Proof. Part (a): for the sake of contradiction, assume that the coaction map $\nu :\ M\longrightarrow D\otimes _kM$ is injective. Then the iterated coaction map $\nu ^{(n)}:\ M\longrightarrow D^{\otimes n}\otimes _kM$ is also injective for every $n\ge 1$ .

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 $n\ge 1$ such that $\mu ^{(n)}(d_i)=0$ in $D^{\otimes n+1}$ for every $1\le i\le r$ . 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 [Reference Positselski20, Lemma A.2.1]. For a discussion of generalizations and other versions of the comodule and contramodule Nakayama lemmas, see [Reference Positselski24, Lemma 2.1].

Proof. This is an equivalent restatement of Lemma 3.1 for $D=C/\gamma (k)$ .

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 $K=\ker (f)\in C{{-\mathsf{Comod}}}$ , then ${}_\gamma K=\ker ({}_\gamma f)$ . Now if ${}_\gamma K=0$ , then $K=0$ by Lemma 3.2(a). The proof of part (b) is similar (or rather, dual-analogous in the sense of [Reference Positselski36, Section 8.2]). The functor $P\longmapsto {}^\gamma \!P$ is right exact, etc.

Proof. Part (a): if the map ${}_\gamma f$ is injective, then the morphism $f$ is injective by Lemma 3.3(a). Now if the $C$ -comodule $I$ is injective, then $f$ is a split monomorphism in $C{{-\mathsf{Comod}}}$ . Therefore, ${}_\gamma \textrm {coker}(f)=\textrm {coker}({}_\gamma f)$ and Lemma 3.2(a) tell us that $f$ is an isomorphism whenever ${}_\gamma f$ is. The proof of part (b) is dual-analogous.

Proof. Part (a): put $V={}_\gamma M$ and $J=C\otimes _k V$ . Then the injective map $V\longrightarrow J$ induced by the map $\gamma :\ k \longrightarrow C$ 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 $f:\ M\longrightarrow J$ . The morphism $f$ is injective by Lemma 3.3(a). The comodule $J$ can be also described as an injective envelope of the comodule $M$ in the comodule category $C{{-\mathsf{Comod}}}$ . 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 $C{{-\mathsf{Contra}}}$ (cf. [Reference Positselski32, Example 12.3]).

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 2). In the situation at hand with a conilpotent coalgebra $C$ , the assersion of part (a) follows straighforwardly from Lemmas 3.4(a) and 3.5(a). Part (b) follows from Lemmas 3.4(b) and 3.5(b).

Proof. Part (a): to construct a minimal injective coresolution of $M$ , all one needs to do is to iterate the construction of Lemma 3.5(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 injective 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 $M\longrightarrow I^\bullet$ and $M\longrightarrow J^\bullet$ be two minimal injective coresolutions of $M$ . Then there exists a morphism of complexes of $C$ -comodules $f:\ I^\bullet \longrightarrow J^\bullet$ making the triangular diagram $M\longrightarrow I^\bullet \longrightarrow J^\bullet$ commutative. Now we have ${}_\gamma I^\bullet =\textrm {Hom}_C(k,I^\bullet )$ and ${}_\gamma J^\bullet =\textrm {Hom}_C(k,I^\bullet )$ . The map of complexes $\textrm {Hom}_C(k,f):\ \textrm {Hom}_C(k,I^\bullet )\longrightarrow \textrm {Hom}_C(k,J^\bullet )$ induces an isomorphism of the cohomology spaces, as both the complexes compute $\textrm {Ext}_C^*(k,M)$ . Since both the complexes ${}_\gamma I^\bullet$ and ${}_\gamma J^\bullet$ have vanishing differentials by assumption, it follows that $\textrm {Hom}_C(k,f)= {}_\gamma f$ is a termwise isomorphism of complexes of vector spaces. Applying Lemma 3.4(a), we conclude that $f$ is a termwise isomorphism of complexes of $C$ -comodules.

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

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

\begin{equation*} 0 \longrightarrow M \longrightarrow C\otimes _k M \longrightarrow C\otimes _k C_+\otimes _k M \longrightarrow C\otimes _k C_+\otimes _k C_+\otimes _k M \longrightarrow \dotsb \end{equation*}

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

(5) \begin{equation} M \longrightarrow C_+\otimes _k M \longrightarrow C_+\otimes _k C_+\otimes _k M \longrightarrow \dotsb \end{equation}

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

(6) \begin{equation} k \longrightarrow C_+ \longrightarrow C_+\otimes _k C_+ \longrightarrow C_+\otimes _k C_+\otimes _k C_+ \longrightarrow \dotsb \end{equation}

[Reference Positselski and Vishik39, Section 1.1], [Reference Positselski19, 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 $\textrm {Ext}_C^*(k,k)$ . 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 $\textrm {Ext}_C^*(k,k)$ on $\textrm {Ext}^*_C(k,M)$ . Thus, the isomorphism $\textrm {Ext}_C^*(k,k)\simeq \textrm {Ext}_{C^{\mathrm{op}}}^*(k,k)$ 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 [Reference Positselski19, Section 2.4]. An alternative proof of the first assertion of the proposition was suggested in [Reference Positselski19, Section 2.5].

Second proof. Let $C{{-\mathsf{comod}}}$ and ${{\mathsf{comod}-}} C$ 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.

First, the fully faithful, exact inclusions of abelian categories $C{{-\mathsf{comod}}}\longrightarrow C{{-\mathsf{Comod}}}$ and ${{\mathsf{comod}-}} C\longrightarrow {{\mathsf{Comod}-}} C$ induce isomorphisms on the Ext spaces. Moreover, the triangulated functors between bounded above derived categories $\mathsf{D}^-(C{{-\mathsf{comod}}})\longrightarrow \mathsf{D}^-(C{{-\mathsf{Comod}}})$ and $\mathsf{D}^-({{\mathsf{comod}-}} C)\longrightarrow \mathsf{D}^-({{\mathsf{Comod}-}} C)$ 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 [Reference Kashiwara and Schapira12, opposite version of Proposition 1.7.11], [Reference Kashiwara and Schapira13, opposite version of Theorem 13.2.8], [Reference Keller and Hazewinkel14, opposite version of Theorem 12.1(b)], or [Reference Positselski and Schnürer37, 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 finite-dimensional $C$ -subcomodule $M'\subset M$ such that the composition $M'\longrightarrow M\longrightarrow L$ is surjective.

Second, 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 [Reference Positselski36, Lemma 3.1(c)]. Consequently, there is a natural anti-equivalence of abelian categories

\begin{equation*} N\longmapsto N^*\;:\; ({{\mathsf{comod}-}} C)^{\mathsf{op}} \longrightarrow C{{-\mathsf{comod}}} \end{equation*}

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

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) \begin{equation} \textrm {Hom}_k(V^*,W^*)\simeq \textrm {Hom}_k(W,V)^{**}, \end{equation}

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

Now, for any two right $C$ -comodules $L$ and $M$ , the vector space $\textrm {Hom}_{C^{\mathrm{op}}}(L,M)$ is the kernel of (the difference of) a natural pair of maps $\textrm {Hom}_k(L,M)\rightrightarrows \textrm {Hom}_k(L,\ M\otimes _k C)$ ,

(9) \begin{equation} 0\longrightarrow \textrm {Hom}_{C^{\mathrm{op}}}(L,M)\longrightarrow \textrm {Hom}_k(L,M)\,\rightrightarrows \textrm {Hom}_k(L,\ M\otimes _kC). \end{equation}

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

(10) \begin{equation} 0\longrightarrow \textrm {Hom}^C(P,Q)\longrightarrow \textrm {Hom}_k(P,Q)\,\rightrightarrows \textrm {Hom}_k(\textrm {Hom}_k(C,P),Q). \end{equation}

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

(11) \begin{equation} 0\longrightarrow \textrm {Hom}^C(M^*,L^*)\longrightarrow \textrm {Hom}_k(M^*,L^*)\,\rightrightarrows \textrm {Hom}_k(\textrm {Hom}_k(C,M^*),L^*). \end{equation}

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 $L$ .

Proof. This observation goes back to the discussion in [Reference Positselski20, Section A.1.2]. The point is that the functor $N\longmapsto N^*:\ {{\mathsf{Comod}-}} C \longrightarrow C{{-\mathsf{Contra}}}$ 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) \begin{equation} 0 \longrightarrow M \longrightarrow J^0 \longrightarrow J^1 \longrightarrow J^2 \longrightarrow \dotsb \end{equation}

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

(13) \begin{equation} 0\longleftarrow \textrm {Hom}_k(M,k)\longleftarrow \textrm {Hom}_k(J^0,k)\longleftarrow \textrm {Hom}_k(J^1,k)\longleftarrow \dotsb, \end{equation}

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

Now let $L$ be a another right $C$ -comodule. Then one can compute the vector spaces $\textrm {Ext}^n_{C^{\mathrm{op}}}(L,M)$ as the cohomology spaces of the complex obtained by applying the functor $\textrm {Hom}_{C^{\mathrm{op}}}(L,{-})$ to the coresolution (12). Similarly, one can compute the vector spaces $\textrm {Ext}^{C,n}(M^*,L^*)$ as the cohomology spaces of the complex obtained by applying the functor $\textrm {Hom}^C({-},L^*)$ to the resolution (13). Taking into account Lemma 4.2, we obtain a natural isomorphism of complexes

\begin{equation*} \textrm {Hom}^C((J^\bullet )^*,L^*)\simeq \textrm {Hom}_{C^{\mathrm{op}}}(L,J^\bullet )^{**} \end{equation*}

inducing a natural isomorphism of the cohomology spaces

\begin{equation*} \textrm {Ext}^{C,n}(M^*,L^*)\simeq \textrm {Ext}^n_{C^{\mathrm{op}}}(L,M)^{**} \end{equation*}

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

Proof. The map $\eta _{N,U}$ is defined by the formula

\begin{equation*} \eta _{N,U}(x\otimes u)(f)=f(x_{(0)})x_{(1)}\otimes u, \end{equation*}

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 $\nu :\ N\longrightarrow N\otimes _kC$ .

To prove the isomorphism assertion for a finitely cogenerated injective $C$ -comodule $N$ , it suffices to consider the case $N=V\otimes _kC$ , where $\dim _kV\lt \infty$ . In this case, $N^*=C^*\otimes _k V^*$ is a free left $C^*$ -module, so $\textrm {Hom}_{C^*}(N^*,\ C\otimes _kU)=\textrm {Hom}_k(V^*,\ C\otimes _kU)= V\otimes _kC\otimes _kU=N\otimes _kU$ .

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

\begin{equation*} \textrm {Hom}_{C^*}(N^*,\ C\otimes _kU) \longrightarrow \textrm {Hom}_k(N^*,U) \end{equation*}

induced by the counit map $\epsilon :\ C\longrightarrow k$ . 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

\begin{equation*} N\otimes _kU\overset \eta \longrightarrow \textrm {Hom}_{C^*}(N^*,\ C\otimes _kU) \longrightarrow \textrm {Hom}_k(N^*,U) \end{equation*}

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

Proof of Theorem 5.1. The argument resembles the proofs of the comodule Ext comparison theorems in [Reference Positselski35, Sections 5.4 and 5.7] (see specifically [Reference Positselski35, Theorem 5.21]). Notice first of all that the functor $\Upsilon$ is always fully faithful by [Reference Sweedler44, 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 $\textrm {Ext}^i_{C^*}(L,J)=0$ for all left $C$ -comodules $L$ , all injective left $C$ -comodules $J$ , and all integers $1\le i\le n$ . Equivalently, this means that the space $\textrm {Ext}^i_{C^*}(L,\ C\otimes _kU)$ should vanish for all $k$ -vector spaces $U$ and $1\le i\le n$ .

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 [Reference Eklof and Trlifaj6, Lemma 1], the problem reduces to showing that $\textrm {Ext}^i_{C^*}(k,\ C\otimes _kU)=0$ for all $k$ -vector spaces $U$ .

Let

(14) \begin{align} 0 \longrightarrow k \longrightarrow V_0\otimes _k C \longrightarrow V_1\otimes _k C \longrightarrow V_2\otimes _k C \longrightarrow \dotsb \\ \nonumber \longrightarrow V_{n-1}\otimes _k C \longrightarrow V_n\otimes _k C \overset {\tau _n} \longrightarrow V_{n+1}\otimes _k C \longrightarrow \dotsb \end{align}

be a minimal injective/cofree coresolution of the right $C$ -comodule $k$ , as per Corollary 3.6(a) and Proposition 3.7(a). Here $V_i$ , $i\ge 0$ , are some $k$ -vector spaces. Computing the Ext spaces $\textrm {Ext}_{C^{\mathrm{op}}}^*(k,k)$ using the injective coresolution (14), we obtain isomorphisms $V_i\simeq \textrm {Ext}_{C^{\mathrm{op}}}^i(k,k)$ (in particular, $V_0\simeq k$ ). Furthermore, Proposition 4.1 provides isomorphisms $\textrm {Ext}_{C^{\mathrm{op}}}^i(k,k)\simeq \textrm {Ext}_C^i(k,k)$ . So the vector space $V_i$ is finite-dimensional for $0\le i\le n$ by assumption.

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

(15) \begin{align} 0\longleftarrow k\longleftarrow C^*\otimes _kV_0^*\longleftarrow C^*\otimes _kV_1^*\longleftarrow C^*\otimes _kV_2^* \longleftarrow \dotsb \\ \nonumber \longleftarrow C^*\otimes _kV_{n-1}^*\longleftarrow C^*\otimes _kV_n^* \longleftarrow (V_{n+1}\otimes _k C)^*\longleftarrow \dotsb \end{align}

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

\begin{equation*} 0\longrightarrow V_0\otimes _k J\longrightarrow V_1\otimes _k J\longrightarrow \dotsb \longrightarrow V_n\otimes _k J\longrightarrow \textrm {Hom}_{C^*}((V_{n+1}\otimes _k C)^*,\ J), \end{equation*}

which for $J=C\otimes _kU$ turns into

(16) \begin{align} 0 \longrightarrow V_0\otimes _k C\otimes _k U \longrightarrow V_1\otimes _k C\otimes _k U \longrightarrow \dotsb \\ \nonumber \longrightarrow V_n\otimes _k C\otimes _k U\overset \theta \longrightarrow \textrm {Hom}_{C^*}((V_{n+1}\otimes _k C)^*,\ C\otimes _kU). \end{align}

In the cohomological degrees $\le n$ , the complex (16) can be simply obtained by applying the vector space tensor product functor ${-}\otimes _kU$ to the coresolution (14). This follows from Lemma 5.2. Consequently, the complex (16) is exact in the cohomological degrees $0\lt i\lt n$ . It is only the complicated rightmost term of (16) that remains to be dealt with.

Finally, the same Lemma 5.2 provides a commutative square diagram of $k$ -linear maps

(17)

where the leftmost vertical isomorphism is $\eta =\eta _{V_n\otimes _kC,\ U}$ , the rightmost vertical monomorphism is $\eta =\eta _{V_{n+1}\otimes _kC,\ U}$ , 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 $\theta :\ V_n\otimes _k C\otimes _k U\longrightarrow \textrm {Hom}_{C^*}((V_{n+1}\otimes _k C)^*,\ C\otimes _kU)$ coincides with the kernel of the map $\tau _n\otimes _kU\;:\; V_n\otimes _k C\otimes _k U\longrightarrow V_{n+1}\otimes _k C\otimes _k U$ (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.

Proof. This is a purely formal consequence of Theorem5.1. See Lemma A.3 in the appendix for the case $n=0$ and Lemma A.4(a) for the general case.

Proof. This is [Reference Jacobson11, Section IX.5].

Proof. Notice first of all the isomorphism $\textrm {Ext}_{C^{\mathrm{op}}}^*(k,k)\simeq \textrm {Ext}_C^*(k,k)$ provided by Proposition 4.1. Furthermore, let $A$ be an augmented $k$ -algebra with an augmentation $\alpha :\ A\longrightarrow k$ and the augmentation ideal $A_+=\ker (\alpha )\subset A$ . Then the vector space $\textrm {Ext}^1_A(k,k)\simeq \textrm {Tor}_1^A(k,k)^*\simeq \textrm {Ext}^1_{A^{\mathrm{op}}}(k,k)$ is computed as the kernel of the $k$ -linear map

\begin{equation*} A_+^* \longrightarrow (A_+\otimes _k A_+)^* \end{equation*}

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

(19) \begin{equation} C_+^{**} \longrightarrow (C_+^*\otimes _k C_+^*)^* \end{equation}

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 $\textrm {Ext}_C^1(k,k)$ is naturally isomorphic to the kernel of the comultiplication map

\begin{equation*} C_+ \longrightarrow C_+\otimes _kC_+. \end{equation*}

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

(20) \begin{equation} C_+^{**} \longrightarrow (C_+\otimes _kC_+)^{**}. \end{equation}

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

To compute the dimension cardinality, put $V=\textrm {Ext}^1_C(k,k)$ , and notice that $C$ is a subcoalgebra of the cofree conilpotent (tensor) coalgebra $\textrm {Ten}(V)=\bigoplus _{n=0}^\infty V^{\otimes n}$ cospanned by $V$ [Reference Positselski26, Lemma 5.2] (see [Reference Positselski36, 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 $\textrm {Ext}^1_{C^*}(k,k)$ , which is in turn a subspace in $C_+^{**}$ . Thus, the dimension cardinalities of all the three vector spaces $V^{**}$ , $\textrm {Ext}^1_{C^*}(k,k)$ , and $C_+^{**}$ are equal to each other. If $V$ is finite-dimensional, then the map (18) is an isomorphism by Theorem5.1 (for $n=1$ ).

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 $\textrm {Hom}_{C^*}({-},k)$ , we obtain the desired injective map of $k$ -vector spaces

\begin{equation*} \xi _N\;:\; (N_\gamma )^{**}=\textrm {Hom}_{C^*}((N_\gamma )^*,k) \longrightarrow \textrm {Hom}_{C^*}(N^*,k). \end{equation*}

If the coalgebra $C$ is conilpotent and finitely cogenerated, then the contramodule forgetful functor $C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ from the category of left $C$ -contramodules to the category of left $C^*$ -modules is fully faithful [Reference Positselski30, Theorem 2.1] (see Theorem7.4 below). Consequently, we have $\textrm {Hom}_{C^*}(N^*,k)=\textrm {Hom}^C(N^*,k)$ .

It remains to observe that the natural map

(22) \begin{equation} (N_\gamma )^{**} \longrightarrow \textrm {Hom}^C(N^*,k) \end{equation}

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

\begin{equation*} \textrm {Hom}^C(N^*,k)\simeq \textrm {Hom}_k({}^\gamma (N^*),k) \simeq \textrm {Hom}_k((N_\gamma )^*,k)=(N_\gamma )^{**}, \end{equation*}

because ${}^\gamma (N^*)\simeq (N_\gamma )^*$ . The latter isomorphism is a particular case of the formula (4) in Section 3. Alternatively, one can obtain the isomorphism (22) as a particular case of the isomorphism (7) from Lemma 4.2 for $L=k$ and $M=N$ .

Proof. Similarly to the proof of Theorem5.1, we consider a minimal injective/cofree coresolution (14) of the right $C$ -comodule $k$ . The assumption of the proposition implies that the vector spaces $V_i\simeq \textrm {Ext}^i_C(k,k)$ are finite-dimensional for all $0\le i\le n-1$ . Applying the dual vector space functor $\textrm {Hom}_k({-},k)$ to the complex (14), we obtain a resolution of the left $C^*$ -module $k$ ,

(24) \begin{align} 0\longleftarrow k\longleftarrow C^*\otimes _kV_0^*\longleftarrow C^*\otimes _kV_1^*\longleftarrow C^*\otimes _kV_2^* \longleftarrow \dotsb \\ \nonumber \longleftarrow C^*\otimes _kV_{n-1}^*\longleftarrow (V_n\otimes _k C)^* \longleftarrow (V_{n+1}\otimes _k C)^*\longleftarrow \dotsb \end{align}

(cf. formula (15), which holds under the stricter assumptions of Theorem5.1).

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

(25) \begin{align} 0 \longrightarrow V_0 \longrightarrow V_1 \longrightarrow \dotsb \longrightarrow V_{n-1}\\ \nonumber \longrightarrow \textrm {Hom}_{C^*}((V_n\otimes _k C)^*,\ k) \longrightarrow \textrm {Hom}_{C^*}((V_{n+1}\otimes _k C)^*,k). \end{align}

By Lemma 6.5, the complex (25) is naturally isomorphic to the complex

(26) \begin{equation} 0 \longrightarrow V_0 \longrightarrow V_1 \longrightarrow \dotsb \longrightarrow V_{n-1} \longrightarrow V_n^{**} \longrightarrow V_{n+1}^{**} \end{equation}

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, $\textrm {Ext}^n_{C^*}(k,k)\simeq V_n^{**}\simeq \textrm {Ext}^n_C(k,k)^{**}$ .

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

(27) \begin{equation} 0 \longrightarrow k \longrightarrow J^0 \longrightarrow J^1 \longrightarrow J^2 \longrightarrow \dotsb \end{equation}

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

(28) \begin{equation} 0\longleftarrow k\longleftarrow (J^0)^*\longleftarrow (J^1)^*\longleftarrow (J^2)^* \longleftarrow \dotsb \end{equation}

is a (nonprojective) resolution of the left $C^*$ -module $k$ . However, the coresolution (27) is homotopy equivalent (as a complex of injective right $C$ -comodules) to the minimal coresolution (14). Hence, the resolution (28) is homotopy equivalent (as a complex of left $C^*$ -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 $\textrm {Ext}^i_{C^*}(k,k)$ for $0\le i\le n$ using the nonprojective resolution (28), obtaining natural isomorphisms

\begin{equation*} \textrm {Ext}^i_{C^*}(k,k)\simeq H^i\textrm {Hom}_{C^*}((J^\bullet )^*,k) \simeq H^i((J^\bullet _\gamma )^{**})\simeq (H^i(J^\bullet _\gamma ))^{**} \simeq \textrm {Ext}^i_C(k,k)^{**} \end{equation*}

(with the second isomorphism provided by Lemma 6.5) for all $0\le i\le n$ .

Proof of Theorem 6.1. The map $\textrm {Ext}^n_C(L,M)\longrightarrow \textrm {Ext}^n_{C^*}(L,M)$ is injective for all left $C$ -comodules $L$ and $M$ in our assumptions by Corollary 5.3. The dimension cardinality assertion of the theorem is provided by Proposition 6.3 (for $n=1$ ) or Proposition 6.6 (for $n\ge 2$ ) together with Lemma 6.2. The nonsurjectivity assertion follows from the dimension inequality.

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 [Reference Positselski20, Lemma A.2.3].

Proof. The map $\zeta _{N,U}$ is defined by the formula

\begin{equation*} \zeta _{N,U}(f\otimes g)(x)=f(x_{(0)})g(x_{(-1)}) \end{equation*}

for all $f\in N^*$ , $g\in \textrm {Hom}_k(C,U)$ , 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 $\nu :\ N\longrightarrow C\otimes _k N$ . 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 $N=C\otimes _kV$ , where $\dim _kV\lt \infty$ . In this case, $N^*=V^*\otimes _k C^*$ is a free right $C^*$ -module, so $N^*\otimes _{C^*}\textrm {Hom}_k(C,U)=V^*\otimes _k\textrm {Hom}_k(C,U)=\textrm {Hom}_k(V,\textrm {Hom}_k(C,U)) =\textrm {Hom}_k(C\otimes _kV,\ U)$ .

Proof. Moreover, the forgetful functor $C{{-\mathsf{Contra}}}\longrightarrow R{{-\mathsf{Mod}}}$ (defined as the composition $C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}\longrightarrow R{{-\mathsf{Mod}}}$ ) 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 [Reference Positselski30, Theorem 2.1]. For a discussion of related results with further references, see [Reference Positselski24, Section 3.8].

Proof of Theorem 7.2. The argument resembles the proofs of the contramodule Ext comparison theorems in [Reference Positselski35, Sections 5.4 and 5.7] (see specifically [Reference Positselski35, Theorem 5.20]). Notice first of all that the coalgebra $C$ is finitely cogenerated in the assumptions of Theorem7.2, so the functor $\Theta$ is fully faithful by Theorem7.4.

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 $\textrm {Ext}^i_{C^*}(P,Q)=0$ for all projective left $C$ -contramodules $P$ and all integers $1\le i\le n-1$ .