A bounded below, noncontractible, acyclic complex of projective modules

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings $S$, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings $S$ are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper.


Introduction
Bounded above acyclic complexes of projective objects are contractible.So are bounded below acyclic complexes of injective objects.On the other hand, there is an easy, thematic example of a doubly unbounded, acyclic, noncontractible complex of finitely generated projective-injective modules over the algebra of dual numbers R = k[ǫ]/(ǫ 2 ) (over any field k): (1) We refer to [9,Prologue], [10,] and the references therein for a discussion of the role of the complex (1) in the context of derived Koszul duality and derived categories of the second kind.Do there exist bounded below, noncontractible, acyclic complexes of projective modules; and if so, under what rings?Dual-analogously, are there any bounded above, noncontractible, acyclic complexes of injective modules?These questions were posed, in the context of potential applications to the Finitistic Dimension Conjecture, in the recent preprint of Shaul [16].According to [16,Theorem 5.1], nonexistence of such complexes of projective/injective modules over a two-sided Noetherian ring S with a dualizing complex would imply finiteness of the finitistic dimensions of S.
The aim of the present paper is to show that, over certain rather big rings S, such complexes do exist.The examples of rings S which we obtain are certainly noncommutative and non-Noetherian.The more explicit ones among them are rings of column-finite or row/column-zero-convergent infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables.
On the other hand, in the world of coalgebras over fields, we demonstrate examples of bounded above, acyclic, noncontractible complexes of injective comodules and bounded below, acyclic, noncontractible complexes of projective contramodules over certain cocommutative coalgebras dual to algebras of formal power series in infinitely many variables.These examples go back to [6, Section 0.2.7],where they were very briefly discussed in the context of semi-infinite homological algebra and derived comodule-contramodule correspondence.
Almost all the examples presented in this paper are based on one idea, namely, that of the dual Koszul complex of the ring of polynomials in infinitely many variables.A straightforward realization of this idea is possible in the worlds of comodules and contramodules, but we need an additional trick with a passage to infinite matrices in order to produce examples of complexes of projective/injective modules.The only exception is the (much simpler) universal example, communicated to the author by A. Canonaco.We reproduce it at the end of the paper in Example 8. 4.
The approach to the Finitistic Dimension Conjecture developed in [15,16] goes back to Rickard's paper [14], where it was shown that if the injective modules over a finite-dimensional algebra generate its unbounded derived category as a triangulated category with coproducts, then the finitistic dimension is finite.A counterexample in [14,Theorem 3.5] shows that for the ring of commutative polynomials in infinitely many variables, the generation property fails.Our examples in this paper follow in the footsteps of [6, Section 0.2.7] and [14,Theorem 3.5].We also provide some details of the claims in [6, Section 0.2.7] which were skipped in the book [6].
Acknowledgement.This paper was inspired by Liran Shaul's talk at the Algebra seminar in Prague, organized by Jan Trlifaj.I want to thank both the speaker and the organizer of the seminar.I also wish to thank Alberto Canonaco for communicating his example to me and giving a kind permission to reproduce it here (see Example 8.4).The author is supported by the GA ČR project 23-05148S and the Czech Academy of Sciences (RVO 67985840).

Projective, Flat, and Injective Bounded Acyclicity Problems
The general convention in this paper is that complexes are presumed to be cohomologically graded, so the differential raises the degree.A complex In this notation, it is a standard fact that every bounded above acyclic complex of projective modules/objects (in an abelian or exact category) is contractible, and every bounded below acyclic complex of injective modules/objects is contractible.When we occasionally consider homologically graded complexes, we use the notation with lower indices, P • = (P n , d n : P n → P n−1 ).
Let S be an associative ring.The two "wrong-sided bounded projective/injective acyclicity problems" posed in [16,] are: • Is every bounded above acyclic complex of injective S-modules contractible?
• Is every bounded below acyclic complex of projective S-modules contractible?In addition to the above two, we would like to ask a similar question about flat S-modules.Here one has to be careful: even a two-sided bounded acyclic complex of flat modules need not be contractible.However, such a complex is always pure acyclic, or in other words, has flat modules of cocycles.Thus we ask: • Is every bounded below acyclic complex of flat S-modules pure acyclic?Given a ring S and a left S-module M, the character module M + = Hom Z (M, Q/Z) is a right S-module.The following lemma is well-known.Lemma 1.1.A left S-module F is flat if and only if the right S-module F + is injective.
The next proposition explains the connection between the injective, flat, and projective wrong-sided bounded acyclicity questions, and shows that presenting a counterexample to the "projective" question is enough.Proposition 1.2.Given a ring S, consider the following three properties: (1) Every bounded above acyclic complex of injective right S-modules is contractible.
(2) Every bounded below acyclic complex of flat left S-modules is pure acyclic.
(3) Every bounded below acyclic complex of projective left S-modules is contractible.Then the implications (1) =⇒ ( 2 be a bounded below acyclic complex of flat left S-modules.Then, by the direct implication of Lemma 1.1, is a bounded above acyclic complex of injective right S-modules.A complex of injective modules is contractible if and only if its modules of cocycles are injective.If this is the case for the complex F •,+ , then the inverse implication of Lemma 1.1 tells that the modules of cocycles of the complex F • are flat; so F • is a pure acyclic complex of flat modules. (2) =⇒ (3) By Neeman's theorem [5, Theorem 8.6 (iii) ⇒ (i)], any pure acyclic complex of projective modules is contractible.(Cf.[16,proof of Theorem A.7].)

The Injective Construction of Acyclic Complex of Projectives
Let k be a field, (x α ) α∈A be an infinite set of variables, and R = k[x α : α ∈ A] be the commutative ring of polynomials in the variables x α over k.Endow the onedimensional vector space k over k with the R-module structure by the obvious rule: all the elements x α ∈ R act by zero in k.
Theorem 2.1 (Rickard).For any injective R-module J and all integers n ≥ 0, one has Ext n R (J, k) = 0. Proof.For a countably infinite set of variables x α , this is formulated and proved in [14,Theorem 3.5].The general case of a possibly uncountable index set A is similar.One represents A as the union of its finite subsets B ⊂ A, so the ring R the direct limit of the related polynomial rings R B in finitely many variables, considers the direct limit of finite Koszul complex indexed by the finite subsets B ⊂ A, etc. (Cf. the proof of Theorem 3.1 below for some further details.) Let A be an additive category and M ∈ A be an object.Then we denote by add(M) the full subcategory in A formed by the direct summands of finite direct sums of copies of M. The following lemma is a straightforward category-theoretic generalization of a well-known module-theoretic observation going back to Dress [3].Lemma 2.2.Let A be an idempotent-complete additive category and M ∈ A be an object.
(a) Let S = Hom A (M, M) op be the opposite ring to the endomorphism ring of the object M ∈ A; so the ring S acts on the object M on the right.Then the covariant functor Hom A (M, −) : A −→ S-Mod restricts to an equivalence of additive categories Hom A (M, −) : add(M) ≃ S-mod proj between the full subcategory add(M) ⊂ A and the full subcategory of finitely generated projective left S-modules S-mod proj in the category of left S-modules S-Mod.
(b) Let S = Hom A (M, M) be the endomorphism ring of the object M ∈ A; so the ring S acts on the object M on the left.Then the contravariant functor Hom A (−, M) : A op −→ S-Mod restricts to an anti-equivalence of additive categories Hom A (−, M) : add(M) op ≃ S-mod proj between the full subcategory add(M) ⊂ A and the full subcategory of finitely generated projective left S-modules S-mod proj ⊂ S-Mod.
The following corollary sums up the "injective coresolution construction of a bounded below acyclic complex of projective modules".
Corollary 2.3.Let R = k[x α : α ∈ A] be the ring of polynomials in infinitely many variables over a field k, and let be an injective coresolution of the one-dimensional R-module k.Let J be an injective R-module such that the R-module J n is a direct summand of J for all n ≥ 0. Let be the complex obtained by applying the functor Hom R (J, −) to the truncated coresolution (2).Then (3) is a bounded below, noncontractible, acyclic complex of finitely generated projective left modules over the ring S = Hom R (J, J) op .
Proof.The complex (3) is acyclic by Theorem 2.1.The left S-module Hom R (J, J n ) is a direct summand of the left S-module Hom R (J, J) = S for every n ≥ 0, since the R-module J n is a direct summand of J.So (3) is even a complex of cyclic projective left S-modules (i.e., projective S-modules with one generator).It remains to explain why the complex of S-modules (3) is not contractible.For this purpose, one observes that, given a full subcategory B in an additive category A, a complex C • in B is contractible in A if and only if it is contractible in B. Indeed, any contracting homotopy for C • as a complex in A would be a collection of morphisms in A between objects from B, which means a collection of morphisms in B.
By Lemma 2.2(a) (for A = R-Mod and M = J), the functor Hom R (J, −) is an equivalence of categories add(J) ≃ S-mod proj .The truncated coresolution (2), is a noncontractible (since nonacyclic) complex in R-Mod with the terms belonging to add(J), so it is a noncontractible complex in add(J).Applying the equivalence of additive categories add(J) ≃ S-mod proj , we obtain a noncontractible complex (3) in S-mod proj , which is consequently also noncontractible in S-Mod.It is important for this argument that the functor Hom R (J, −) : add(J) −→ S-Mod is fully faithful.

Dual Rickard's Acyclicity Theorem
The aim of this section is to prove the following dual version of Rickard's theorem [14,Theorem 3.5].
be the ring of polynomials in infinitely many variables over a field k.As above, we endow the one-dimensional k-vector space k with the obvious R-module structure.Then, for any flat R-module P and all integers n ≥ 0, one has Ext n R (k, P ) = 0. Proof.For every α ∈ A, consider the two-term Koszul complex of free R-modules with one generator (4) situated in the cohomological degrees −1 and 0. For every finite subset of indices B ⊂ A, denote by K B • (R) the tensor product, taken over the ring R, of the complexes (4) with α ∈ B. As a finite subset B ⊂ A varies, the complexes K B • (R) form an inductive system, indexed by the poset of all finite subsets B ⊂ A ordered by inclusion. Put module with a set of generators of the cardinality equal to the cardinality of the set A for all n > 0. (More invariantly, K n (R) is the free R-module spanned by the set of all subsets in A of the finite cardinality n).
For any finite subset B ⊂ A, the complex K B • (A) is a finite resolution of the R-module R/ α∈B x α R by finitely generated free R-modules.Passing to the direct limit, one can easily see that The following three lemmas are straightforward or standard.
Lemma 3.2.Let R be an associative ring, Ξ be a directed poset, and (F ξ ) ξ∈Ξ be an inductive system of projective R-modules whose direct limit F = lim − →ξ∈Ξ F ξ is also a projective R-module.Let P be an arbitrary R-module.Then the higher derived inverse limit functors vanish on the projective system Hom R (F ξ , P ) ξ∈Ξ , − →ξ∈Ξ F ξ is projective (but keeping the conditions that the R-modules F ξ are projective), one would have lim ← − i ξ∈Ξ Hom R (F ξ , P ) = Ext i R (F, P ) for every i ≥ 0. Lemma 3.3.Let A be an infinite set and (V B ) B⊂A be a projective system of abelian groups, indexed by the poset of all finite subsets B ⊂ A ordered by inclusion.Assume that there exists an integer n ≥ 0 such that V B = 0 whenever the cardinality of B exceeds n.Then the whole derived inverse limit functor vanishes on the projective system Proof.This is a special case of the assertion that the derived functors of inverse limit are preserved by the passage to a cofinal subsystem.This can be deduced from fact that the derived inverse limits vanish on so-called weakly flabby (faiblement flasque) projective systems [4,Théorème 1.8].A stronger result that the derived inverse limit (in an abelian category with exact product functors) only depends on the pro-object represented by the given projective system can be found in [13,Corollary 7.3.7].
Lemma 3.4.Let Ξ be a directed poset and (C • ξ ) ξ∈Ξ be a projective system of complexes of abelian groups with C n ξ = 0 for n < 0. Then there are two spectral sequences ′ E pq r and ′′ E pq r , starting from the pages with the differentials ′ d pq r : ′ E p,q r −→ ′ E p+r,q−r+1 r and ′′ d pq r : ′′ E p,q r −→ ′′ E p+r,q−r+1 r , converging to the associated graded groups to two different filtrations ′ F p E n and ′′ F p E n on the same graded abelian group E n , n = p + q.
Proof.These are called "the two hypercohomology spectral sequences" (for the derived functor of inverse limit); cf.[2, Section XVII.3].The groups E n are the cohomology groups of the complex obtained by applying the derived functor of inverse limit to the whole complex of projective systems (C • ξ ).Now we can finish the proof of the theorem.By the definition, we have For every n ≥ 0, Lemma 3.2 (with the poset Ξ of all finite subsets B ⊂ A, finitely generated free R-modules F B , and an infinitely generated free R-module F ) tells that lim ← − i B⊂A Hom R (K B n (R), P ) = 0 for all i ≥ 1.On the other hand, the complex Hom R (K B • (R), P ) has its only nonzero cohomology module situated in the cohomological degree n equal to the cardinality of B (as P is a flat module over the ring = 0 for all n ≥ 0.

The Projective Construction of Acyclic Complex of Projectives
Now we are ready to present the "projective resolution construction of a bounded below acyclic complex of projective modules".
be the ring of polynomials in infinitely many variables over a field k, and let be a projective resolution of the one-dimensional R-module k.Let P be a projective R-module such that the R-module P n is a direct summand of P for all n ≥ 0. Let be the complex obtained by applying the contravariant functor Hom R (−, P ) to the truncated resolution (5).Then (6) is a bounded below, noncontractible, acyclic complex of finitely generated projective left modules over the ring S = Hom R (P, P ).
Proof.The complex ( 6) is acyclic by Theorem 3.1.The left S-module Hom R (P n , P ) is a direct summand of the left S-module Hom R (P, P ) = S for every n ≥ 0, since the R-module P n is a direct summand of P .So ( 6) is even a complex of cyclic projective left S-modules.
The proof of the assertion that the complex of S-modules (6) is not contractible is similar to the argument in the proof of Corollary 2.3.By Lemma 2.2(b) (for A = R-Mod and M = P ), the functor Hom R (−, P ) is an anti-equivalence of categories add(P ) op ≃ S-mod proj .The truncated resolution (5), is a noncontractible (since nonacyclic) complex in R-Mod with the terms belonging to add(P ), so it is a noncontractible complex in add(P ).Applying the anti-equivalence of additive categories add(P ) op ≃ S-mod proj , we obtain a noncontractible complex (6) in S-mod proj , which is consequently also noncontractible in S-Mod.It is important for this argument that the contravariant functor Hom R (−, P ) : add(P ) op −−→ S-Mod is fully faithful.

Brief Preliminaries on Coalgebras
In this section and the next two, we consider comodules and contramodules over coassociative, counital coalgebras C over a field k.We refer to the book [17] and the survey papers [7, Section 1], [10, Sections 3 and 8] for background material on coalgebras, comodules, and contramodules.
For any coalgebra C, there are locally finite Grothendieck abelian categories of left and right C-comodules C-Comod and Comod-C, and a locally presentable abelian category of left C-contramodules C-Contra.There are enough injective objects in C-Comod, and they are precisely the direct summands of the cofree left C-comodules C⊗ k V (where V ranges over the k-vector spaces).Dual-analogously, there are enough projective objects in C-Contra, and they are precisely the direct summands of the free left C-contramodules Hom k (C, V ) (where V ∈ k-Vect).
The additive categories of injective left C-comodules and projective left C-contramodules are naturally equivalent, (7) Ψ The equivalence is provided by the restrictions of the adjoint functors for a more detailed discussion.
In fact, we are only interested in one special kind of coalgebras, namely, the symmetric coalgebra Sym(U) of a k-vector space U. To define the symmetric coalgebra, consider the tensor coalgebra Ten(U) = ∞ n=0 U ⊗n , as defined, e. g., in [10, Section 2.3] (where the notation is slightly different).The tensor coalgebra is the cofree conilpotent coalgebra cospanned by U [10, Remark 3.2]; it is also naturally graded.The symmetric coalgebra is simplest defined as the graded subcoalgebra in Ten(U) whose grading components Sym n (U) ⊂ Ten n (U) = U ⊗n are the subspaces of symmetric tensors Sym n (U) ⊂ U ⊗n in the tensor powers of the vector space U.So the whole symmetric coalgebra is [10,Section 8.3], coalgebras C can be described (and in fact, defined) in terms of their vector space dual algebras C * = Hom k (C, k), which carry natural linearly compact ( = pseudocompact) topologies.In particular, if U is a finite-dimensional k-vector space with a basis x * 1 , . . ., x * m , then the dual algebra Sym(U) * to the symmetric coalgebra Sym(U) is the topological algebra of formal Taylor power series Sym(U) Generally speaking, for an infinite-dimensional k-vector space W , one has Sym(W ) = lim − →U⊂W Sym(U) and Sym(W ) * = lim ← −U⊂W Sym(U) * , where U ranges over the finite-dimensional vector subspaces of W . So, if {x * α : α ∈ A} is a k-vector space basis of W , indexed by some set A, then Sym(W , where B ranges over the finite subsets of A. Here, given two finite subsets ] in the projective system takes x α to x α for all α ∈ B ′ and x β to 0 for all β ∈ B ′′ \ B ′ .Such rings ] are the "commutative rings of formal power series in infinitely many variables" that we are interested in.

Comodule and Contramodule Acyclicity Theorems
As above, we denote by W an infinite-dimensional k-vector space with a basis {x As in the proof of Theorem 3.1, we start with considering the two-term Koszul complex of free R B -modules with one generator (8) situated in the cohomological degrees −1 and 0 (where α ∈ B).Denote by K B • ( R B ) the tensor product, taken over the ring R B , of the complexes (8).As the elements {x α : α ∈ B} form a regular sequence in the formal power series ring R B , the complex The (augmented) Koszul complex K B • ( R B ) −→ k is a complex of linearly compact topological k-vector spaces; so it can be obtained by applying the vector space dualization functor Hom k (−, k) to a certain complex of discrete vector spaces.The latter complex has the form where m = dim U B and Λ n (V ), n ≥ 0, denotes the exterior powers of a vector space V .The complex ( 9) is an injective/cofree C B -comodule coresolution of the trivial one-dimensional comodule k over the conilpotent coalgebra C B = Sym(U B ).
Passing to the direct limit of the finite complexes ( 9) over all the finite subsets B ⊂ A, we obtain a bounded below complex (10) The complex (10) is an injective/cofree C-comodule coresolution of the trivial onedimensional comodule k over the conilpotent coalgebra C = Sym(W ).
One can easily check that the coresolutions ( 9) and ( 10) are well-defined and functorial for any k-vector spaces U (in place of U B ) and W , and do not depend on the choice of any bases in the vector spaces.In fact, the differential Sym(W Applying the vector space dualization functor Hom k (−, k) to the complex (10), we obtain a bounded above complex (11) 11) is a projective/free C-contramodule resolution of the trivial onedimensional C-contramodule k.
Applying the functor Φ C = C⊙ C − to the truncated C-contramodule resolution (11), we obtain a bounded above complex of injective/cofree C-comodules Applying the functor Ψ C = Hom C (C, −) to the truncated C-comodule coresolution (10), we obtain a bounded below complex of projective/free C-contramodules For any infinite-dimensional k-vector space W , the complex of cofree comodules (12) is acyclic (i.e., its cohomology spaces vanish in all the degrees).
Proof.This was stated in [6, Section 0.2.7] (as a part of introductory/preliminary material for the book).The proof is not difficult.
The complex (12) is the direct limit of its subcomplexes taken over the directed poset of all finite subsets B ⊂ A. The complex (14), which is a complex of comodules over the subcoalgebra C B = Sym(U B ) of the coalgebra C = Sym(W ), can be obtained by applying the cotensor product functor C B C − to the complex (12) (see [7,] or [6, Section 0.2.1 or 1.2.1]).The complex ( 14) is not acyclic, but its cohomology spaces gradually vanish as the size of the finite subset B ⊂ A grows.Indeed, applying the vector space dualization functor Hom k (−, k) to the finite complex (9), we obtain a finite Koszul complex that was denoted above by K B • ( R B ).It has the form (15) and can be viewed as a projective/free C B -contramodule resolution of the trivial one-dimensional C B -contramodule k.
Applying the functor Φ C B = C B ⊙ C B − to the truncated C B -contramodule resolution (15), we obtain a finite complex of injective/cofree C B -comodules (16) The only cohomology space of the complex ( 16) is the one-dimensional k-vector space Λ m (U B ) * situated in the cohomological degree −m, i. e., at the rightmost term.Consider the direct sum decomposition W = U B ⊕ V B , where V B ⊂ W is the subspace with the basis {x * α : α ∈ A \ B}.Consider the graded dual vector space to the exterior algebra ∞ n=0 Λ n (V B ), and view it as a complex (17) 0 with zero differential.Then the complex ( 14) is the tensor product, taken over the field k, of the complexes ( 16) and (17).Accordingly, the cohomology spaces of the complex ( 14) are concentrated in the cohomological degrees ≤ −m, where m is the cardinality of the set B.
As the size of the subset B ⊂ A grows, the cohomology of the complex ( 14) move away and disappear at the cohomological degree −∞.So the direct limit (12) of the complexes ( 14) is acyclic.Theorem 6.2.For any infinite-dimensional k-vector space W , the complex of free contramodules (13) is acyclic (i.e., its cohomology spaces vanish in all the degrees).
Proof.This was also stated in [6, Section 0.2.7].The proof is only slightly more complicated than the proof of the previous theorem, in that one needs to deal with inverse limits.However, we have done all the preparatory work already.
The complex (13) is the inverse limit of its quotient complexes taken over the directed poset of all finite subsets B ⊂ A. The complex (18), which is a complex of contramodules over the subcoalgebra C B ⊂ C, can be obtained by applying the Cohom functor Cohom C (C B , −) to the complex (13) (see [7, Sections 2.5-2.6] or [6, Section 0.2.4 or 3.2.1]).
Similarly to the previous proof, the complex (18) is not acyclic, but its cohomology spaces gradually vanish as the size of the finite subset B ⊂ A grows.For the sake of completeness of the exposition, let us start with applying the functor Ψ C B = Hom C B (C B , −) to the truncated C B -comodule coresolution (9).We obtain a finite complex of projective/free C B -contramodules The only cohomology space of the complex (19) is the one-dimensional k-vector space Λ m (U B ) situated in the cohomological degree m, i. e, at the rightmost term.In fact, the complex of contramodules (19) can be obtained by applying the vector space dualization functor Hom k (−, k) to the complex of comodules (16).
Consider the exterior algebra ∞ n=0 Λ n (V B ), where as in the previous proof W = U B ⊕ V B , and view it as a complex (20) with zero differential.Then the complex (18) is the complex of k-vector space morphisms, Hom k (−, −), from the complex ( 16) into the complex (20).Accordingly, the cohomology spaces of the complex (18) are concentrated in the cohomological degrees ≥ m.
The rest of the argument proceeds along the lines of the proof of Theorem 3.1, based on Lemmas 3.2-3.4.As mentioned above, the complex (13) is the inverse limit of the complexes (18) taken over the directed poset Ξ of all finite subsets B ⊂ A with respect to inclusion.At every cohomological degree n ≥ 0, Lemma 3.2 (for R = k, F B = C B , and Denote by C • B the complex (18).By Lemma 3.3, we have lim = 0 for all i ≥ 0 and n ≥ 0. Now in the context of Lemma 3.4 we have ′ E pq 2 = 0 for all p, q ≥ 0, and ′′ E pq 1 = 0 for all q ≥ 1.Therefore, E n = 0 and H n (lim = 0 for all n ≥ 0.

Two Contramodule Constructions of Acyclic Complexes of Projectives
We have essentially already constructed the promised bounded above, noncontractible, acyclic complex of injective comodules and bounded below, noncontractible, acyclic complex of projective contramodules over the cocommutative coalgebra C = Sym(W ).Let us state this as a corollary.
Corollary 7.1.Let W be an infinite-dimensional vector space over a field k and C = Sym(W ) be the symmetric coalgebra.Then (a) the complex (12) is a bounded above, noncontractible, acyclic complex of injective comodules over C; (b) the complex (13) is a bounded below, noncontractible, acyclic complex of projective contramodules over C.
Proof.Part (a): the complex ( 12) is acyclic by Theorem 6.1.It remains to explain why the complex of C-comodules (12) is not contractible.
The truncated resolution (11), is a noncontractible (since nonacyclic) complex in the abelian category C-Contra with the terms belonging to the full subcategory of projective objects C-Contra proj , so it is a noncontractible complex in C-Contra proj .Applying the equivalence of additive categories Φ C : C-Contra proj ≃ C-Comod inj (7), we obtain a noncontractible complex (12) in C-Comod inj , which is consequently also noncontractible in C-Comod.Part (b): the complex ( 13) is acyclic by Theorem 6.2.It remains to explain why the complex of C-contramodules (13) is not contractible.
The truncated coresolution (10), is a noncontractible (since nonacyclic) complex in the abelian category C-Comod with the terms belonging to the full subcategory of injective objects C-Comod inj , so it is a noncontractible complex in C-Comod inj .Applying the equivalence of additive categories Ψ C : C-Comod inj ≃ C-Contra proj (7), we obtain a noncontractible complex (13) in C-Contra proj , which is consequently also noncontractible in C-Contra.
be the complex obtained by applying the functor Hom C (J, −) to the truncated coresolution (10).Then (21) is a bounded below, noncontractible, acyclic complex of finitely generated projective left modules over the ring S = Hom C (J, J) op .
Proof.For any C-comodule M, we have Hom . Thus the complex (21) can be obtained by applying the vector space Hom functor Hom k (W, −) to the complex (13), and it follows from Theorem 6.2 that the complex (21) is acyclic.Furthermore, by construction, the C-comodule J 0 = C is a direct summand of J, while the C-comodules J n = C ⊗ k Λ n (W ) are isomorphic to J for n ≥ 1. Hence the left S-module Hom C (J, J 0 ) is a direct summand of Hom C (J, J) = S, and the left S-modules Hom C (J, J n ) are isomorphic to S for n ≥ 1.So (21) is even a complex of cyclic projective left S-modules.
The assertion that the complex of S-modules (21) is not contractible is provable similarly to the argument in the proof of Corollary 2.3.By Lemma 2.2(a) (for A = C-Comod and M = J), the functor Hom C (J, −) is an equivalence of categories add(J) ≃ S-mod proj .The truncated coresolution (10), is a noncontractible (since nonacyclic) complex in C-Comod with the terms belonging to add(J), so it is a noncontractible complex in add(J).Applying the equivalence of additive categories add(J) ≃ S-mod proj , we obtain a noncontractible complex (21) in S-mod proj , which is consequently also noncontractible in S-Mod.It is important for this argument that the functor Hom C (J, −) : add(J) −→ S-Mod is fully faithful.
Alternatively, put P = Hom k (C, W ) ∈ C-Contra proj .Then the co-contra correspondence (7) restricts to an equivalence of additive categories add(J) ≃ add(P) taking J to P. Hence the ring S can be alternatively described as S = Hom C (P, P) op .The complex of left S-modules (21) can be constructed by applying the functor Hom C (P, −) to the complex of C-contramodules (13), whose terms belong to add(P).
Then the noncontractibility argument can be based on Corollary 7.1(b) and the fact that the functor Hom C (P, −) is an equivalence of categories add(P) ≃ S-mod proj (by Lemma 2.2(a) for A = C-Contra and M = P).Once again, it is important for this argument that the functor Hom C (P, −) : add(P) −→ S-Mod is fully faithful.In fact, the whole functor Hom C (P, −) : C-Contra −→ S-Mod is fully faithful (on the whole abelian category C-Contra) by [11,Theorem 6.10].The latter conclusion is based on the observations that P is the coproduct of dim W copies of the projective generator C * = Hom k (C, k) of the abelian category C-Contra, and C * is abstractly κ-small in C-Contra for C = Sym(W ) if κ is the successor cardinality of dim W .
Thus the complex (22) can be obtained by applying the contravariant vector space Hom functor Hom k (−, W * ) to the complex (12), and it follows from Theorem 6.1 that the complex (22) is acyclic.Furthermore, by construction, the C-comodule P 0 = C * is is a direct summand of P, while the C-contramodules P n = Hom k (C, Λ n (W ) * ) are isomorphic to P for n ≥ 1. Hence the left S-module Hom C (P 0 , P) is a direct summand of Hom C (P, P) = S, and the left S-modules Hom C (P n , P) are isomorphic to S for n ≥ 1.So (22) is even a complex of cyclic projective left S-modules.
The assertion that the complex of S-modules (22) is not contractible is provable similarly to the argument in the proof of Corollary 4.1.By Lemma 2.2(b) (for A = C-Contra and M = P), the functor Hom C (−, P) is an anti-equivalence of categories add(P) op ≃ S-mod proj .The truncated resolution (11), is a noncontractible (since nonacyclic) complex in C-Contra with the terms belonging to add(P), so it is a noncontractible complex in add(P).Applying the antiequivalence of additive categories add(P) op ≃ S-mod proj , we obtain a noncontractible complex (22), which is consequently also noncontractible in S-Mod.It is important Let D denote the indexing set of a basis {y δ : δ ∈ D} in the k-vector space W * .The cardinality |D| of the set D is equal to |k| |A| , where |k| is the cardinality of the field k and |A| is the cardinality of the set A. Then the ring S 7.3 = Hom C (J 7.3 , J 7.3 ) = Hom C (P 7.3 , P 7.3 ) is the ring of infinite, column-zero-convergent D×D matrices with entries from the topological commutative formal power series ring R in infinitely many variables indexed by A.
Endowed with its natural topology, the ring S 7.2 becomes a complete, separated right linear topological ring (i.e., a topological ring with a base of neighborhoods of zero formed by open right ideals).Such topological rings were discussed in the papers [8,11,12].Moreover, the ring S 7.2 is topologically left perfect in the sense of [12,Section 14] (as one can see from the discussion in [11,Section 7.3] or [12,Section 5] together with [8,Example 12.3]).Similarly, the ring S 7.3 is a complete, separated, left linear topological ring which is, moreover, topologically right perfect.In other words, the ring S 7.3 is the endomorphism ring of a module with perfect decomposition [12, Section 10], while S 7.2 is the opposite ring to the endomorphism ring of such a module.
Viewed as abstract rings, both the rings S 7. The ring S 7.3 differs from the opposite ring to S 7.2 for the only reason that the cardinality of the set D is larger than that of the set A. One can employ the bigger cofree comodule J 7.3 = C ⊗ k W * in lieu of the cofree comodule J 7.2 = C ⊗ k W in the construction of Corollary 7.2 (while leaving the rest of the construction unchanged).This will produce a pair of opposite rings S and S op , both of them semiregular, both of them satisfying all the claims of Conclusion 8.1.
Remark 8.2.None of the rings S 4.1 , S 7.2 , and S 7.3 is Noetherian on either side.Indeed, consider the ring T = Hom k (k (E) , k (E) ) of infinite, column-finite E × E matrices with entries from the field k (where E is a set and k (E) is the k-vector space with a basis indexed by E).Then T is a quotient ring of S 4.1 (for E = A) and of S 7.3 (for E = D), while the opposite ring to T is a quotient ring of S 7.2 (for E = A).The ring T is von Neumann regular, so it cannot be left or right Noetherian (as any one-sided Noetherian von Neumann regular ring is semisimple Artinian).The commutative ring R = k[x α : α ∈ A] of polynomials in infinitely many variables over a field k is not Noetherian, but it is coherent.Hence the class of flat R-modules is closed under infinite products in R-Mod, and it follows that the R-module Hom R (Q, P ) is flat for any projective R-module Q and flat R-module P .Thus the complex (6) from Corollary 4.1 is a bounded below, acyclic complex of flat equations z n z n+1 = 0, and it remains to let f : S uni −→ S be the homomorphism taking x n to z n for every n ≥ 0.
While the example in Example 8.4 is certainly simpler (to construct and prove its properties) than the examples in Corollaries 2.3, 4.1, 7.2, and 7.3, no example of a bounded below, noncontractible, acyclic complex of projective modules (or of a bounded above, noncontractible, acyclic complex of injective modules) can be too simple.The results of [16, Appendix A] demonstrate this.
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic Email address: positselski@math.cas.cz , Flat, and Injective Bounded Acyclicity Problems 3 2.The Injective Construction of Acyclic Complex of Projectives 4 3. Dual Rickard's Acyclicity Theorem 5 4. The Projective Construction of Acyclic Complex of Projectives functor Φ C being the left adjoint and Ψ C the right adjoint.The functors Ψ C and Φ C are constructed as Ψ C (M) = Hom C (C, M) and Φ C (P) = C ⊙ C P for all M ∈ C-Comod and P ∈ C-Contra.Here ⊙ C : Comod-C × C-Contra −→ k-Vect is the functor of contratensor product over a coalgebra C, while Hom C denotes the Hom functor in the comodule category C-Comod.The equivalence of additive categories (7) is called the (underived ) comodule-contramodule correspondence.We refer to [7, Sections 1.2 and 3.1], [10, Sections 8.6-8.7], or [6, Sections 0.2.6 and 5.1] * α : α ∈ A} indexed by a set A. Given a finite set B, we let R B = k[[x α : α ∈ B]] be the (topological) ring of commutative formal Taylor power series in finitely many variables indexed by B. Furthermore, we put R = lim ← −B⊂A R B (with the transition maps described in the previous section).So, denoting by U B ⊂ W the finite-dimensional vector subspace spanned by {x * α : α ∈ B}, we have R B = Sym(U B ) * and R = lim ← −B⊂A k[[x α : α ∈ B]] = Sym(W ) * .Let us also introduce the notation C B = Sym(U B ) and C = Sym(W ) for the symmetric coalgebras.
of the map induced by the comultiplication map Sym(W ) −→ Sym(W ) ⊗ k W and the map induced by the multiplication map W Now let us present the two contramodule constructions of bounded below, noncontractible, acyclic complexes of projective modules.Recall the notation Hom C (−, −) for the Hom spaces in the category C-Comod.The notation Hom C (−, −) stands for the Hom spaces in the category C-Contra.Corollary 7.2.Let W be an infinite-dimensional vector space over a field k and C = Sym(W ) be the symmetric coalgebra.Let0 −−→ k −−→ J 0 −−→ J 1 −−→ J 2 −−→ • • •be a notation for the injective coresolution (10) of the trivial one-dimensional C-comodule k.Denote by J the cofree C-comodule C ⊗ k W cospanned by W .Let

Remark 8 . 3 .
A simpler construction of rings and complexes of modules satisfying Conclusion 8.1 than the one discussed above exists (see Example 8.4 below).But the following naïve attempt at constructing an example for Conclusion 8.1(b) fails.