and
${}^{\gamma}\\!P={}^{E}\\!P$. The similar notation for a right $C$-comodule
$N$ is $N_{\gamma}=N_{E}$, and for a right $C$-contramodule $Q$ it is
$Q^{\gamma}=Q^{E}$. Endowing the one-dimensional $k$-vector space $k$ with the
(left and right) $C$-comodule and $C$-contramodule structures defined in terms
of $\gamma$, one has natural isomorphisms of $k$-vector spaces

${}_{\gamma}M\simeq\operatorname{Hom}_{C}(k,M)\quad\text{and}\quad\operatorname{Hom}_{k}({}^{\gamma}\\!P,k)\simeq\operatorname{Hom}^{C}(P,k).$

One can further compute the vector space ${}^{\gamma}\\!P$ as the
_contratensor product_ $k\odot_{C}P$ (see [22, Section 3.1] for the
definition), but we will not need to use this fact.

######  Lemma 2.2.

(a) Let $(C,\gamma)$ be a conilpotent coaugmented coalgebra and $M\neq 0$ be a
left $C$-comodule. Then ${}_{\gamma}M\neq 0$.

(b) Let $(C,\gamma)$ be a conilpotent coaugmented coalgebra and $P\neq 0$ be a
left $C$-contramodule. Then ${}^{\gamma}\\!P\neq 0$.

###### Proof.

This is an equivalent restatement of Lemma 2.1 for $D=C/\gamma(k)$. ∎

For a conilpotent coalgebra $C$, the subcomodule ${}_{\gamma}M$ of a
$C$-comodule $M$ can be also described as the socle (i. e., the maximal
semisimple subcomodule) of $M$, and the quotient contramodule
${}^{\gamma}\\!P$ of a $C$-contramodule $P$ can be described as the cosocle
(i. e., the maximal semisimple quotient contramodule) of $P$.

######  Lemma 2.3.

Let $(C,\gamma)$ be a conilpotent coaugmented coalgebra. Then

(a) a morphism of left $C$-comodules $f\colon L\longrightarrow M$ is injective
if and only if the induced map of vector spaces
${}_{\gamma}f\colon{}_{\gamma}L\longrightarrow{}_{\gamma}M$ is injective;

(b) a morphism of left $C$-contramodules $f\colon P\longrightarrow Q$ is
surjective if and only if the induced map of vector spaces
${}^{\gamma}\\!f\colon{}^{\gamma}\\!P\longrightarrow{}^{\gamma}Q$ is
surjective.

###### Proof.

Part (a): if the map $f$ is injective, then so is the map ${}_{\gamma}f$,
since ${}_{\gamma}N$ is a vector subspace in $N$ for every left $C$-comodule
$N$. Conversely, the functor $N\longmapsto{}_{\gamma}N$ is left exact; so if
$K=\ker(f)\in C{\operatorname{\mathsf{--Comod}}}$, then
${}_{\gamma}K=\ker({}_{\gamma}f)$. Now if ${}_{\gamma}K=0$, then $K=0$ by
Lemma 2.2(a). The proof of part (b) is similar (or rather, dual-analogous in
the sense of [34, Section 8.2]). The functor $P\longmapsto{}^{\gamma}\\!P$ is
right exact, etc. ∎

######  Lemma 2.4.

Let $(C,\gamma)$ be a conilpotent coaugmented coalgebra. Then

(a) a morphism of injective left $C$-comodules $f\colon I\longrightarrow J$ is
an isomorphism if and only if the induced map of vector spaces
${}_{\gamma}f\colon{}_{\gamma}I\longrightarrow{}_{\gamma}J$ is an isomorphism;

(b) a morphism of projective left $C$-contramodules $f\colon P\longrightarrow
Q$ is an isomorphism if and only if the induced map of vector spaces
${}^{\gamma}\\!f\colon{}^{\gamma}\\!P\longrightarrow{}^{\gamma}Q$ is an
isomorphism.

###### Proof.

Part (a): if the map ${}_{\gamma}f$ is injective, then the morphism $f$ is
injective by Lemma 2.3(a). Now if the $C$-comodule $I$ is injective, then $f$
is a split monomorphism in $C{\operatorname{\mathsf{--Comod}}}$. Therefore,
${}_{\gamma}\operatorname{coker}(f)=\operatorname{coker}({}_{\gamma}f)$, and
Lemma 2.2(a) tells that $f$ is an isomorphism whenever ${}_{\gamma}f$ is. The
proof of part (b) is dual-analogous. ∎

######  Lemma 2.5.

Let $(C,\gamma)$ be a conilpotent coaugmented coalgebra. Then

(a) for any left $C$-comodule $M$ there exists a cofree left $C$-comodule $J$
together with an injective $C$-comodule morphism $f\colon M\longrightarrow J$
such that the induced map
${}_{\gamma}f\colon{}_{\gamma}M\longrightarrow{}_{\gamma}J$ is an isomorphism
of $k$-vector spaces;

(b) for any left $C$-contramodule $Q$ there exists a free left
$C$-contramodule $P$ together with a surjective $C$-contramodule morphism
$f\colon P\longrightarrow Q$ such that the induced map
${}^{\gamma}\\!f\colon{}^{\gamma}\\!P\longrightarrow{}^{\gamma}Q$ is an
isomorphism of $k$-vector spaces.

###### Proof.

Part (a): put $V={}_{\gamma}M$ and $J=C\otimes_{k}V$. Then the injective map
$V\longrightarrow J$ induced by the map $\gamma\colon k\longrightarrow C$ is a
$C$-comodule morphism $M\supset{}_{\gamma}M\longrightarrow J$. Since the
cofree comodules are injective, this morphism can be extended to a
$C$-comodule morphism $f\colon M\longrightarrow J$. The morphism $f$ is
injective by Lemma 2.3(a). The comodule $J$ can be also described as an
injective envelope of the comodule $M$ in the comodule category
$C{\operatorname{\mathsf{--Comod}}}$. The proof of part (b) is dual-analogous;
the contramodule $P$ can be also described as a projective cover of the
contramodule $Q$ in contramodule category
$C{\operatorname{\mathsf{--Contra}}}$ (cf. [30, Example 12.3]). ∎

######  Corollary 2.6.

Let $C$ be a conilpotent (coaugmented) coalgebra. Then

(a) a $C$-comodule is injective if and only if is cofree;

(b) a $C$-contramodule is projective if and only if it is free.

###### Proof.

Generally speaking, over a coalgebra $C$ over a field $k$, the injective
comodules are the direct summands of the cofree ones, and the projective
contramodules are the direct summands of the free ones (see Section 1). In the
situation at hand with a conilpotent coalgebra $C$, the assersion of part (a)
follows straighforwardly from Lemmas 2.4(a) and 2.5(a). Part (b) follows from
Lemmas 2.4(b) and 2.5(b). ∎

Let $K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of left
$C$-comodules. We will say that the complex
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is _minimal_ if the
differential in the complex of vector spaces
${}_{\gamma}K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ vanishes.
Similarly, a complex of left $C$-contramodules
$Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is said to be _minimal_ if
the differential in the complex of vector spaces
${}^{\gamma}Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ vanishes.

Let $M$ be a $C$-comodule. An injective coresolution $M\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the comodule $M$ is said
to be _minimal_ if the complex of comodules
$J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is minimal. Dually, a
projective resolution
$Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow P$ of a
$C$-contramodule $P$ is said to be _minimal_ if the complex of contramodules
$Q_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is minimal.

######  Proposition 2.7.

The following assertions hold for any conilpotent coalgebra $C$.

(a) Any $C$-comodule $M$ admits a minimal injective coresolution. A minimal
injective coresolution of $M$ is unique up to a (nonunique) isomorphism.

(b) Any $C$-contramodule $P$ admits a minimal projective resolution. A minimal
projective resolution of $P$ is unique up to a (nonunique) isomorphism.

###### Proof.

Part (a): to construct a minimal injective coresolution of $M$, all one needs
to do is to iterate the construction of Lemma 2.5(a). Pick an injective
$C$-comodule $J^{0}$ together with an injective morphism of $C$-comodules
$M\longrightarrow J^{0}$ such that the induced map
${}_{\gamma}M\longrightarrow{}_{\gamma}J^{0}$ is an isomorphism. Then the
induced map ${}_{\gamma}J^{0}\longrightarrow{}_{\gamma}(J^{0}/M)$ is zero.
Pick an injective $C$-comodule $J^{1}$ together with an injetive morphism of
$C$-comodules $J^{0}/M\longrightarrow J^{1}$ such that the induced map
${}_{\gamma}(J^{0}/M)\longrightarrow{}_{\gamma}J^{1}$ is an isomorphism, etc.

To prove uniqueness, let $M\longrightarrow
I^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and $M\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be two minimal injective
coresolutions of $M$. Then there exists a morphism of complexes of
$C$-comodules $f\colon
I^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ making the triangular
diagram $M\longrightarrow
I^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ commutative. Now we have
${}_{\gamma}I^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Hom}_{C}(k,I^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
${}_{\gamma}J^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\operatorname{Hom}_{C}(k,I^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
The map of complexes
$\operatorname{Hom}_{C}(k,f)\colon\operatorname{Hom}_{C}(k,I^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}_{C}(k,J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
induces an isomorphism of the cohomology spaces, as both the complexes compute
$\operatorname{Ext}_{C}^{*}(k,M)$. Since both the complexes
${}_{\gamma}I^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
${}_{\gamma}J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ have vanishing
differentials by assumption, it follows that
$\operatorname{Hom}_{C}(k,f)={}_{\gamma}f$ is a termwise isomorphism of
complexes of vector spaces. Applying Lemma 2.4(a), we conclude that $f$ is a
termwise isomorphism of complexes of $C$-comodules.

The proof of part (b) is dual-analogous. ∎

##  3\. The Left and Right Comodule and Contramodule Ext Comparison

Let $(C,\gamma)$ be a coaugmented coalgebra over a field $k$ (as defined in
Section 2). Then the one-dimensional $k$-vector space $k$ can be endowed with
left and right $C$-comodule and $C$-contramodule structures defined in terms
of $\gamma$. We recall the notation $\operatorname{Ext}_{C}^{*}({-},{-})$ for
the Ext spaces in the category of left $C$-comodules
$C{\operatorname{\mathsf{--Comod}}}$ and
$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}({-},{-})$ for the Ext spaces in the
category of right $C$-comodules ${\operatorname{\mathsf{Comod--}}}C$, as well
as the notation $\operatorname{Ext}^{C,*}({-},{-})$ for the Ext spaces in the
category of left $C$-contramodules $C{\operatorname{\mathsf{--Contra}}}$.

The condition that the vector spaces $\operatorname{Ext}_{C}^{n}(k,k)$ be
finite-dimensional plays a key role in this paper. In this section we explain
that this condition is left-right symmetric: it holds for a coalgebra $C$ if
and only if it holds for $C^{\mathrm{op}}$. Furthermore, the vector space
$\operatorname{Ext}_{C}^{n}(k,k)$ is finite-dimensional if and only if the
vector space $\operatorname{Ext}^{C,n}(k,k)$ is. The following proposition is
certainly essentially well-known; we spell out the details here for the sake
of completeness of the exposition.

######  Proposition 3.1.

Let $(C,\gamma)$ be a coaugmented coalgebra over $k$. Then there are natural
isomorphisms of Ext spaces

$\operatorname{Ext}_{C}^{n}(k,k)\simeq\operatorname{Ext}_{C^{\mathrm{op}}}^{n}(k,k)\qquad\text{for
all\/ $n\geq 0$}.$

Moreover, in fact, the Ext-algebras $\operatorname{Ext}_{C}^{*}(k,k)$ and
$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}(k,k)$ are naturally opposite to each
other (as graded algebras),

$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}(k,k)\simeq\operatorname{Ext}_{C}^{*}(k,k)^{\mathrm{op}}.$

The module analogue of Proposition 3.1 is well-known: for any augmented
associative algebra $A$ over a field $k$, there are natural isomorphisms of
$k$-vector spaces
$\operatorname{Ext}^{n}_{A}(k,k)\simeq\operatorname{Tor}^{A}_{n}(k,k)^{*}\simeq\operatorname{Ext}^{n}_{A^{\mathrm{op}}}(k,k)$,
making $\operatorname{Ext}_{A^{\mathrm{op}}}^{*}(k,k)$ the graded algebra with
the opposite multiplication to $\operatorname{Ext}_{A}^{*}(k,k)$.

###### First proof.

Put $C_{+}=C/\gamma(k)$. For any left $C$-comodule $M$, the reduced cobar
coresolution

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muC\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}C_{+}\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}C_{+}\otimes_{k}C_{+}\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

is an injective coresolution of $M$ in the abelian category
$C{\operatorname{\mathsf{--Comod}}}$. Applying the functor
$\operatorname{Hom}_{C}(k,{-})$, we obtain a cobar complex

(5) $M\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\otimes_{k}C_{+}\otimes_{k}M\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

computing the vector spaces $\operatorname{Ext}_{C}^{*}(k,M)$. In particular,
the vector spaces $\operatorname{Ext}_{C}^{*}(k,k)$ are computed by the cobar
complex

(6) $k\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\otimes_{k}C_{+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\otimes_{k}C_{+}\otimes_{k}C_{+}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

[36, Section 1.1], [19, Section 2.1].

Reordering the tensor factors inside every term of the complex in the opposite
way identifies the complex (6) for a coalgebra $C$ with the similar complex
for the opposite coalgebra $C^{\mathrm{op}}$. This suffices to prove the first
assertion of the proposition.

To prove the second assertion, one needs to observe that the obvious (free
associative) multiplication on the cobar complex (6) induces a multiplication
on its cohomology spaces that is precisely opposite to the composition
multiplication on $\operatorname{Ext}_{C}^{*}(k,k)$. Moreover, the obvious
left action of the DG-algebra (6) on the complex (5) induces a left action of
the cohomology algebra of (6) on the cohomology of (5) corresponding to the
natural graded right module structure over $\operatorname{Ext}_{C}^{*}(k,k)$
on $\operatorname{Ext}^{*}_{C}(k,M)$. Thus the isomorphism
$\operatorname{Ext}_{C}^{*}(k,k)\simeq\operatorname{Ext}_{C^{\mathrm{op}}}^{*}(k,k)$
obtained by comparing the cobar complexes (6) for $C$ and $C^{\mathrm{op}}$
makes the multiplication on the latter Ext-algebra opposite to the one on the
former one.

This argument was sketched in [19, Section 2.4]. An alternative proof of the
first assertion of the proposition was suggested in [19, Section 2.5]. ∎

###### Second proof.

Let $C{\operatorname{\mathsf{--comod}}}$ and
${\operatorname{\mathsf{comod--}}}C$ denote the abelian categories of _finite-
dimensional_ left and right $C$-comodules, respectively. One makes two
observations, which, taken together, imply both the assertions of the
proposition.

Firstly, the fully faithful, exact inclusions of abelian categories
$C{\operatorname{\mathsf{--comod}}}\longrightarrow
C{\operatorname{\mathsf{--Comod}}}$ and
${\operatorname{\mathsf{comod--}}}C\longrightarrow{\operatorname{\mathsf{Comod--}}}C$
induce isomorphisms on the Ext spaces. Moreover, the triangulated functors
between bounded above derived categories
$\mathsf{D}^{-}(C{\operatorname{\mathsf{--comod}}})\longrightarrow\mathsf{D}^{-}(C{\operatorname{\mathsf{--Comod}}})$
and
$\mathsf{D}^{-}({\operatorname{\mathsf{comod--}}}C)\longrightarrow\mathsf{D}^{-}({\operatorname{\mathsf{Comod--}}}C)$
induced by these inclusions of abelian categories are fully faithful. This is
a particular case of a well-known general result for abelian or even exact
categories (see [12, opposite version of Proposition 1.7.11], [13, opposite
version of Theorem 13.2.8], [14, opposite version of Theorem 12.1(b)], or [35,
Proposition 2.1]). The point is that for any finite-dimensional $C$-comodule
$L$, any $C$-comodule $M$, and any surjective $C$-comodule morphism
$M\longrightarrow L$, there exists a $C$-subcomodule $M^{\prime}\subset M$
such that the composition $M^{\prime}\longrightarrow M\longrightarrow L$ is
surjective.

Secondly, for any finite-dimensional right $C$-comodule $N$, the dual vector
space $N^{*}$ is naturally a finite-dimensional left $C$-comodule. Here it
helps to notice that for any such $N$ there exists a finite-dimensional
subcoalgebra $E\subset C$ such that the $C$-comodule structure on $N$ arises
from an $E$-comodule structure [34, Lemma 3.1(c)]. Consequently, there is a
natural anti-equivalence of abelian categories

$N\longmapsto
N^{*}\colon({\operatorname{\mathsf{comod--}}}C)^{\mathsf{op}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC{\operatorname{\mathsf{--comod}}}$

taking the right $C$-comodule $k$ to the left $C$-comodule $k$. This anti-
equivalence induces the desired anti-isomorphism of the Ext algebras. ∎

######  Lemma 3.2.

Let $C$ be a coalgebra over a field $k$. Let
$M\in{\operatorname{\mathsf{Comod--}}}C$ be a right $C$-comodule and
$L\in{\operatorname{\mathsf{comod--}}}C$ be a finite-dimensional right
$C$-comodule. Then the vector space of left $C$-contramodule homomorpisms
$M^{*}\longrightarrow L^{*}$ is naturally isomorphic to the double dual vector
space to the vector space of right $C$-comodule homomorphisms
$L\longrightarrow M$,

(7)
$\operatorname{Hom}^{C}(M^{*},L^{*})\simeq\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)^{**}.$

This isomorphism identifies the map
$\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)\longrightarrow\operatorname{Hom}^{C}(M^{*},L^{*})$
induced by the contravariant functor of vector space dualization $N\longmapsto
N^{*}\colon{\operatorname{\mathsf{Comod--}}}C\longrightarrow
C{\operatorname{\mathsf{--Contra}}}$ with the natural inclusion
$\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)\longrightarrow\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)^{**}$
of a vector space $\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)$ into its double
dual vector space.

###### Proof.

Notice that for any $k$-vector space $V$ and any finite-dimensional $k$-vector
space $W$ there is a natural isomorphism of $k$-vector spaces

(8)
$\operatorname{Hom}_{k}(V^{*},W^{*})\simeq\operatorname{Hom}_{k}(W,V)^{**},$

where $V^{*}=\operatorname{Hom}_{k}(V,k)$ and
$W^{*}=\operatorname{Hom}_{k}(W,k)$. The isomorphism (8) identifies the map
$\operatorname{Hom}_{k}(W,V)\longrightarrow\operatorname{Hom}_{k}(V^{*},W^{*})$
induced by the contravariant functor $U\longmapsto U^{*}\colon
k{\operatorname{\mathsf{--Vect}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ with the natural inclusion
$\operatorname{Hom}_{k}(W,V)\longrightarrow\operatorname{Hom}_{k}(W,V)^{**}$
of a vector space $\operatorname{Hom}_{k}(W,V)$ into its double dual vector
space.

Now, for any two right $C$-comodules $L$ and $M$, the vector space
$\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)$ is the kernel of (the difference
of) a natural pair of maps
$\operatorname{Hom}_{k}(L,M)\rightrightarrows\operatorname{Hom}_{k}(L,\>\allowbreak
M\otimes_{k}\nobreak C)$,

(9)
$0\longrightarrow\operatorname{Hom}_{C^{\mathrm{op}}}(L,M)\longrightarrow\operatorname{Hom}_{k}(L,M)\,\rightrightarrows\operatorname{Hom}_{k}(L,\>M\otimes_{k}C).$

For any two left $C$-contramodules $P$ and $Q$, the vector space
$\operatorname{Hom}^{C}(P,Q)$ is the kernel of (the difference of) a natural
pair of maps
$\operatorname{Hom}_{k}(P,Q)\rightrightarrows\operatorname{Hom}_{k}(\operatorname{Hom}_{k}(C,P),Q)$,

(10)
$0\longrightarrow\operatorname{Hom}^{C}(P,Q)\longrightarrow\operatorname{Hom}_{k}(P,Q)\,\rightrightarrows\operatorname{Hom}_{k}(\operatorname{Hom}_{k}(C,P),Q).$

In particular, for $P=M^{*}$ and $Q=L^{*}$, we have

(11)
$0\longrightarrow\operatorname{Hom}^{C}(M^{*},L^{*})\longrightarrow\operatorname{Hom}_{k}(M^{*},L^{*})\,\rightrightarrows\operatorname{Hom}_{k}(\operatorname{Hom}_{k}(C,M^{*}),L^{*}).$

It remains to use the natural isomorphism (8) in order to show that the double
dual vector space functor $U\longmapsto U^{**}$ takes the rightmost pair of
parallel morphisms in the sequence (9) to the rightmost pair of parallel
morphisms in the sequence (11) for a finite-dimensional $C$-comodule $L$. ∎

######  Proposition 3.3.

Let $(C,\gamma)$ be a coaugmented coalgebra over $k$. Then the vector space
$\operatorname{Ext}^{C,n}(k,k)$ is naturally isomorphic to the double dual
vector space to the vector space
$\operatorname{Ext}_{C^{\mathrm{op}}}^{n}(k,k)$,

$\operatorname{Ext}^{C,n}(k,k)\simeq\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)^{**}\qquad\text{for
all\/ $n\geq 0$}.$

This isomorphism identifies the map
$\operatorname{Ext}_{C^{\mathrm{op}}}^{n}(k,k)\longrightarrow\operatorname{Ext}^{C,n}(k,k)$
induced by the exact contravariant functor of vector space dualization
$N\longmapsto N^{*}\colon{\operatorname{\mathsf{Comod--}}}C\longrightarrow
C{\operatorname{\mathsf{--Contra}}}$ with the natural inclusion
$\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)^{**}$
of a vector space $\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)$ into its
double dual vector space.

###### Proof.

This observation goes back to the discussion in [20, Section A.1.2]. The point
is that the functor $N\longmapsto
N^{*}\colon{\operatorname{\mathsf{Comod--}}}C\longrightarrow
C{\operatorname{\mathsf{--Contra}}}$ takes injective right $C$-comodules to
projective left $C$-contramodules, which makes it easy to compute the induced
map of the Ext spaces.

For any coalgebra $C$ and any right $C$-comodule $M$, pick an injective
coresolution

(12) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muM\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{2}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb$

of $M$ in the abelian category ${\operatorname{\mathsf{Comod--}}}C$. Applying
the dual vector space functor $\operatorname{Hom}_{k}({-},k)$ to the complex
(12), we obtain the complex

(13) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(M,k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(J^{0},k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(J^{1},k)\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb,$

which is a projective resolution of the left $C$-contramodule
$M^{*}=\operatorname{Hom}_{k}(M,k)$.

Now let $L$ be a another right $C$-comodule. Then one can compute the vector
spaces $\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(L,M)$ as the cohomology
spaces of the complex obtained by applying the functor
$\operatorname{Hom}_{C^{\mathrm{op}}}(L,{-})$ to the coresolution (12).
Similarly, one can compute the vector spaces
$\operatorname{Ext}^{C,n}(M^{*},L^{*})$ as the cohomology spaces of the
complex obtained by applying the functor $\operatorname{Hom}^{C}({-},L^{*})$
to the resolution (13). Taking into account Lemma 3.2, we obtain a natural
isomorphism of complexes

$\operatorname{Hom}^{C}((J^{\text{\smaller\smaller$\scriptstyle\bullet$}})^{*},L^{*})\simeq\operatorname{Hom}_{C^{\mathrm{op}}}(L,J^{\text{\smaller\smaller$\scriptstyle\bullet$}})^{**}$

inducing a natural isomorphism of the cohomology spaces

$\operatorname{Ext}^{C,n}(M^{*},L^{*})\simeq\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(L,M)^{**}$

for all right $C$-comodules $M$, all finite-dimensional right $C$-comodules
$L$, and all integers $n\geq 0$. Moreover, it follows from the second
assertion of the lemma that the map
$\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(L,M)\longrightarrow\operatorname{Ext}^{C,n}(M^{*},L^{*})$
induced by the functor $N\longmapsto
N^{*}\colon{\operatorname{\mathsf{Comod--}}}C\longrightarrow
C{\operatorname{\mathsf{--Contra}}}$ agrees with the obvious map
$\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(L,M)\longrightarrow\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(L,M)^{**}$.
∎

##  4\. Weakly Finite Koszulity Implies Comodule Ext Isomorphism

The aim of this section is to prove the following theorem.

######  Theorem 4.1.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 0$ be an
integer such that the vector space $\operatorname{Ext}^{i}_{C}(k,k)$ is
finite-dimensional for all $1\leq i\leq n$. Then the map of Ext spaces

$\operatorname{Ext}_{C}^{i}(L,M)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{i}(L,M)$

induced by the inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is an isomorphism for all left
$C$-comodules $L$ and $M$ and all $0\leq i\leq n$.

We start with a lemma describing some vector spaces of morphisms in the
category of left $C^{*}$-modules for a coalgebra $C$. A right $C$-comodule $N$
is said to be _finitely cogenerated_ if it is a subcomodule of a cofree
$C$-comodule $V\otimes_{k}C$ with a finite-dimensional vector space of
cogenerators $V$.

######  Lemma 4.2.

Let $C$ be a coalgebra over $k$, let $N$ be a right $C$-comodule, and let $U$
be a $k$-vector space. Then there is a natural monomorphism of $k$-vector
spaces

$\eta_{N,U}\colon N\otimes_{k}U\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}(N^{*},\>C\otimes_{k}U),$

which is an isomorphism whenever $N$ is a finitely cogenerated injective
$C$-comodule. Here the left $C^{*}$-module structure on
$N^{*}=\operatorname{Hom}_{k}(N,k)$ is obtained by dualizing the right
$C^{*}$-module structure on $N$ (or equivalently, by applying the forgetful
functor to the natural left $C$-contramodule structure on $N^{*}$), while the
left $C^{*}$-module structure on $C\otimes_{k}U$ comes from the cofree left
$C$-comodule structure.

###### Proof.

The map $\eta_{N,U}$ is defined by the formula

$\eta_{N,U}(x\otimes u)(f)=f(x_{(0)})x_{(1)}\otimes u,$

for all $x\in N$, $u\in U$, and $f\in N^{*}$. Here $\nu(x)=x_{(0)}\otimes
x_{(1)}\in N\otimes_{k}C$ is a notation for the right $C$-coaction map
$\nu\colon N\longrightarrow N\otimes_{k}C$.

To prove the isomorphism assertion for a finitely cogenerated injective
$C$-comodule $N$, it suffices to consider the case $N=V\otimes_{k}C$, where
$\dim_{k}V<\infty$. In this case, $N^{*}=C^{*}\otimes_{k}V^{*}$ is a free left
$C^{*}$-module, so
$\operatorname{Hom}_{C^{*}}(N^{*},\>C\otimes_{k}U)=\operatorname{Hom}_{k}(V^{*},\>C\otimes_{k}U)=V\otimes_{k}C\otimes_{k}U=N\otimes_{k}U$.

To prove that $\eta_{N,U}$ is an injective map for any $C$-comodule $N$,
consider the $k$-linear map

$\operatorname{Hom}_{C^{*}}(N^{*},\>C\otimes_{k}U)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(N^{*},U)$

induced by the counit map $\epsilon\colon C\longrightarrow k$. Notice that the
counit is _not_ a $C$-comodule morphism, and consequently not a $C^{*}$-module
morphism, but only a linear map of $k$-vector spaces. The composition

$N\otimes_{k}U\overset{\eta}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\operatorname{Hom}_{C^{*}}(N^{*},\>C\otimes_{k}U)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(N^{*},U)$

is the natural injective map of $k$-vector spaces
$N\otimes_{k}U\longrightarrow\operatorname{Hom}_{k}(N^{*},U)$ (defined for any
two vector spaces $U$ and $W=N$). Since the composition is a monomorphism, so
is the map $\eta=\eta_{N,U}$. ∎

Let $C=(C,\gamma)$ be a coaugmented coalgebra. For every $m\geq 0$, denote by
$F_{m}C\subset C$ the kernel of the composition $C\longrightarrow C^{\otimes
m+1}\longrightarrow(C/\gamma(k))^{\otimes m+1}$, where $\mu^{(m)}\colon
C\longrightarrow C^{\otimes m+1}$ is the iterated comultiplication map and
$C^{\otimes m+1}\longrightarrow(C/\gamma(k))^{\otimes m+1}$ is the natural
surjection. So one has $F_{-1}C=0$ and $F_{0}C=\gamma(k)\subset C$. One can
check that $F$ is a comultiplicative filtration on $C$, that is,

$\mu(F_{m}C)\subset\sum\nolimits_{p+q=m}F_{p}C\otimes_{k}F_{q}C\subset
C\otimes_{k}C$

for all $m\geq 0$ [36, Section 3.1].

Let $L$ be a left $C$-comodule. Denote by $F_{m}L\subset L$ the full preimage
of the subspace $F_{m}C\otimes_{k}L\subset C\otimes_{k}L$ under the coaction
map $\nu\colon L\longrightarrow C\otimes_{k}L$. So one has $F_{-1}L=0$ and
$F_{0}L={}_{\gamma}L$ (in the notation of Section 2). One can check that $F$
is a comultiplicative filtration on $L$ compatible with the filtration $F$ on
$C$, that is,

$\nu(F_{m}L)\subset\sum\nolimits_{p+q=m}F_{p}C\otimes_{k}F_{q}L\subset
C\otimes_{k}L$

for all $m\geq 0$ [19, Section 4.1].

In other words, this means that $F_{m}L$ is a $C$-subcomodule of $L$ for every
$m\geq 0$ _and_ the successive quotient $C$-comodules $F_{m}L/F_{m-1}L$ are
trivial (i. e., the coaction of $C$ in them is induced by the coaugmentation
$\gamma$). In fact, one has $F_{m}L/F_{m-1}L={}_{\gamma}(L/F_{m-1}L)\subset
L/F_{m-1}L$ for every $m\geq 1$.

By the definition, a coaugmented coalgebra $C$ is conilpotent if and only if
$C=\bigcup_{m=0}^{\infty}F_{m}C$ (i. e., the increasing filtration $F$ on $C$
is exhaustive). In this case, it follows that
$L=\bigcup_{m=0}^{\infty}F_{m}L$, i. e., the increasing filtration $F$ on any
$C$-comodule is exhaustive as well.

###### Proof of Theorem 4.1.

The argument resembles the proofs of the comodule Ext comparison theorems in
[33, Sections 5.4 and 5.7] (see specifically [33, Theorem 5.21]). Notice first
of all that the functor $\Upsilon$ is always fully faithful by [41,
Propositions 2.1.1–2.1.2 and Theorem 2.1.3(e)].

According to Lemma A.5 from the appendix, in order to prove the theorem it
suffices to show that $\operatorname{Ext}^{i}_{C^{*}}(L,J)=0$ for all left
$C$-comodules $L$, all injective left $C$-comodules $J$, and all integers
$1\leq i\leq n$. Equivalently, this means that the space
$\operatorname{Ext}^{i}_{C^{*}}(L,\>C\otimes_{k}U)$ should vanish for all
$k$-vector spaces $U$ and $1\leq i\leq n$.

According to the discussion preceding this proof, the $C$-comodule $L$ has a
natural increasing filtration with (direct sums of) the trivial $C$-comodule
$k$ as the successive quotients. Using the Eklof lemma [6, Lemma 1], the
problem reduces to showing that
$\operatorname{Ext}^{i}_{C^{*}}(k,\>C\otimes_{k}U)=0$ for all $k$-vector
spaces $U$.

Let

(14) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{2}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n}\otimes_{k}C\overset{\tau_{n}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}V_{n+1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

be a minimal injective/cofree resolution of the right $C$-comodule $k$, as per
Corollary 2.6(a) and Proposition 2.7(a). Here $V_{i}$, $i\geq 0$, are some
$k$-vector spaces. Computing the Ext spaces
$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}(k,k)$ using the injective
coresolution (14), we obtain isomorphisms
$V_{i}\simeq\operatorname{Ext}_{C^{\mathrm{op}}}^{i}(k,k)$ (in particular,
$V_{0}\simeq k$). Furthermore, Proposition 3.1 provides isomorphisms
$\operatorname{Ext}_{C^{\mathrm{op}}}^{i}(k,k)\simeq\operatorname{Ext}_{C}^{i}(k,k)$.
So the vector space $V_{i}$ is finite-dimensional for $0\leq i\leq n$ by
assumption.

Applying the dual vector space functor $\operatorname{Hom}_{k}({-},k)$ to the
complex of right $C$-comodules (14), we obtain a resolution of the left
$C^{*}$-module $k$,

(15) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{0}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{2}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{n-1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{n}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(V_{n+1}\otimes_{k}C)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

The exact complex of left $C^{*}$-modules (15) starts and proceeds up to the
homological degree $n$ as a resolution by finitely generated projective
$C^{*}$-modules before turning into a resolution by some arbitrary
$C^{*}$-modules (in fact, projective left $C$-contramodules) from the
homological degree $n+1$ on.

For every $0\leq i\leq n+1$ and any left $C^{*}$-module $J$, the space
$\operatorname{Ext}^{i}_{C^{*}}(k,J)$ can be computed as the degree $i$
cohomology space of the complex obtained by applying the functor
$\operatorname{Hom}_{C^{*}}({-},J)$ to the resolution (15) of the left
$C^{*}$-module $k$ (see Lemma A.1). We are only interested in $0\leq i\leq n$
now, so let us write down the resulting complex in the cohomological degrees
from $0$ to $n+1$. It has the form

$0\longrightarrow V_{0}\otimes_{k}J\longrightarrow
V_{1}\otimes_{k}J\longrightarrow\dotsb\longrightarrow
V_{n}\otimes_{k}J\longrightarrow\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},\>J),$

which for $J=C\otimes_{k}U$ turns into

(16) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\otimes_{k}C\otimes_{k}U\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\otimes_{k}C\otimes_{k}U\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n}\otimes_{k}C\otimes_{k}U\overset{\theta}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},\>C\otimes_{k}U).$

In the cohomological degrees $\leq n$, the complex (16) can be simply obtained
by applying the vector space tensor product functor ${-}\otimes_{k}U$ to the
coresolution (14). This follows from Lemma 4.2. Consequently, the complex (16)
is exact in the cohomological degrees $0<i<n$. It is only the complicated
rightmost term of (16) that remains to be dealt with.

Finally, the same Lemma 4.2 provides a commutative square diagram of
$k$-linear maps

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.43976pt\raise-40.20358pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},\>C\otimes_{k}U)}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where the leftmost vertical isomorphism is
$\eta=\eta_{V_{n}\otimes_{k}C,\>U}$, the rightmost vertical monomorphism is
$\eta=\eta_{V_{n+1}\otimes_{k}C,\>U}$, the horizontal arrows are induced by
the differential $\tau_{n}$ in the coresolution (14), and the diagonal
composition $\theta$ is the rightmost differential in the complex (16). Now it
is clear from the diagram (17) that the kernel of the map $\theta\colon
V_{n}\otimes_{k}C\otimes_{k}U\longrightarrow\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},\>C\otimes_{k}U)$
coincides with the kernel of the map $\tau_{n}\otimes_{k}U\colon
V_{n}\otimes_{k}C\otimes_{k}U\longrightarrow V_{n+1}\otimes_{k}C\otimes_{k}U$
(because the rightmost vertical map $\eta$ is injective). Thus exactness of
the coresolution (14) implies exactness of the complex (16) in the
cohomological degree $n$ as well. ∎

######  Corollary 4.3.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 0$ be an
integer such that the vector space $\operatorname{Ext}_{C}^{i}(k,k)$ is
finite-dimensional for all $1\leq i\leq n$. Then the map of Ext spaces

$\operatorname{Ext}_{C}^{i}(L,M)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{i}(L,M)$

induced by the comodule inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is injective for all left $C$-comodules
$L$ and $M$ and all $0\leq i\leq n+1$.

###### Proof.

This is a purely formal consequence of Theorem 4.1. See Lemma A.3 in the
appendix for the case $n=0$, and Lemma A.4(a) for the general case. ∎

##  5\. Comodule Ext Isomorphism Implies Weakly Finite Koszulity

The aim of this section is to prove the following theorem.

######  Theorem 5.1.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 1$ be the
minimal integer for which the vector space $\operatorname{Ext}_{C}^{n}(k,k)$
is infinite-dimensional. Then the map
$\operatorname{Ext}^{n}_{C}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(k,k)$
induced by the comodule inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is injective, but _not_ surjective. In
fact, if $\lambda$ is the dimension cardinality of the $k$-vector space
$\operatorname{Ext}^{n}_{C}(k,k)$ and $\kappa$ is the cardinality of the field
$k$, then the dimension cardinality of $\operatorname{Ext}^{n}_{C^{*}}(k,k)$
is equal to $\kappa^{\kappa^{\lambda}}$.

The following lemma computing the dimension cardinality of the dual vector
space is a classical result.

######  Lemma 5.2.

Let $V$ be an infinite-dimensional vector space of dimension cardinality
$\lambda$ over a field $k$ of cardinality $\kappa$. Then the dimension
cardinality of the dual $k$-vector space $V^{*}=\operatorname{Hom}_{k}(V,k)$
is equal to $\kappa^{\lambda}$.

###### Proof.

This is [11, Section IX.5]. ∎

Let $C$ be a conilpotent coalgebra. Then the vector space
$\operatorname{Ext}^{1}_{C}(k,k)$ can be interpreted as the _space of
cogenerators_ of the coalgebra $C$ (we refer to [24, Lemma 5.2] for a
discussion). So we will say that a conilpotent coalgebra $C$ is _finitely
cogenerated_ if the vector space $\operatorname{Ext}^{1}_{C}(k,k)$ is finite-
dimensional.

For any coaugmented coalgebra $(C,\gamma)$, the dual linear map
$\gamma^{*}\colon C^{*}\longrightarrow k$ to the coaugmentation $\gamma\colon
k\longrightarrow C$ defines an augmentation on the algebra $C^{*}$.
Accordingly, the one-dimensional vector space $k$ can be endowed with left and
right $C^{*}$-module structures provided by the augmentation $\gamma^{*}$.
These $C^{*}$-module structures on $k$ can be also viewed as coming from the
left and right $C$-comodule structures on $k$ induced by the coaugmentation
$\gamma$. The same module structures also come from the left and right
$C$-contramodule structures on $k$ induced by $\gamma$.

We start with a discussion of the case $n=1$ in Theorem 5.1.

######  Proposition 5.3.

Let $C$ be a conilpotent coalgebra over a field $k$. Then there is a natural
$k$-vector space monomorphism

(18)
$\operatorname{Ext}_{C}^{1}(k,k)^{**}\,\rightarrowtail\,\operatorname{Ext}_{C^{*}}^{1}(k,k).$

The dimension cardinality of the vector space
$\operatorname{Ext}_{C^{*}}^{1}(k,k)$ is equal to that of the vector space
$\operatorname{Ext}_{C}^{1}(k,k)^{**}$.

###### Proof.

Notice first of all the isomorphism
$\operatorname{Ext}_{C^{\mathrm{op}}}^{*}(k,k)\simeq\operatorname{Ext}_{C}^{*}(k,k)$
provided by Proposition 3.1. Furthermore, let $A$ be an augmented $k$-algebra
with an augmentation $\alpha\colon A\longrightarrow k$ and the augmentation
ideal $A_{+}=\ker(\alpha)\subset A$. Then the vector space
$\operatorname{Ext}^{1}_{A}(k,k)\simeq\operatorname{Tor}_{1}^{A}(k,k)^{*}\simeq\operatorname{Ext}^{1}_{A^{\mathrm{op}}}(k,k)$
is computed as the kernel of the $k$-linear map

$A_{+}^{*}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(A_{+}\otimes_{k}A_{+})^{*}$

dual to the multiplication map $A_{+}\otimes_{k}A_{+}\longrightarrow A_{+}$.
In particular, the vector space $\operatorname{Ext}^{1}_{C^{*}}(k,k)$ is
naturally isomorphic to the kernel of the map

(19) $C_{+}^{**}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(C_{+}^{*}\otimes_{k}C_{+}^{*})^{*}$

dual to the multiplication map $C_{+}^{*}\otimes_{k}C_{+}^{*}\longrightarrow
C_{+}^{*}$ (where $C_{+}=C/\gamma(k)$).

On the other hand, from the cobar complex (6) one can immediately see that the
vector space $\operatorname{Ext}_{C}^{1}(k,k)$ is naturally isomorphic to the
kernel of the comultiplication map

$C_{+}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC_{+}\otimes_{k}C_{+}.$

Hence the double dual vector space $\operatorname{Ext}_{C}^{1}(k,k)^{**}$ is
naturally isomorphic to the kernel of the map

(20) $C_{+}^{**}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(C_{+}\otimes_{k}C_{+})^{**}.$

Notice that $(C_{+}^{*}\otimes_{k}C_{+}^{*})$ is naturally a subspace in
$(C_{+}\otimes_{k}C_{+})^{*}$; hence $(C_{+}^{*}\otimes_{k}C_{+}^{*})^{*}$ is
a quotient space of $(C_{+}\otimes_{k}C_{+})^{**}$. One can also observe that
the cokernel of the map $(C_{+}\otimes_{k}C_{+})^{*}\longrightarrow C_{+}^{*}$
is naturally a quotient space of the cokernel of the map
$C_{+}^{*}\otimes_{k}C_{+}^{*}\longrightarrow C_{+}^{*}$. Comparing the maps
(19) and (20), one immediately obtains the desired monomorphism (18).

To compute the dimension cardinality, put $V=\operatorname{Ext}^{1}_{C}(k,k)$,
and notice that $C$ is a subcoalgebra of the cofree conilpotent (tensor)
coalgebra $\operatorname{Ten}(V)=\bigoplus_{n=0}^{\infty}V^{\otimes n}$
cospanned by $V$ [24, Lemma 5.2] (see [34, Sections 2.3 and 3.3] for an
introductory discussion). Hence, if $V$ is infinite-dimensional, then the
dimension cardinalities of $V$ and $C_{+}$ are equal to each other. Now
$V^{**}$ is a subspace in $\operatorname{Ext}^{1}_{C^{*}}(k,k)$, which is in
turn a subspace in $C_{+}^{**}$. Thus the dimension cardinalities of all the
three vector spaces $V^{**}$, $\operatorname{Ext}^{1}_{C^{*}}(k,k)$, and
$C_{+}^{**}$ are equal to each other. If $V$ is finite-dimensional, then the
map (18) is an isomorphism by Theorem 4.1 (for $n=1$). ∎

######  Example 5.4.

Let $C$ be an infinitely cogenerated conilpotent coalgebra. Interpreted in the
light of Lemma A.3 from the appendix, Proposition 5.3 tells that there exists
a short exact sequence of left $C^{*}$-modules

(21) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

with the two-dimensional $C^{*}$-module $M$ _not_ coming from any $C$-comodule
(while the one-dimensional $C^{*}$-module $k$, of course, comes from the one-
dimensional $C$-comodule $k$) via the comodule inclusion functor $\Upsilon$.
Let us explain how to construct a short exact sequence (21) explicitly.

Let $C$ be a conilpotent coalgebra. Recall the notation $F_{0}C=\gamma(k)$ and
$F_{1}C=\ker(C\to C_{+}\otimes_{k}C_{+})$ from Section 4. Then we have a
natural direct sum decomposition $F_{1}C=F_{0}C\oplus V$, where
$V=\ker(C_{+}\to C_{+}\otimes_{k}C_{+})=\operatorname{Ext}^{1}_{C}(k,k)$.
Choose a linear function $f\colon V^{*}\longrightarrow k$. The composition of
linear maps $C^{*}\twoheadrightarrow(F_{1}C)^{*}\twoheadrightarrow
V^{*}\overset{f}{\longrightarrow}k$ defines a linear function $\tilde{f}\colon
C^{*}\longrightarrow k$.

Let $M$ be the two-dimensional $k$-vector space with the basis vectors $e_{1}$
and $e_{2}$. Define the left action of $C^{*}$ in $M$ by the formulas
$ae_{1}=\gamma^{*}(a)e_{1}$ and $ae_{2}=\gamma^{*}(a)e_{2}+\tilde{f}(a)e_{1}$
for all $a\in C^{*}$ (where $\gamma^{*}\colon C^{*}\longrightarrow k$ is the
dual map to the coaugmentation $\gamma$). Then $ke_{1}\subset M$ is a
$C^{*}$-submodule of $M$ isomorphic to $\Upsilon(k)$ and $M/ke_{1}$ is a
quotient module of $M$ also isomorphic to $\Upsilon(k)$. The $C^{*}$-module
$M$ itself belongs to the essential image of the functor $\Upsilon$ if and
only if the linear function $f\colon V^{*}\longrightarrow k$ comes from a
vector in $V$.

Before passing to the case $n>1$ in Theorem 5.1, we need a preparatory lemma.

######  Lemma 5.5.

Let $(C,\gamma)$ be a coaugmented coalgebra over $k$ and $N$ be a right
$C$-comodule. Then there is a natural monomorphism of $k$-vector spaces

$\xi_{N}\colon(N_{\gamma})^{**}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}(N^{*},k)$

from the double dual vector space to the vector subspace $N_{\gamma}\subset N$
to the vector space of left $C^{*}$-module morphisms $N^{*}\longrightarrow k$.
The map $\xi_{N}$ is an isomorphism whenever the coalgebra $C$ is conilpotent
and finitely cogenerated.

###### Proof.

The inclusion $N_{\gamma}\longrightarrow N$ is a morphism of right
$C$-comodules. Consequently, the $k$-vector space dual map
$N^{*}\longrightarrow(N_{\gamma})^{*}$ is an epimorphism of left
$C^{*}$-modules (and in fact, of left $C$-contramodules), where the
$C^{*}$-module (or $C$-contramodule) structure on $(N_{\gamma})^{*}$ is
induced by the (co)augmentation. Applying the functor
$\operatorname{Hom}_{C^{*}}({-},k)$, we obtain the desired injective map of
$k$-vector spaces

$\xi_{N}\colon(N_{\gamma})^{**}=\operatorname{Hom}_{C^{*}}((N_{\gamma})^{*},k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}(N^{*},k).$

If the coalgebra $C$ is conilpotent and finitely cogenerated, then the
contramodule forgetful functor
$C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ from the category of left
$C$-contramodules to the category of left $C^{*}$-modules is fully faithful
[28, Theorem 2.1] (see Theorem 6.4 below). Consequently, we have
$\operatorname{Hom}_{C^{*}}(N^{*},k)=\operatorname{Hom}^{C}(N^{*},k)$.

It remains to observe that the natural map

(22) $(N_{\gamma})^{**}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}^{C}(N^{*},k)$

is an isomorphism for any coaugmented coalgebra $(C,\gamma)$. Indeed, we have

$\operatorname{Hom}^{C}(N^{*},k)\simeq\operatorname{Hom}_{k}({}^{\gamma}(N^{*}),k)\simeq\operatorname{Hom}_{k}((N_{\gamma})^{*},k)=(N_{\gamma})^{**},$

because ${}^{\gamma}(N^{*})\simeq(N_{\gamma})^{*}$. The latter isomorphism is
a particular case of the formula (4) in Section 2. Alternatively, one can
obtain the isomorphism (22) as a particular case of the isomorphism (7) from
Lemma 3.2 for $L=k$ and $M=N$. ∎

######  Proposition 5.6.

Let $C$ be a finitely cogenerated conilpotent coalgebra over a field $k$ and
$n\geq 2$ an integer such that the vector space
$\operatorname{Ext}^{i}_{C}(k,k)$ is finite-dimensional for all $1\leq i\leq
n-1$. Then there is a natural $k$-vector space isomorphism

(23)
$\operatorname{Ext}_{C}^{n}(k,k)^{**}\simeq\operatorname{Ext}_{C^{*}}^{n}(k,k).$

###### Proof.

Similarly to the proof of Theorem 4.1, we consider a minimal injective/cofree
coresolution (14) of the right $C$-comodule $k$. The assumption of the
proposition implies that the vector spaces
$V_{i}\simeq\operatorname{Ext}^{i}_{C}(k,k)$ are finite-dimensional for all
$0\leq i\leq n-1$. Applying the dual vector space functor
$\operatorname{Hom}_{k}({-},k)$ to the complex (14), we obtain a resolution of
the left $C^{*}$-module $k$,

(24) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{0}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{2}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{n-1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(V_{n}\otimes_{k}C)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(V_{n+1}\otimes_{k}C)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

(cf. formula (15), which holds under the stricter assumptions of Theorem 4.1).

Lemma A.1 from the appendix tells that, for every $0\leq i\leq n$ and any left
$C^{*}$-module $M$, the space $\operatorname{Ext}^{i}_{C^{*}}(k,M)$ can be
computed as the degree $i$ cohomology space of the complex obtained by
applying the functor $\operatorname{Hom}_{C^{*}}({-},M)$ to the resolution
(24) of the left $C^{*}$-module $k$. For $M=k$, the resulting complex turns
into

(25) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\\\ \mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}((V_{n}\otimes_{k}C)^{*},\>k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},k).$

By Lemma 5.5, the complex (25) is naturally isomorphic to the complex

(26) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\\\ \mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n}^{**}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n+1}^{**}$

obtained by applying the functor $(({-})_{\gamma})^{**}$ to the coresolution
(14). As the latter coresolution was chosen to be minimal, the differential in
the complex (26) (or, which is the same, in the complex (25)) vanishes. Thus
$\operatorname{Ext}^{n}_{C^{*}}(k,k)\simeq
V_{n}^{**}\simeq\operatorname{Ext}^{n}_{C}(k,k)^{**}$.

This proves existence of _an_ isomorphism (23). To show that this isomorphism
is _natural_ , consider an arbitrary injective coresolution

(27) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muJ^{2}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb$

of the right $C$-comodule $k$. Then

(28) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu(J^{0})^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu(J^{1})^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu(J^{2})^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

is a (nonprojective) resolution of the left $C^{*}$-module $k$. However, the
coresolution (27) is homotopy equivalent (as a complex of injective left
$C$-comodules) to the minimal coresolution (14). Hence the resolution (28) is
homotopy equivalent (as a complex of right $C^{*}$-modules) to the initially
projective resolution (24). Consequently, the natural map (47) from Lemma A.1
for the nonprojective resolution (28) is naturally isomorphic to the similar
map for the initially projective resolution (24).

Thus one can compute $\operatorname{Ext}^{i}_{C^{*}}(k,k)$ for $0\leq i\leq n$
using the nonprojective resolution (28), obtaining natural isomorphisms

$\operatorname{Ext}^{i}_{C^{*}}(k,k)\simeq
H^{i}\operatorname{Hom}_{C^{*}}((J^{\text{\smaller\smaller$\scriptstyle\bullet$}})^{*},k)\simeq
H^{i}((J^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma})^{**})\simeq(H^{i}(J^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\gamma}))^{**}\simeq\operatorname{Ext}^{i}_{C}(k,k)^{**}$

(with the second isomorphism provided by Lemma 5.5) for all $0\leq i\leq n$. ∎

###### Proof of Theorem 5.1.

The map
$\operatorname{Ext}^{n}_{C}(L,M)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(L,M)$
is injective for all left $C$-comodules $L$ and $M$ in our assumptions by
Corollary 4.3. The dimension cardinality assertion of the theorem is provided
by Proposition 5.3 (for $n=1$) or Proposition 5.6 (for $n\geq 2$) together
with Lemma 5.2. The nonsurjectivity assertion follows from the dimension
inequality. ∎

######  Remark 5.7.

Arguing more carefully, one can show that, in the assumptions of Proposition
5.6, the isomorphism (23) identifies the map
$\operatorname{Ext}^{n}_{C}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(k,k)$
induced by the comodule inclusion functor $\Upsilon\colon
C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ with the natural inclusion
$\operatorname{Ext}^{n}_{C}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(k,k)^{**}$
of a vector space $\operatorname{Ext}^{n}_{C}(k,k)$ into its double dual
vector space. For this purpose, one can start with comparing the proofs of
Propositions 3.3 and 5.6 in order to observe that the forgetful functor
$\Theta\colon C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ induces an isomorphism
$\operatorname{Ext}^{C,n}(k,k)\simeq\operatorname{Ext}_{C^{*}}^{n}(k,k)$ in
the assumptions of Proposition 5.6. Then the result of Proposition 3.3
identifies the map
$\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{*}}(k,k)$
induced by the dual vector space functor $N\longmapsto
N^{*}\colon{\operatorname{\mathsf{Comod--}}}C\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ with the natural inclusion
$\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)\longrightarrow\operatorname{Ext}^{n}_{C^{\mathrm{op}}}(k,k)^{**}$.
It remains to restrict the consideration to finite-dimensional comodules and
use the observations from the second proof of Proposition 3.1.

##  6\. Weakly Finite Koszulity Implies Contramodule Ext Isomorphism

Before stating the main theorem of this section, let us have a discussion of
_separated_ and _nonseparated_ contramodules.

Let $C=(C,\gamma)$ be a coaugmented coalgebra and $P$ be a left
$C$-contramodule. For every $m\geq 0$, denote by $F^{m}P\subset P$ the image
of the subspace
$\operatorname{Hom}_{k}(C/F_{m-1}C,P)\subset\operatorname{Hom}_{k}(C,P)$ under
the contraaction map $\pi\colon\operatorname{Hom}_{k}(C,P)\longrightarrow P$.
So one has $F^{0}P=P$ and $F^{1}P=\ker(P\twoheadrightarrow{}^{\gamma}\\!P)$
(in the notation of Sections 2 and 4). One can check that $F$ is a
contramultiplicative decreasing filtration on $P$ compatible with the
increasing filtration $F$ on $C$, that is

$\pi(\operatorname{Hom}_{k}(C/F_{q-1}C,F^{p}P))\subset F^{p+q}P$

for all $p$, $q\geq 0$.

In other words, this means that $F^{m}P$ is a $C$-subcontramodule of $P$ for
every $m\geq 0$ _and_ the successive quotient $C$-contramodules
$F^{m}P/F^{m+1}P$ are trivial (i. e., the contraaction of $C$ in them is
induced by the coaugmentation $\gamma$). In fact, one has
$F^{m}P/F^{m+1}P={}^{\gamma}(F^{m}P)$ for every $m\geq 0$.

The problem with the decreasing filtration $F$ on $P$ is that it is _not_ in
general separated (not even when $C$ is a conilpotent coalgebra); see the
counterexample in [22, Section 1.5] (going back to [39, Example 2.5], [45,
Example 3.20], and [20, Section A.1.1]). The next lemma explains how this
problem can be partially rectified (for a conilpotent coalgebra $C$).

We say that a vector space $P$ endowed with a decreasing filtration $F$ is
_separated_ if the natural map to the projective limit

$\lambda_{P,F}\colon P\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\varprojlim\nolimits_{m\geq 1}P/F^{m}P$

is injective, and that $P$ is _complete_ if the map $\lambda_{P,F}$ is
surjective. A contramodule $P$ over a conilpotent coalgebra $C$ is said to be
_separated_ (respectively, _complete_) if it is separated (resp., complete)
with respect to its natural decreasing filtration $F$ constructed above.

Notice that, for any subcontramodule $Q$ in a contramodule $P$, one has
$F^{m}Q\subset F^{m}P$ for all $m\geq 0$. Thus any subcontramodule of a
separated contramodule is separated. For a free contramodule
$P=\operatorname{Hom}_{k}(C,V)$, one has
$F^{m}P=\operatorname{Hom}_{k}(C/F_{m-1}C,V)$. Hence all free (and therefore,
all projective) contramodules over a conilpotent coalgebra are separated and
complete.

######  Lemma 6.1.

Let $C$ be a conilpotent coalgebra. Then

(a) any $C$-contramodule is complete;

(b) any $C$-contramodule can be presented as the quotient contramodule of a
separated contramodule by a separated subcontramodule.

###### Proof.

Part (b) holds because any contramodule is a quotient of a free contramodule,
which is separated, and any subcontramodule of a separated contramodule is
separated. For part (a), see [20, Lemma A.2.3]. ∎

The aim of this section is to prove the following theorem.

######  Theorem 6.2.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 1$ be an
integer such that the vector space $\operatorname{Ext}^{i}_{C}(k,k)$ is
finite-dimensional for all $1\leq i\leq n$. Then the map of the Ext spaces

(29) $\operatorname{Ext}^{C,i}(P,Q)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{i}(P,Q)$

induced by the forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is an isomorphism for all left
$C$-contramodules $P$, all _separated_ left $C$-contramodules $Q$, and all
$0\leq i\leq n-1$. The map (29) is also an isomorphism for _all_ left
$C$-contramodules $P$ and $Q$ and all $0\leq i\leq n-2$.

The following lemma, describing some tensor products of $C^{*}$-modules for a
coalgebra $C$, is a partial dual version of Lemma 4.2.

######  Lemma 6.3.

Let $C$ be a coalgebra over $k$, let $N$ be a left $C$-comodule, and let $U$
be a $k$-vector space. Then there is a natural map of $k$-vector spaces

$\zeta_{N,U}\colon N^{*}\otimes_{C^{*}}\operatorname{Hom}_{k}(C,U)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(N,U)$

which is an isomorphism whenever $N$ is a finitely cogenerated injective
$C$-comodule. Here the left $C^{*}$-module structure on
$\operatorname{Hom}_{k}(C,U)$ comes from the free left $C$-contramodule
structure, or equivalently, is induced by the right $C^{*}$-module structure
on $C$ (coming from the right $C$-comodule structure on $C$).

###### Proof.

The map $\zeta_{N,U}$ is defined by the formula

$\zeta_{N,U}(f\otimes g)(x)=f(x_{(0)})g(x_{(-1)})$

for all $f\in N^{*}$, $g\in\operatorname{Hom}_{k}(C,U)$, and $x\in N$. Here
$\nu(x)=x_{(-1)}\otimes x_{(0)}\in C\otimes_{k}N$ is a notation for the left
$C$-coaction map $\nu\colon N\longrightarrow C\otimes_{k}N$. So $f(x_{(0)})\in
k$ and $g(x_{(-1)})\in U$.

To prove the isomorphism assertion for a finitely cogenerated injective
$C$-comodule $N$, it suffices to consider the case $N=C\otimes_{k}V$, where
$\dim_{k}V<\infty$. In this case, $N^{*}=V^{*}\otimes_{k}C^{*}$ is a free
right $C^{*}$-module, so
$N^{*}\otimes_{C^{*}}\operatorname{Hom}_{k}(C,U)=V^{*}\otimes_{k}\operatorname{Hom}_{k}(C,U)=\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(C,U))=\operatorname{Hom}_{k}(C\otimes_{k}V,\>U)$.
∎

The most difficult part of the proof of Theorem 6.2 is the case of $n=1$. This
is a previously known result [28]. Notice that, according to the following
theorem, no separatedness assumptions are needed in Theorem 6.2 for $n=1$.

######  Theorem 6.4.

Let $C$ be a finitely cogenerated conilpotent coalgebra over a field $k$. Then
the contramodule forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is fully faithful.

###### Proof.

Moreover, the forgetful functor
$C{\operatorname{\mathsf{--Contra}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ (defined as the composition
$C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$) is fully faithful for any dense subring
$R\subset C^{*}$ in the natural pro-finite-dimensional (otherwise known as
linearly compact or pseudocompact) topology of the $k$-algebra $C^{*}$. See
[28, Theorem 2.1]. For a discussion of related results with further
references, see [22, Section 3.8]. ∎

###### Proof of Theorem 6.2.

The argument resembles the proofs of the contramodule Ext comparison theorems
in [33, Sections 5.4 and 5.7] (see specifically [33, Theorem 5.20]). Notice
first of all that the coalgebra $C$ is finitely cogenerated in the assumptions
of Theorem 6.2, so the functor $\Theta$ is fully faithful by Theorem 6.4.

To prove the first assertion of the theorem, let $Q$ be a separated left
$C$-contramodule. According to Lemma A.6, it suffices to show that
$\operatorname{Ext}^{i}_{C^{*}}(P,Q)=0$ for all projective left
$C$-contramodules $P$ and all integers $1\leq i\leq n-1$.

Lemma 6.1(a) tells that the $C$-contramodule $Q$ is complete with respect to
its natural decreasing filtration $F$; so it is separated and complete under
our assumptions. According to the discussion preceding the lemma, the
successive quotient contramodules $F^{m}Q/F^{m-1}Q$ have trivial
$C$-contramodule structures (i. e., their $C$-contramodule structures are
induced by the coaugmentation $\gamma$). Using the dual Eklof lemma [6,
Proposition 18], the question reduces to showing that
$\operatorname{Ext}_{C^{*}}^{i}(P,T)=0$ for all $k$-vector spaces $T$ with
trivial $C$-contramodule structures.

Any vector space $T$ is a direct summand of the dual vector space $W^{*}$ to
some vector space $W$. Thus is suffices to show that the vector space
$\operatorname{Ext}_{C^{*}}^{i}(P,W^{*})\simeq\operatorname{Tor}^{C^{*}}_{i}(W,P)^{*}$
vanishes. Finally, $W$ is a direct sum of copies of the one-dimensional vector
space $k$, and Tor commutes with direct sums. So we need to show that
$\operatorname{Tor}^{C^{*}}_{i}(k,P)=0$ for any projective left
$C$-contramodule $P$ and $1\leq i\leq n-1$ (and the trivial right
$C^{*}$-module structure on $k$). Equivalently, this means that the space
$\operatorname{Tor}^{C^{*}}_{i}(k,\operatorname{Hom}_{k}(C,U))$ should vanish
for all $k$-vector spaces $U$ and $1\leq i\leq n-1$.

This time, we consider a minimal injective/cofree coresolution of the left
$C$-comodule $k$,

(30) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{2}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{n-1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{n}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muC\otimes_{k}V_{n+1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb,$

and apply the dual vector space functor $\operatorname{Hom}_{k}({-},k)$ to it
in order to obtain an initially projective resolution of the right
$C^{*}$-module $k$,

(31) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{0}^{*}\otimes_{k}C^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{1}^{*}\otimes_{k}C^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{2}^{*}\otimes_{k}C^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{n-1}^{*}\otimes_{k}C^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{n}^{*}\otimes_{k}C^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(C\otimes_{k}V_{n+1})^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

Here the vector space $V_{i}\simeq\operatorname{Ext}_{C}^{i}(k,k)$ is finite-
dimensional for $0\leq i\leq n$ by assumption. For every $0\leq i\leq n+1$ any
any left $C^{*}$-module $P$, the space $\operatorname{Tor}^{C^{*}}_{i}(k,P)$
can be computed as the degree $i$ homology space of the complex obtained by
applying the functor ${-}\otimes_{C^{*}}P$ to the resolution (31) of the right
$C^{*}$-module $k$ (see Lemma A.2). We are only interested in $0\leq i\leq
n-1$ now, so we only write down the resulting complex in the cohomological
degrees from $0$ to $n$. It has the form

$0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{0}^{*}\otimes_{k}P\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{1}^{*}\otimes_{k}P\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{n-1}^{*}\otimes_{k}P\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muV_{n}^{*}\otimes_{k}P,$

which for $P=\operatorname{Hom}_{k}(C,U)$ turns into

(32) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(V_{0},\operatorname{Hom}_{k}(C,U))\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(V_{1},\operatorname{Hom}_{k}(C,U))\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(V_{n-1},\operatorname{Hom}_{k}(C,U))\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu\operatorname{Hom}_{k}(V_{n},\operatorname{Hom}_{k}(C,U)).$

By Lemma 6.3, the complex (32) (in the cohomological degrees $\leq n$) can be
simply obtained by applying the contravariant vector space Hom functor
$\operatorname{Hom}_{k}({-},U)$ to the coresolution (30). Consequently, the
complex (32) is exact in the cohomological degrees $0<i\leq n-1$. Therefore,
$\operatorname{Tor}^{C^{*}}_{i}(k,\operatorname{Hom}_{k}(C,U))=0$ for $0<i\leq
n-1$, as desired.

Having proved the first assertion of the theorem, we can now easily deduce the
second one. For a nonseparated left $C$-contramodule $Y$, by Lemma A.6 we only
need to show that $\operatorname{Ext}^{i}_{C^{*}}(P,Y)=0$ for all projective
left $C$-contramodules $P$ and $1\leq i\leq n-2$. Lemma 6.1(b) provides a
short exact sequence $0\longrightarrow Q^{\prime}\longrightarrow
Q^{\prime\prime}\longrightarrow Y\longrightarrow 0$ with separated
$C$-contramodules $Q^{\prime}$ and $Q^{\prime\prime}$. In the long exact
sequence
$\dotsb\longrightarrow\operatorname{Ext}^{i}_{C^{*}}(P,Q^{\prime\prime})\longrightarrow\operatorname{Ext}^{i}_{C^{*}}(P,Y)\longrightarrow\operatorname{Ext}^{i+1}_{C^{*}}(P,Q^{\prime})\longrightarrow\dotsb$
we have
$\operatorname{Ext}^{i}_{C^{*}}(P,Q^{\prime\prime})=0=\operatorname{Ext}^{i+1}_{C^{*}}(P,Q^{\prime})$,
hence $\operatorname{Ext}^{i}_{C^{*}}(P,Y)=0$. ∎

######  Corollary 6.5.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 0$ be an
integer such that the vector space $\operatorname{Ext}_{C}^{i}(k,k)$ is
finite-dimensional for all $1\leq i\leq n$. Then the map of the Ext spaces

(33) $\operatorname{Ext}^{C,i}(P,Q)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{i}(P,Q)$

induced by the forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is injective for all left
$C$-contramodules $P$, all _separated_ left $C$-contramodules $Q$, and all
$0\leq i\leq n$. The map (33) is also injective for _all_ left
$C$-contramodules $P$ and $Q$ and all $0\leq i\leq n-1$.

###### Proof.

No separatedness assumption is needed in the obvious case $n=0$. For $n\geq
1$, this is a purely formal consequence of Theorem 6.2. See Lemma A.3 for the
case $n=1$ (when the separatedness assumption is not needed, either, by virtue
of Theorem 6.4) and Lemma A.4(a) for the general case. ∎

##  7\. Contramodule Ext Isomorphism Implies Weakly Finite Koszulity

The aim of this section is to prove the following theorem. Together with the
results of the previous Sections 4–6, this will allow us to finish the proof
of Theorem 0.1 from the introduction.

######  Theorem 7.1.

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\geq 1$ be the
minimal integer for which the vector space $\operatorname{Ext}^{n}_{C}(k,k)$
is infinite-dimensional. Let $T$ be an infinite-dimensional $k$-vector space
endowed with the trivial left $C$-contramodule structure (provided by the
coaugmentation $\gamma\colon C\longrightarrow k$). Then the map
$\operatorname{Ext}^{C,n}(T,k)\longrightarrow\operatorname{Ext}_{C^{*}}^{n}(T,k)$
induced by the contramodule forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is _not_ injective. Consequently, there
exists a projective left $C$-contramodule $P$ such that the map
$\operatorname{Ext}^{C,n-1}(P,k)\longrightarrow\operatorname{Ext}_{C^{*}}^{n-1}(P,k)$
induced by the functor $\Theta$ is (injective, but) _not_ surjective.

We start with a discussion of the case $n=1$ in Theorem 7.1. This is the only
case when $\operatorname{Ext}^{C,n-1}(P,k)\neq 0$.

######  Example 7.2.

Let $C$ be an infinitely cogenerated conilpotent coalgebra over $k$, and let
$T$ be an infinite-dimensional vector space endowed with the trivial left
$C$-contramodule structure. Then we claim that there exists a nonsplit short
exact sequence of left $C$-contramodules

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muQ\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muT\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

that splits as a short exact sequence of left $C^{*}$-modules. Consequently,
the splitting map $q\colon Q\longrightarrow k$ is a morphism of
$C^{*}$-modules but _not_ a morphism of $C$-contramodules. So the forgetful
functor $\Theta$ is not full.

To construct the desired short exact sequence, put $C_{+}=C/\gamma(k)$ and
$V=\ker(C_{+}\to C_{+}\otimes_{k}C_{+})=\operatorname{Ext}^{1}_{C}(k,k)$, as
in Example 5.4. So we have $F_{0}C=\gamma(k)$ and $F_{1}C=F_{0}C\oplus V$.
Notice that $V^{*}\otimes_{k}T$ is naturally a proper vector subspace in
$\operatorname{Hom}_{k}(V,T)$, as both the vector spaces $V$ and $T$ are
infinite-dimensional. Essentially,
$V^{*}\otimes_{k}T\subset\operatorname{Hom}_{k}(V,T)$ is the subspace of all
linear maps $V\longrightarrow T$ of finite rank. Choose a linear function
$\phi\colon\operatorname{Hom}_{k}(V,T)\longrightarrow k$ vanishing on
$V^{*}\otimes_{k}T$.

Let the underlying vector space of $Q$ be simply the direct sum $k\oplus T$.
Define the contraaction map
$\pi\colon\operatorname{Hom}_{k}(C,Q)\longrightarrow Q$ as follows. The
component $\pi_{k,T}\colon\operatorname{Hom}_{k}(C,k)\longrightarrow T$ of the
map $\pi$ is zero. The component
$\pi_{k,k}\colon\operatorname{Hom}_{k}(C,k)\longrightarrow k$ is induced by
the coaugmentation $\gamma$. The component
$\pi_{T,T}\colon\operatorname{Hom}_{k}(C,T)\longrightarrow T$ is also induced
by $\gamma$; specifically, it is defined as the composition
$\operatorname{Hom}_{k}(C,T)\longrightarrow\operatorname{Hom}_{k}(k,T)\longrightarrow
T$ of the surjective map
$\operatorname{Hom}_{k}(\gamma,T)\colon\operatorname{Hom}_{k}(C,T)\longrightarrow\operatorname{Hom}_{k}(k,T)$
and the identity isomorphism $\operatorname{Hom}_{k}(k,T)\simeq T$.

Finally, the component
$\pi_{T,k}\colon\operatorname{Hom}_{k}(C,T)\longrightarrow k$ of the map $\pi$
is defined as the composition
$\operatorname{Hom}_{k}(C,T)\longrightarrow\operatorname{Hom}_{k}(F_{1}C,T)\longrightarrow\operatorname{Hom}_{k}(V,T)\longrightarrow
k$ of the surjective map
$\operatorname{Hom}_{k}(C,T)\longrightarrow\operatorname{Hom}_{k}(F_{1}C,T)$
induced by the inclusion $F_{1}C\longrightarrow C$, the direct summand
projection
$\operatorname{Hom}_{k}(F_{1}C,T)=\operatorname{Hom}_{k}(F_{0}C\oplus
V,\>T)\longrightarrow\operatorname{Hom}_{k}(V,T)$, and the chosen linear
function $\phi\colon\operatorname{Hom}_{k}(V,T)\longrightarrow k$.

Then the direct sum decomposition of vector spaces $Q=k\oplus T$ still holds
as a direct sum decomposition in the module category
$C^{*}{\operatorname{\mathsf{--Mod}}}$, since the composition of the embedding
$C^{*}\otimes_{k}T\longrightarrow\operatorname{Hom}_{k}(C,T)$ with the map
$\pi_{T,k}\colon\operatorname{Hom}_{k}(C,T)\longrightarrow k$ is the zero map.
But the $C$-contramodule structure on $Q$ is nontrivial (not induced by the
coaugmentation $\gamma$), so $Q$ is _not_ isomorphic to $k\oplus T$ in
$C{\operatorname{\mathsf{--Contra}}}$.

In order to pass from some object $Q$ in Example 7.2 to a projective object
$P$ in the context Theorem 7.1 for $n=1$, we will use Lemma A.7 from the
appendix. In the case $n>1$, we need the following simple coalgebra-theoretic
lemma.

######  Lemma 7.3.

Let $(C,\gamma)$ be a coaugmented coalgebra, $N$ be a right $C$-comodule, and
$T$ be a $k$-vector space. Then there is a natural isomorphism of $k$-vector
spaces

$\operatorname{Hom}^{C}(\operatorname{Hom}_{k}(N,T),k)\simeq\operatorname{Hom}_{k}(N_{\gamma},T)^{*}.$

###### Proof.

One has
${}^{\gamma}\\!\operatorname{Hom}_{k}(N,T)\simeq\operatorname{Hom}_{k}(N_{\gamma},T)$
by formula (4) from Section 2. Hence

$\operatorname{Hom}^{C}(\operatorname{Hom}_{k}(N,T),k)\simeq\operatorname{Hom}_{k}({}^{\gamma}\\!\operatorname{Hom}_{k}(N,T),k)\simeq\operatorname{Hom}_{k}(\operatorname{Hom}_{k}(N_{\gamma},T),k)$

as desired (cf. Lemma 5.5). ∎

######  Proposition 7.4.

Let $C$ be a finitely cogenerated conilpotent coalgebra over a field $k$ and
$n\geq 2$ be the minimal integer for which the vector space
$\operatorname{Ext}^{n}_{C}(k,k)$ is infinite-dimensional. Let $T$ be an
infinite-dimensional $k$-vector space endowed with the trivial left
$C$-contramodule structure (provided by the coaugmentation $\gamma$). Then the
map
$\operatorname{Ext}^{C,n}(T,k)\longrightarrow\operatorname{Ext}_{C^{*}}^{n}(T,k)$
induced by the contramodule forgetful functor $\Theta\colon
C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ is surjective, but _not_ injective.
Consequently, one has $\operatorname{Ext}_{C^{*}}^{n-1}(P,k)\neq 0$ for the
projective/free left $C$-contramodule $P=\operatorname{Hom}_{k}(C,T)$.

###### Proof.

Our strategy of proving the first assertion of the proposition is to
explicitly compute the map of Ext spaces in question, and see that it is
surjective but not injective. As in the proof of Theorem 4.1, we choose a
minimal injective/cofree coresolution (14) of the right $C$-comodule $k$,

(34) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muk\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{2}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n+1}\otimes_{k}C\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

The vector space
$V_{i}\simeq\operatorname{Ext}^{i}_{C^{\mathrm{op}}}(k,k)\simeq\operatorname{Ext}^{i}_{C}(k,k)$
is finite-dimensional for $0\leq i\leq n-1$ and infinite-dimensional for $i=n$
by assumption.

Applying the contravariant vector space Hom functor
$\operatorname{Hom}_{k}({-},T)$ to the complex (34), we obtain a minimal
projective resolution of the left $C$-contramodule $T$,

(35) $0\longleftarrow
T\longleftarrow\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(V_{0},T))\longleftarrow\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(V_{1},T))\longleftarrow\dotsb\\\
\longleftarrow\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(V_{n-1},T))\longleftarrow\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(V_{n},T))\\\
\longleftarrow\operatorname{Hom}_{k}(C,\operatorname{Hom}_{k}(V_{n+1},T))\longleftarrow\dotsb$

On the other hand, applying the dual vector space functor
$\operatorname{Hom}_{k}({-},k)$ to the complex (34), we obtain a resolution of
the left $C^{*}$-module $k$

(36) $0\mskip 1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5muk\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{0}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{2}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{n-1}^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(V_{n}\otimes_{k}C)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5mu(V_{n+1}\otimes_{k}C)^{*}\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip 1.5mu\dotsb$

as in formula (24) from the proof of Proposition 5.6. Applying the vector
space tensor product functor ${-}\otimes_{k}T$ to the complex (36), we get an
initially projective resolution of the left $C^{*}$-module $T$,

(37) $0\longleftarrow T\mskip
1.5mu\leftarrow\joinrel\relbar\joinrel\relbar\mskip
1.5muC^{*}\otimes_{k}V_{0}^{*}\otimes_{k}T\longleftarrow
C^{*}\otimes_{k}V_{1}^{*}\otimes_{k}T\longleftarrow\dotsb\\\ \longleftarrow
C^{*}\otimes_{k}V_{n-1}^{*}\otimes_{k}T\longleftarrow(V_{n}\otimes_{k}C)^{*}\otimes_{k}T\longleftarrow(V_{n+1}\otimes_{k}C)^{*}\otimes_{k}T.$

For any left $C$-contramodule $Y$, the projective resolution (35) of the
trivial left $C$-contramodule $T$ can be used in order to compute the vector
spaces $\operatorname{Ext}^{C,i}(T,Y)$ in all cohomological degrees $i\geq 0$.
On the other hand, the resolution (37) of the trivial left $C^{*}$-module $T$
is only initially projective. By Lemma A.1, for any left $C^{*}$-module $Y$,
the resolution (37) can be used in order to compute the vector spaces
$\operatorname{Ext}_{C^{*}}^{i}(T,Y)$ in the cohomological degrees $0\leq
i\leq n$.

Now we have a natural injective morphism of complexes of left $C^{*}$-modules
from the complex (37) to the complex (35), acting by the identity map on the
leftmost terms $T$. This morphism of resolutions can be used in order to
compute the maps $\operatorname{Ext}^{C,i}(T,Y)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{i}(T,Y)$ in the cohomological degrees $0\leq
i\leq n$ for any left $C$-contramodule $Y$.

Put $Y=k$. Applying the contramodule Hom functor
$\operatorname{Hom}^{C}({-},k)$ to the projective resolution (35) of the
trivial left $C$-contramodule $T$, we obtain a complex

(38) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{0},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{1},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{n-1},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{n},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{n+1},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb$

By Lemma 7.3, the complex (38) is naturally isomorphic to the complex obtained
by applying the functor $\operatorname{Hom}_{k}(({-})_{\gamma},T)^{*}$ to the
coresolution (34). As the latter coresolution was chosen to be minimal, the
differential in the complex (38) vanishes.

Applying the module Hom functor $\operatorname{Hom}_{C^{*}}({-},k)$ to the
initially projective resolution (37) of the trivial left $C^{*}$-module $T$,
we obtain a complex

(39) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{n-1}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,\operatorname{Hom}_{C^{*}}((V_{n}\otimes_{k}C)^{*},k))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},k)).$

The complex (39) can be also obtained by applying the covariant vector space
Hom functor $\operatorname{Hom}_{k}(T,{-})$ to the complex

(40) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\\\ \mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}((V_{n}\otimes_{k}C)^{*},k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{C^{*}}((V_{n+1}\otimes_{k}C)^{*},k)$

produced by applying the module Hom functor
$\operatorname{Hom}_{C^{*}}({-},k)$ to the resolution (36) of the trivial
$C^{*}$-module $k$ (see formula (25) in the proof of Proposition 5.6).

By Lemma 5.5, the complex (40) is naturally isomorphic to the complex

(41) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{0}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{1}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n-1}\\\ \mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n}^{**}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muV_{n+1}^{**}$

obtained by applying the functor $(({-})_{\gamma})^{**}$ to the coresolution
(34). Once again, as the latter coresolution was chosen to be minimal, the
differential in the complex (41) (or, which is the same, in the complex (40))
vanishes. We have computed the complex (39) as having the form

(42) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,V_{0})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,V_{1})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,V_{n-1})\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,V_{n}^{**})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(T,V_{n+1}^{**}),$

or, which is the same,

(43) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{0}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muT^{*}\otimes_{k}V_{n-1}\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(V_{n}^{*}\otimes_{k}T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(V_{n+1}^{*}\otimes_{k}T)^{*}$

and zero differential.

Now it is finally clear from the formulas (38) and (43) that the map

(44) $\operatorname{Ext}^{C,n}(T,k)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}_{C^{*}}^{n}(T,k)$

is isomorphic to the natural map

(45) $\operatorname{Hom}(V_{n},T)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(V_{n}^{*}\otimes_{k}T)^{*},$

which can be obtained by applying the dual vector space functor
$\operatorname{Hom}_{k}({-},k)$ to the natural embedding

(46) $V_{n}^{*}\otimes_{k}T\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(V_{n},T).$

The map (46) is always injective, but for infinite-dimensional vector spaces
$V_{n}$ and $T$ it is not surjective. Thus the map (45), and consequently the
desired map (44), is surjective but not injective.

To deduce the second assertion of the proposition, we apply Lemma A.4(b) to
the contramodule forgetful functor $\Phi=\Theta$, the left $C$-contramodule
$Y=k$, and the natural epimorphism of left $C$-contramodules
$P=\operatorname{Hom}_{k}(C,T)\longrightarrow T$ with a kernel $X$. By the
first assertion of the proposition, there exists a nonzero extension class
$\beta\in\operatorname{Ext}^{C,n}(T,Y)$ annihilated by the map (44). The class
$\beta$ is also annihilated by the map
$\operatorname{Ext}^{C,n}(T,Y)\longrightarrow\operatorname{Ext}^{C,n}(P,Y)=0$
(as $n\geq 2$ and the $C$-contramodule $P$ is projective). Finally, the map
$\operatorname{Ext}^{C,n-1}(X,Y)\longrightarrow\operatorname{Ext}_{C^{*}}^{n-1}(X,Y)$
is injective by Corollary 6.5 (since the left $C$-contramodule $Y=k$ is
separated). So Lemma A.4(b) tells that the map
$0=\operatorname{Ext}^{C,n-1}(P,Y)\longrightarrow\operatorname{Ext}_{C^{*}}^{n-1}(P,Y)$
cannot be surjective. ∎

###### Proof of Theorem 7.1.

The case $n=1$ is covered by Example 7.2 with Lemma A.7. The case $n\geq 2$ is
treated in Proposition 7.4. ∎

###### Proof of Theorem 0.1.

Follows from Theorems 4.1, 5.1, 6.2, 7.1 (applied to the coalgebras $C$ and
$C^{\mathrm{op}}$) and Propositions 3.1, 3.3. Specifically:

(iv) $\Longrightarrow$ (i) is Theorem 4.1;

(i) $\Longrightarrow$ (iv) is the first assertion of Theorem 5.1;

(iv) $\Longrightarrow$ (ii) is Theorem 4.1 for the coalgebra $C^{\mathrm{op}}$
together with Proposition 3.1;

(ii) $\Longrightarrow$ (iv) is the first assertion of Theorem 5.1 for the
coalgebra $C^{\mathrm{op}}$ together with Proposition 3.1;

(iv) $\Longrightarrow$ (iii) is the first assertion of Theorem 6.2;

(iii) $\Longrightarrow$ (iv) is the second assertion of Theorem 7.1;

(iv) $\Longleftrightarrow$ (v) is Proposition 3.3.

The last assertion of Theorem 0.1 is provided by the second assertion of
Theorem 6.2. ∎

##  8\. Half-Bounded Derived Full-and-Faithfulness

The aim of this short section is to finish the proof of Theorem 0.2 from the
introduction. The argument is based on the following result from the preprint
[29].

######  Proposition 8.1.

(a) Let $\mathsf{A}$ and $\mathsf{B}$ be abelian categories, and
$\Upsilon\colon\mathsf{B}\longrightarrow\mathsf{A}$ be a fully faithful exact
functor. Assume that there are enough injective objects in the abelian
category $\mathsf{A}$, and the functor $\Upsilon$ has a right adjoint functor
$\Gamma\colon\mathsf{A}\longrightarrow\mathsf{B}$. Then the induced
triangulated functor between the bounded derived categories
$\Upsilon^{\mathsf{b}}\colon\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
is fully faithful if and only if the induced triangulated functor between the
bounded below derived categories
$\Upsilon^{+}\colon\mathsf{D}^{+}(\mathsf{B})\longrightarrow\mathsf{D}^{+}(\mathsf{A})$
is fully faithful.

(b) Let $\mathsf{A}$ and $\mathsf{B}$ be abelian categories, and
$\Theta\colon\mathsf{B}\longrightarrow\mathsf{A}$ be a fully faithful exact
functor. Assume that there are enough projective objects in the abelian
category $\mathsf{A}$, and the functor $\Theta$ has a left adjoint functor
$\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$. Then the induced
triangulated functor between the bounded derived categories
$\Theta^{\mathsf{b}}\colon\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
is fully faithful if and only if the induced triangulated functor between the
bounded above derived categories
$\Theta^{+}\colon\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
is fully faithful.

###### Proof.

Part (b) is [29, Proposition 6.5 (c) $\Leftrightarrow$ (d)]. The point is that
any one of the functors $\Theta^{\mathsf{b}}$ and $\Theta^{-}$ is fully
faithful if and only if the higher left derived functors
$\mathbb{L}_{n}\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$, $n\geq 1$ of
the functor $\Delta$ vanish on the essential image of the functor $\Theta$.
Part (a) is the dual assertion. ∎

###### Proof of Theorem 0.2.

Follows from Theorem 0.1 and Proposition 8.1. Specifically:

(i) $\Longleftrightarrow$ (vii) is Theorem 0.1(i) $\Leftrightarrow$ (iv);

(iii) $\Longleftrightarrow$ (vii) is Theorem 0.1(ii) $\Leftrightarrow$ (iv);

(v) $\Longleftrightarrow$ (vii) is Theorem 0.1(iii) $\Leftrightarrow$ (iv) and
the last assertion of Theorem 0.1;

(vii) $\Longleftrightarrow$ (viii) is Theorem 0.1(iv) $\Leftrightarrow$ (v);

(i) $\Longleftrightarrow$ (ii) is Proposition 8.1(a);

(iii) $\Longleftrightarrow$ (iv) is Proposition 8.1(a);

(v) $\Longleftrightarrow$ (vi) is Proposition 8.1(b).

In order to apply Proposition 8.1, it only needs to be explained why the
functor $\Upsilon\colon C{\operatorname{\mathsf{--Comod}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ has a right adjoint and why the functor
$\Theta\colon C{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ has a left adjoint. In fact, these are
quite general properties of coalgebras, as the assumption that $C$ is
conilpotent is not needed here.

For any coalgebra $C$ over a field $k$, the comodule inclusion functor