Jacobi-Zariski long nearly exact sequences for associative algebras
Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, Andrea Solotar
aa r X i v : . [ m a t h . K T ] S e p Jacobi-Zariski long nearly exact sequences forassociative algebras
Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos,and Andrea Solotar ∗ Abstract
For an extension of associative algebras B ⊂ A over a field and an A -bimodule X , we obtain a Jacobi-Zariski long nearly exact sequence relatingthe Hochschild homologies of A and B , and the relative Hochschild homology,all of them with coefficients in X . This long sequence is exact twice in three.There is a spectral sequence which converges to the gap of exactness, whichterms at page are Tor p + qB e ( X, ( A/B ) ⊗ pB ) for p, q > . MSC2020: 18G25, 16E40, 16E30, 18G15
Keywords:
Hochschild, homology, relative, Jacobi-Zariski Introduction
Let k be a field. A long sequence of vector spaces and linear maps is called nearlyexact if it is a complex which is exact twice in three, see Definition 4.1. Its homologyat the spots where it is not exact is a graded vector space called the gap of thesequence.Let B ⊂ A be an extension of associative k -algebras. Let X be an A -bimodule.In this paper we show that there is a Jacobi-Zariski long nearly exact sequencerelating the Hochschild homologies of A and B , and the relative one, all of themwith coefficients in X . Moreover, there is a spectral sequence converging to thegap of this Jacobi-Zariski nearly exact sequence. Its first page is given by E p,q = Tor B e q ( X, ( A/B ) ⊗ B p ) for p, q > and zeros elsewhere.If A/B is flat as a B -bimodule, A. Kaygun in [16, 17] has obtained a Jacobi-Zariski long exact sequence. The long nearly exact sequence that we work out inthis paper specialises to Kaygun’s long exact sequence. Indeed, if A/B is flat thenthe spectral sequence converges to and the nearly exact sequence is actually exact.On the other hand, we improve the Jacobi-Zariski long exact sequence of [7]. Inthis paper we do not assume that the bounded extension B ⊂ A splits.The Jacobi-Zariski sequence - also called transitivity sequence - is originatedin commutative algebra, see for instance [1, p. 61] or [15]. Given a sequence of ∗ This work has been supported by the projects UBACYT 20020130100533BA, PIP-CONICET 11220150100483CO, USP-COFECUB. The third mentioned author was supportedby the thematic project of FAPESP 2014/09310-5, and acknowledges support from the“Brazilian-French Network in Mathematics” and from the CNPq research grant 302003/2018-5. The fourth mentioned author is a research member of CONICET (Argentina) and a SeniorAssociate at ICTP. wo morphisms between three commutative rings, there is an associated long exactsequence relating the Andr´e-Quillen homology groups, see also [11]. In characteristiczero, Andr´e-Quillen homology is a direct summand of Hochschild homology definedin [13], see [21] and [2]. Moreover, under flatness hypotheses, it is isomorphic toHarrison homology with degree shifted by 1. An example of use of the Jacobi-Zariskiexact sequence in commutative algebra can be found in [20]. Concerning the originof the name of Jacobi-Zariski given to the exact sequence for commutative rings,see [1, p. 102].In [7], the Jacobi-Zariski sequence in non commutative algebra has been usefulin relation to Han’s conjecture (see [12]). Moreover, it enables to give formulas forthe change of dimension of Hochschild (co)homology when deleting or adding aninert arrow to the quiver of an algebra in [8, 9].The aim of this work is to develop the theory of the Jacobi-Zariski long nearlyexact sequences for arbitrary extensions of algebras. In a forthcoming work, we willuse it in relation to Han’s conjecture for a bounded extension of finite dimensionalalgebras which is non necessarily split. Moreover, the Jacobi-Zariski long nearly ex-act sequence will be also useful for describing the change of dimension of Hochschild(co)homology when adding or deleting non inert arrows.Below we summarise the contents of this paper.In Section 2 we provide a brief account of relative Hochschild homology asdefined by G. Hochschild in [14]. Then we introduce the normalised relative barresolution of an algebra A with respect to a subalgebra B . Up to our knowledge, thisresolution has been considered only once previously in [10, p.56]. If there exists atwo-sided ideal M such that A = B ⊕ M , then the normalised relative bar resolutionis the one in [7, Theorem 2.3], see also [5, Lemma 2.1]. When B = k it is the usualnormalised bar resolution.We provide a direct proof of the existence of the normalised relative bar res-olution. It relies on the choice of a k -linear section σ to the canonical projection π : A → A/B . The proof enables to set up tools and techniques for the rest of thepaper. Actually the differential does not depend on the choice of σ but it is notprovided by a simplicial module (see for instance [18, p. 44]).The normalised relative bar resolution enables us to provide a short nearly exactsequence that we call fundamental in Section 3. It is a short sequence of complexeswhich is exact except may be in its middle complex, where it has a “gap complex”.For an A -bimodule X , this fundamental sequence is based on the complex obtainedfrom the normalised relative bar resolution, as well as on the usual complexes of A and B for Hochschild homology with coefficients in X .In Section 4 we first prove a general result: to an arbitrary short nearly ex-act sequence of complexes, we associate a long nearly exact sequence in homol-ogy. Specialising this procedure to the fundamental sequence, we obtain the aimedJacobi-Zariski long nearly exact sequence.Moreover, we show that the homology of the gap complex of an arbitrary shortnearly exact sequence is precisely the gap of the associated long nearly exact se-quence. As before, we apply this to the Jacobi-Zariski long nearly exact sequencethat we have obtained.Section 5 is devoted to approximate the gap of the Jacobi-Zariski sequence.This is achieved by firstly describing the gap complex of the fundamental sequence.Then, by choosing a linear section σ to the canonical projection π : A → A/B , weprovide a filtration of the gap complex. In the associated graded complex of thisfiltration, S = σ ( A/B ) plays an important role when endowed with the transported tructure of B -bimodule isomorphic to A/B . We show that the homology of thequotients of the filtration are the required Tor vector spaces.In the last section, as mentioned, we retrieve previous results from [16, 17] andwe improve the Jacobi-Zariski sequence from [7]. Normalised relative bar resolution
Let k be a field and let B ⊂ A be an extension of k -algebras. The category of A -modules is exact with respect to short exact sequences of A -modules which aresplit as sequences of B -modules, see [22, 3]. The relative projective A -modulesare described for instance in [7], they are A -direct summands of induced modulesfrom B . A relative projective resolution of an A -module is made with relativeprojectives and has a B -contracting homotopy, see [14, p. 250]. This way for anyright A -module X and left A -module Y , the vector spaces Tor A | B ∗ ( X, Y ) are welldefined.The extension of algebras B ⊂ A provides an extension of the enveloping alge-bras B e ⊂ A e . Let X be a left A e -module, that is an A -bimodule. G. Hochschilddefined in [14] H ∗ ( A | B, X ) =
Tor A e | B e ∗ ( X, A ) as the relative Hochschild homology of X .As pointed out in [7], it comes down to the same to consider the extensions B ⊗ A op ⊂ A e or B e ⊂ A e for the above definitions.We will now introduce the normalised relative bar resolution. Its existence in[10] is based on the use of the reference [19], which relevance is not clear for us. Remark 2.1
In general
A/B has no associative multiplicative structure. To getaround this, we consider a k -section σ to π : A → A/B the canonical B -bimoduleprojection, that is πσ = 1 . Observe that in general σ cannot be chosen to bea B -bimodule map. Through σ there is a non associative product in A/B : if α, α ′ ∈ A/B , we consider π ( σ ( α ) σ ( α ′ )) ∈ A/B . We will use this to define thedifferentials below.
Remark 2.2
Each summand of the differential in the next proposition is not welldefined with respect to tensor products over B . Nevertheless, their alternate sum iswell defined. In other words the next resolution is not given by a simplicial moduleas defined for instance in [18, p. 44]. Proposition 2.3
Let B ⊂ A be an extension of algebras. There is a relativeresolution of A which we call the normalised relative bar resolution: · · · d → A ⊗ B ( A/B ) ⊗ B m ⊗ B A d → · · · d → A ⊗ B A/B ⊗ B A d → A ⊗ B A d → A → where the last d is the product of A and d ( a ⊗ α ⊗ · · · ⊗ α n − ⊗ a n ) = a σ ( α ) ⊗ α ⊗ · · · ⊗ α n − ⊗ a n + n − X i =1 ( − i a ⊗ α ⊗ · · · ⊗ π ( σ ( α i ) σ ( α i +1 )) ⊗ · · · ⊗ α n − ⊗ a n +( − n − a ⊗ α ⊗ · · · ⊗ σ ( α n − ) a n . The differential d does not depend on the section σ . roof. We outline the main steps and tools of the proof. • The bimodules involved are induced bimodules, hence they are relative pro-jective, see for instance [7]. • A somehow intricate but straightforward computation shows that d is welldefined with respect to tensor products over B . It uses repeatedly the fol-lowing key fact: if b ∈ B and α ∈ A/B , then there exists c ∈ B suchthat σ ( bα ) = bσ ( α ) + c . Indeed, π is a morphism of B -bimodules, hence π ( σ ( bα ) − bσ ( α )) = 0 . The right analogous is also needed, namely thereexists c ′ ∈ B such that σ ( αb ) = σ ( α ) b + c ′ . The other important point touse is that the tensor products are over B at the codomain. • The maps d are clearly A -bimodule morphisms. • There is a contracting homotopy s given by s ( a ⊗ α ⊗ · · · ⊗ α n − ⊗ a n ) = 1 ⊗ π ( a ) ⊗ α ⊗ · · · ⊗ α n − ⊗ a n . Note first that s is well defined with respect to tensor products over B . Next,to check that sd + ds = 1 , use that if a ∈ A then there exists c ∈ B suchthat σπ ( a ) = a + c .This contracting homotopy is a B − A -morphism. • We indicate two ways to verify that d = 0 .The first one relies on the fact that in each degree the codomain of s isgenerated by Im s as an A -bimodule. In the inductive step of the proof oneshows that d s = 0 , hence d = 0 .The second one is by doing the computation of d , using repeatedly that if α and α ′ are in A/B , then there exists c ∈ B such that σ ( π ( σ ( α ) σ ( α ′ ))) = σ ( α ) σ ( α ′ ) + c. • Finally let σ ′ be another k -section. If α ∈ A/B then there exists c α ∈ B suchthat σ ′ ( α ) = σ ( α )+ c α . Using this, an elaborate but not difficult computationshows that d does not depend on the section. ⋄ Corollary 2.4
Let B ⊂ A be an extension of k -algebras, let σ be a k -linear sectionof π : A → A/B , and let X be an A -bimodule. H ∗ ( A | B, X ) is the homology of the chain complex C ∗ ( A | B, X ) : · · · b A | B → X ⊗ B e ( A/B ) ⊗ B m b A | B → · · · b A | B → X ⊗ B e A/B b A | B → X B → where X B = X ⊗ B e B = X/ h bx − xb i = H ( B, X ) and b A | B ( x ⊗ α ⊗ · · · ⊗ α n ) = xσ ( α ) ⊗ α ⊗ · · · ⊗ α n + n − X i =1 ( − i x ⊗ α ⊗ · · · ⊗ π ( σ ( α i ) σ ( α i +1 )) ⊗ · · · ⊗ α n +( − n σ ( α n ) x ⊗ α ⊗ · · · ⊗ α n − . The differential b A | B does not depend on the choice of the linear section σ . Fundamental short nearly exact sequence for relativehomology
In this section we associate to an extension of algebras a fundamental short nearlyexact sequence of complexes.
Definition 3.1
Let → C ∗ ι → D ∗ κ → E ∗ → be a sequence of positively graded chain complexes of vector spaces. It is called short nearly exact if ι is injective, κ is surjective and κι = 0 .The complex ( Ker κ/ Im ι ) ∗ is its gap complex . Lemma 3.2
Let (3.1) be a short nearly exact sequence of positively graded com-plexes, and consider it as a double complex after the standard change of signs. Thespectral sequence obtained by filtering the double complex by rows converges to thehomology of the gap complex.
Proof.
At page , the spectral sequence given by the filtration by rows has asingle column at p = 1 which is the gap complex, its boundaries are induced bythose of D ∗ . Hence in page we also have a single column at p = 1 , its valuesare the homology of the gap complex. The differentials come from or go to , sothese spaces live forever and the spectral sequence converges to them. ⋄ Let A be a k -algebra and let X be an A -bimodule. We denote C ∗ ( A, X ) : · · · b A → X ⊗ A ⊗ m b A → · · · b A → X ⊗ A ⊗ A b A → X ⊗ A b A → X → the usual complex which computes the Hochschild homology H ∗ ( A, X ) . To obtaina nearly exact sequence of complexes, we consider a slightly modified truncatedcomplex in degrees and , without changing degrees, as follows: ˇ C ∗ ( A, X ) : · · · b A → X ⊗ A ⊗ m b A → · · · b A → X ⊗ A ⊗ A b A → Ker b A → . This is well defined since Im b A ⊂ Ker b A . This complex still computes H ∗ ( A, X ) except in degree . Similarly we consider ˇ C ∗ ( A | B, X ) : · · · b A | B → X ⊗ B e A/B ⊗ B n b A | B → · · · b A | B → X ⊗ B e A/B ⊗ B A/B b A | B → Ker b A | B → which still computes H ∗ ( A | B, X ) in positive degrees. Theorem 3.3
Let B ⊂ A be an extension of k -algebras and let X be an A -bimodule. There is a short sequence of positively graded chain complexes → ˇ C ∗ ( B, X ) ι → ˇ C ∗ ( A, X ) κ → ˇ C ∗ ( A | B, X ) → which is nearly exact, except in degree where κ is not necessarily surjective. Itwill be called the fundamental sequence . roof. The map ι is clear, and is trivially a map of complexes. For n ≥ we set κ ( x ⊗ a ⊗ · · · ⊗ a n ) = x ⊗ π ( a ) ⊗ · · · ⊗ π ( a n ) which is surjective and verifies κι = 0 . It remains to prove that κ is a map ofcomplexes.In degree zero of the original complexes, let κ : X → X B be given by κ ( x ) = x .Notice that we have X ι =1 → X κ → X B but κι = 0 . This is the reason for havingconsidered the above truncated modification of the original complexes.To obtain κ | : Ker b A = ˇ C ( A, X ) −→ ˇ C ( A | B, X ) =
Ker b A | B we prove that the following diagram is commutative X ⊗ A b A (cid:15) (cid:15) κ / / X ⊗ B e A/B b A | B (cid:15) (cid:15) X κ / / X B Recall that σ is a chosen linear section of the canonical projection π : A → A/B .We have • κb A ( x ⊗ a ) = κ ( xa − ax ) = ( xa − ax ) . • b A | B κ ( x ⊗ a ) = b A | B ( x ⊗ π ( a )) = xσ ( π ( a )) − σ ( π ( a )) x. There exists c ∈ B such that σ ( π ( a )) = a + c . Hence b A | B κ ( x ⊗ a ) = xa + xc − ax − cx. Finally observe that since c ∈ B , we have that xc = cx as elements of X B .We infer the existence of the restriction κ | : Ker b A → Ker b A | B . At degree ofthe original complexes we have κι = 0 , hence the compositionKer b B ι | → Ker b A κ | → Ker b A | B is also zero. However κ | : Ker b A → Ker b A | B is not surjective in general.Next we verify that κ is a morphism of complexes.In degree we have κb A ( x ⊗ a ⊗ a ) = κ ( xa ⊗ a − x ⊗ a a + a x ⊗ a )= xa ⊗ π ( a ) − x ⊗ π ( a a ) + a x ⊗ π ( a ) . While b A | B κ ( x ⊗ a ⊗ a ) = b A | B ( x ⊗ π ( a ) ⊗ π ( a ))= xσ ( π ( a )) ⊗ π ( a ) − x ⊗ π ( σ ( π ( a )) σ ( π ( a )) + σ ( π ( a )) x ⊗ π ( a ) . We have σ ( π ( a )) = a + c and σ ( π ( a )) = a + c , for c , c ∈ B. This way he last expression becomes: xa ⊗ π ( a ) + xc ⊗ π ( a ) − x ⊗ π ( a a + a c + c a + c c )+ a x ⊗ π ( a ) + c x ⊗ π ( a )= xa ⊗ π ( a ) + xc ⊗ π ( a ) − x ⊗ π ( a a ) − x ⊗ π ( a c ) − x ⊗ π ( c a )+ a x ⊗ π ( a ) + c x ⊗ π ( a )= xa ⊗ π ( a ) + xc ⊗ π ( a ) − x ⊗ π ( a a ) − x ⊗ π ( a ) c − x ⊗ c π ( a )+ a x ⊗ π ( a ) + c x ⊗ π ( a )= xa ⊗ π ( a ) + xc ⊗ π ( a ) − x ⊗ π ( a a ) − c x ⊗ π ( a ) − xc ⊗ π ( a )+ a x ⊗ π ( a ) + c x ⊗ π ( a )= xa ⊗ π ( a ) − x ⊗ π ( a a ) + a x ⊗ π ( a ) . Observe that we made use in an essential way that at the codomain of κ , the firsttensor product is over B e , and the other ones are over B . The proof in an arbitrarydegree follows the same lines. ⋄ Jacobi-Zariski long nearly exact sequence
In this section we will obtain one of the main results of this work.
Definition 4.1
A complex of vector spaces ending at a fixed n ≥ · · · δ → U m I → V m K → W m δ → U m − I → V m − → · · · δ → U n I → V n K → W n is a long nearly exact sequence if it is exact except perhaps at V ∗ .Its gap is the graded vector space ( Ker K/ Im I ) ∗ . Theorem 4.2
Let → C ∗ ι → D ∗ κ → E ∗ → be a short nearly exact sequence of positively graded chain complexes (see Definition3.1).There is a long nearly exact sequence . . . δ → H m ( C ) I → H m ( D ) K → H m ( E ) δ → H m − ( C ) I → . . . δ → H ( C ) I → H ( D ) K → H ( E ) → . Its gap is isomorphic to H ∗ ( Ker κ/ Im ι ) , namely to the homology of the gap complexof (4.1).If (4.1) is nearly exact except at the lowest degree where κ is not surjective,then the same holds except that H ( D ) K → H ( E ) is not surjective. In other wordsthere is a long nearly exact sequence . . . δ → H m ( C ) I → H m ( D ) K → H m ( E ) δ → H m − ( C ) I → . . . δ → H ( C ) I → H ( D ) K → H ( E ) with gap as before. emark 4.3 The second part of the statement takes into account the specificsituation occurring in degree of Theorem 3.3. Proof.
After the standard change of signs, we consider the short nearly exactsequence (4.1) as a double complex with three columns at p = 0 , and . ByLemma 3.2, filtering by rows gives a spectral sequence converging to H ∗ ( Ker κ/ Im ι ) .On columns and , starting at page , there are zeros. Indeed, ι is injective and κ is surjective.Consider the filtration by columns, let I and K be the horizontal maps at page . At page , first consider the columns p = 0 and p = 2 where we have respectivelyCoker K and Ker I . The differential d : Ker I → Coker K lowers the total degree by . We claim it is invertible. Indeed, at page the column p = 2 consists of Ker d ,while at the same page, column p = 0 is Coker d . Actually these vector spaceslive forever since d at these spots come from or go to . By the analysis of thefiltration by rows of the double complex, we infer that Ker d = 0 and Coker d = 0 ,that is d : Ker I → Coker K is an isomorphism.The morphism δ is then obtained by pre-composing d − with the canonical pro-jection to Coker K and post-composing with the inclusion of Ker I . By constructionKer δ = Im K and Im δ = Ker I .Still at page but at column p = 1 , we have Ker K/ Im I . At these spots d comes from or goes to . Therefore Ker K/ Im I lives forever.We use again that both filtrations converge to the same graded vector space toinfer that H ∗ ( Ker κ/ Im ι ) is isomorphic to ( Ker K/ Im I ) ∗ , that is to the gap of thelong nearly exact sequence.Finally, if (4.1) is nearly exact except in the lowest degree where κ is not sur-jective, then the filtration by columns at page gives in addition Coker κ at thespot (0 , which lives forever. At the second page H ∗ ( Ker κ/ Im ι ) for ∗ ≥ isnot affected and lives forever. The proof that the lowest d from (2 , to (0 , isinvertible remains true. ⋄ Theorem 4.4
Let B ⊂ A be an extension of k -algebras, and let X be an A -bimodule.There is a Jacobi-Zariski long nearly exact sequence in Hochschild homology · · · δ → H m ( B, X ) I → H m ( A, X ) K → H m ( A | B, X ) δ → H m − ( B, X ) I → . . . δ → H ( B, X ) I → H ( A, X ) K → H ( A | B, X ) . Proof.
Theorem 3.3 establishes that the fundamental sequence → ˇ C ∗ ( B, X ) ι → ˇ C ∗ ( A, X ) κ → ˇ C ∗ ( A | B, X ) → is nearly exact except in its lowest degree where κ is not surjective. The secondpart of the previous result provides the proof of the statement. ⋄ Gap of the Jacobi-Zariski long nearly exact sequence
Next we will approximate the gap of the Jacobi-Zariski long nearly exact sequenceof Theorem 4.4. heorem 5.1 Let B ⊂ A be an extension of k -algebras, and let X be an A -bimodule. Let · · · δ → H m ( B, X ) I → H m ( A, X ) K → H m ( A | B, X ) δ → H m − ( B, X ) I → . . . δ → H ( B, X ) I → H ( A, X ) K → H ( A | B, X ) be the Jacobi-Zariski long nearly exact sequence obtained in Theorem 4.4.In degrees ≥ the gap ( Ker K/ Im I ) ∗ is approximated by a spectral sequenceconverging to it, which terms at page are E p,q = Tor B e p + q ( X, ( A/B ) ⊗ B p ) for p, q > and everywhere else.If A and X are finite dimensional, then in degree we have that K is surjectiveand Ker K = Im I . Remark 5.2
We underline that the degrees of the above
Tor vector spaces at page are strictly positive. Before proving the theorem, we will describe the gap complex of the fundamentalsequence (3.2).Let B ⊂ A be an inclusion of k -algebras, π : A → A/B the canonical B -bimodule map and σ : A → B a chosen k -section to π . Let S = Im σ . Then A = B ⊕ S as vector spaces. Of course S and A/B are isomorphic vector spacesvia σ and π | S . Remark 5.3
By transport of structure from
A/B , the vector space S can be en-dowed with a B -bimodule structure as follows. If s ∈ S and b ∈ B , then b.s = σ ( bπ ( s )) and s.b = σ ( π ( s ) b ) . Of course, the resulting B -bimodule S is not a B -subbimodule of A in general.However there exist c and c ′ ∈ B such that b.s = bs + c and s.b = sb + c ′ since π ( b.s − bs ) = 0 = π ( s.b − sb ) . Definition 5.4
In the above situation A = B ⊕ S , let [ S p B q ] be the subspace of A ⊗ n which is the direct sum of the direct summands of A ⊗ n having p tensorandsin S and q tensorands in B , with p + q = n . For instance for n = 3 , [ S B ] = ( S ⊗ S ⊗ B ) ⊕ ( S ⊗ B ⊗ S ) ⊕ ( B ⊗ S ⊗ S ) . We have A ⊗ n = M p + q = np ≥ q ≥ [ S p B q ] . Recall that the map κ : X ⊗ A ⊗ n → X ⊗ B e ( A/B ) ⊗ B n is given by κ ( x ⊗ a ⊗ · · · ⊗ a n ) = x ⊗ π ( a ) ⊗ · · · ⊗ π ( a n ) . efinition 5.5 For n ≥ , let L n, = Ker κ (cid:12)(cid:12) X ⊗ S ⊗ n : X ⊗ S ⊗ n ։ X ⊗ B e ( A/B ) ⊗ B n . (5.1) The index in L n, underlines that there are no B -tensorands involved in its defi-nition. Remark 5.6
Let L ′ n, = Ker ( X ⊗ ( A/B ) ⊗ n ։ X ⊗ B e ( A/B ) ⊗ B n ) , that is L ′ n, is the subspace which defines the tensor products over B e and B in X ⊗ ( A/B ) ⊗ n .By transport of structure, we have endowed S with a B -bimodule structure suchthat π | S : S → A/B is an isomorphism of B -bimodules. Note that π | S ( L n, ) = L ′ n, . In other words, ( X ⊗ S ⊗ n ) /L n, = X ⊗ B e S ⊗ B n . The next result describes the gap complex of the fundamental sequence (3.2).
Lemma 5.7
In the situation of Theorem 3.3, let → ˇ C ∗ ( B, X ) ι → ˇ C ∗ ( A, X ) κ → ˇ C ∗ ( A | B, X ) → be the fundamental short nearly exact sequence of complexes (3.2).For n ≥ we have Ker κ = L n, ⊕ (cid:18)L p + q = np ≥ q> X ⊗ [ S p B q ] (cid:19) Im ι = X ⊗ B ⊗ n = X ⊗ [ S B n ]( Ker κ/ Im ι ) n = L n, ⊕ (cid:18)L p + q = np> q> X ⊗ [ S p B q ] (cid:19) . Proof.
For n ≥ , consider the vector space decomposition X ⊗ A ⊗ n = M p + q = np ≥ q ≥ X ⊗ [ S p B q ] . If q > , then κ ( X ⊗ [ S p B q ]) = 0 . Hence M p + q = np ≥ q> X ⊗ [ S p B q ] ⊂ Ker κ. If q = 0 , then by definition Ker κ (cid:12)(cid:12) X ⊗ S ⊗ n = L n, . It follows that Ker κ is as stated.On the other hand, clearly Im ι is the direct summand corresponding to p = 0 and q = n ; the result follows. ⋄ Remark 5.8
For n = 1 there is a dead end as follows. According to Theorem 3.3we should consider the sequence → Ker b B ι | → Ker b A κ | → Ker b A | B → hich is the well defined restriction of → X ⊗ B ι → X ⊗ A κ → X ⊗ B e ( A/B ) → . (5.3) The maps ι and κ are still injective and surjective respectively, and Ker κ Im ι = ( X ⊗ B ) ⊕ L , X ⊗ B = L , . In the sequence (5.2) we clearly have that ι | is injective and κ | ι | = 0 . However Ker κ | / Im ι | is intricate to describe. Nevertheless, we will obtain from [8] that if A and X are finite dimensional, the Jacobi-Zariski long nearly exact sequence is exactin degree . Proof of Theorem 5.1.
By Theorem 4.2 the gap of the Jacobi-Zariski long ex-act sequence is the homology of the gap complex of the fundamental sequence.Therefore we focus on the latter.By Lemma 5.7 the vector spaces of the gap complex of the fundamental sequenceare, for n ≥ : ( Ker κ/ Im ι ) n = L n, ⊕ M p + q = np> q> X ⊗ [ S p B q ] . We will show that there is a filtration ( G p ) p> of the gap complex such that G p /G p − (that is the column p at page of the spectral sequence induced by thefiltration) has the stated homology. In each degree the chains of G p have p or lesstensorands from S : G p = M q> X ⊗ [ S i B q ] . Namely ( G p ) n = ( Ker κ/ Im ι ) n for n ≤ p, and ( G p ) n = M i + q = np ≥ i> q> X ⊗ [ S i B q ] for n > p. We recall that the differentials of the gap complex are induced from the differ-entials of the Hochschild complex C ∗ ( A, X ) . The following observations show that G p is indeed a subcomplex:1. If s, s ′ ∈ S , then there exists s ′′ ∈ S and c ∈ B such that ss ′ = s ′′ + c . Inthis case we have that the number of tensorands belonging to S decreases byone when we apply the boundary formulas.2. If s ∈ S and b ∈ B we have sb = c + s.b , where c ∈ B and s.b ∈ S is theright action of B on S by transport of structure, see Remark 5.3.In the boundary formulas, the number of tensorands in S decreases by one inthe summand corresponding to c , and it is maintained in the other. Similarlyfor bs . . If x ∈ X and s ∈ S , then both sx and xs also have less tensorands in S .For p > , we have ( G p /G p − ) q = X ⊗ [ S p B q ] if q > and ( G p /G p − ) = L p, . Actually the three observations above show that the number of tensorands in S decreases by one, except in case 2. In other words considering S with its B -bimodule structure obtained by transport of structure, with “zero internal product”and with “zero action on X ” provides the same complex G p /G p − .This complex has been considered in the proof of [7, Proposition 3.3] where weproved that for q > we have H q ( G p /G p − ) = Tor B e p + q ( X, S ⊗ B p ) while H ( G p /G p − ) = 0 . This finishes the proof in degrees ≥ since the B -bimodules S and A/B are isomorphic, see Remark 5.3.For the convenience of the reader, we next recall the proof that for q > wehave H q ( G p /G p − ) = Tor B e p + q ( X, S ⊗ B p ) while H ( G p /G p − ) = 0 .The standing step is to replace G p /G p − by G ′ p in degree as follows: ( G ′ p ) = X ⊗ S ⊗ p = X ⊗ [ S p B ] while ( G ′ p ) q = ( G p /G p − ) q for q > .The boundaries of G ′ p are the same as those of G p /G p − , this makes sense since L p, ⊂ X ⊗ S ⊗ p . For p > we assert that H q ( G ′ p ) = Tor B e q ( X, S ⊗ B p ) for all q ≥ .We consider below a projective resolution of a B e -module S given in [6, Propo-sition 4.1]. Applying the functor X ⊗ B e − to it gives G ′ , which proves the assertionfor p = 1 .Let q S p = B ⊗ · · · ⊗ B | {z } q ⊗ S ⊗ B ⊗ · · · ⊗ B | {z } p and consider the following complex of free B e -modules: · · · d → M p + q = n +1 p> q> q S p d → · · · d → S ⊕ S d → S d → S → , where the first differential is given by d ( b ⊗ s ⊗ b ′ ) = b.s.b ′ . In greater degrees, thedifferential is the differential of the total complex of the double complex which has q S p at the spot ( q, p ) , with vertical and horizontal differentials q S p → q S p − and q S p → q − S p given respectively by b ⊗ · · · ⊗ b q ⊗ s ⊗ b ′ ⊗ · · · ⊗ b ′ p ( − q +1 [ b ⊗ · · · ⊗ b q ⊗ s.b ′ ⊗ · · · ⊗ b ′ p + P ( − i b ⊗ · · · ⊗ b q ⊗ s ⊗ b ′ ⊗ · · · ⊗ b ′ i b ′ i +1 ⊗ · · · ⊗ b ′ p ] and b ⊗ · · · ⊗ b q ⊗ s ⊗ b ′ ⊗ · · · ⊗ b ′ p P ( − i b ⊗ · · · ⊗ b i b i +1 ⊗ · · · ⊗ b q ⊗ s ⊗ b ′ ⊗ · · · ⊗ b ′ p +( − q b ⊗ · · · ⊗ b q .s ⊗ b ′ ⊗ · · · ⊗ b ′ p or p = 2 , consider the bar resolution of S as a left B -module · · · B ⊗ B ⊗ S → B ⊗ S → S → . As it is well known there is a contracting homotopy t given by t ( b ⊗ · · · ⊗ b n ⊗ s ) =1 ⊗ b ⊗ · · · ⊗ b n ⊗ s which is a right B -module map. Hence we obtain a projectiveresolution of S ⊗ B S by tensoring the bar resolution over B with the above resolutionof [6, Proposition 4.1]. Applying the functor X ⊗ B e − yields G ′ , proving theassertion for p = 2 . The assertion is proved for p > by iterating the process oftensoring by the bar resolution.Back to G p /G p − , from the above we have H ( G ′ p ) = Tor B e ( X, S ⊗ B p ) . Theoriginal complex G p /G p − has L p, in degree , hence the image of the last differ-ential of G ′ p is contained in L p, ⊂ X ⊗ S ⊗ p . Recall that ( X ⊗ S ⊗ p ) /L p, = X ⊗ B e S ⊗ B p = Tor B e ( X, S ⊗ B p ) . The image of the last differential of G ′ p is then L p, , that is the last differential of G p /G p − is surjective and H ( G p /G p − ) = 0 .For p = 1 it is proven in [8, Proposition 3.3] that there is a short exact sequencefor Hochschild cohomology: → H ( A | B, X ) ι → H ( A, X ) κ → H ( B, X ) Let V ′ denote the dual of a vector space V . It is well known that for finitedimensional A and X , we have H ∗ ( A, X ) = ( H ∗ ( A, X ′ )) ′ . The same holds in therelative setting, and the result follows. ⋄ Jacobi-Zariski long exact sequences
The next result is a specialisation of the Jacobi-Zariski nearly exact sequence ofthe previous section, in order to obtain the long exact sequence of A. Kaygun in[16, 17].For the convenience of the reader we provide a proof of the following result.
Lemma 6.1
Let Λ be any k -algebra, and let P and Q be flat Λ -bimodules.The Λ -bimodule P ⊗ Λ Q is flat. Proof.
First we record that if P is a flat bimodule, then it is both left and rightflat. Indeed, let X ֒ → Y be an injection of right modules, and consider the inferredinjection of bimodules Λ ⊗ X ֒ → Λ ⊗ Y . We have an injection P ⊗ Λ e (Λ ⊗ X ) ֒ → P ⊗ Λ e (Λ ⊗ Y ) . We have a natural isomorphism P ⊗ Λ e (Λ ⊗ X ) = X ⊗ Λ P , sending p ⊗ λ ⊗ x to x ⊗ pλ . The result follows.Let now U ֒ → V be an injection of right Λ e -modules, we want to prove that U ⊗ Λ e ( P ⊗ Λ Q ) → V ⊗ Λ e ( P ⊗ Λ Q ) is an injection. Note that we have a naturalisomorphism U ⊗ Λ e ( P ⊗ Λ Q ) = ( U ⊗ Λ P ) ⊗ Λ e Q nduced by the identity of U ⊗ P ⊗ Q . Now since P is left flat, there is an injectionof bimodules U ⊗ Λ P ֒ → V ⊗ Λ P . Applying the exact functor − ⊗ Λ e Q gives theresult. ⋄ We give now an alternative proof of results in [16, 17].
Theorem 6.2
Let B ⊂ A be an extension of k -algebras such that A/B is a flat B -bimodule. Let X be an A -bimodule. There is a Jacobi-Zariski long exact sequence · · · δ → H m ( B, X ) I → H m ( A, X ) K → H m ( A | B, X ) δ → H m − ( B, X ) I → . . . δ → H ( B, X ) I → H ( A, X ) K → H ( A | B, X ) . If A and X are finite dimensional, then the Jacobi-Zariski long exact sequence endsat degree : · · · δ → H m ( B, X ) I → H m ( A, X ) K → H m ( A | B, X ) δ → H m − ( B, X ) I → . . . δ → H ( B, X ) I → H ( A, X ) K → H ( A | B, X ) . Proof.
By Theorem 5.1, there is a spectral sequence converging to the gap of thelong nearly exact sequence in degrees ≥ . The first page of this spectral sequenceis E p,q = Tor B e p + q ( X, ( A/B ) ⊗ B p ) for p, q > and elsewhere. By the previous lemma, for p > we have that the B e -module A/B ⊗ B p is flat. Hence, if q > , we have Tor B e p + q ( X, ( A/B ) ⊗ B p ) = 0 . Conse-quently the first page of the spectral sequence is , so the graded vector space towhich it converges is also .If A and X are finite dimensional, the second part of Theorem 5.1 provides theresult. ⋄ Next we improve the Jacobi-Zariski exact sequence ending at some degree ob-tained in [7, Proposition 3.7] by not assuming that the extension splits.
Definition 6.3 ([7]) Let Λ be a k -algebra and let M be a Λ -bimodule. The bi-module M is tensor nilpotent if there exists n such that M ⊗ Λ n = 0 . The index ofnilpotency of M is the smallest n such that M ⊗ Λ n = 0 . Definition 6.4 ([7]) Let Λ be a k -algebra and let M be a Λ -bimodule. The bi-module is bounded if it is tensor nilpotent, projective on one side, and of finiteprojective dimension as a bimodule. Theorem 6.5
Let B ⊂ A be an extension of k -algebras and let X be an A -bimodule. Assume that A/B is a bounded B -bimodule, with index of nilpotency n and projective dimension u .There is a Jacobi-Zariski long exact sequence · · · δ → H m ( B, X ) I → H m ( A, X ) K → H m ( A | B, X ) δ → H m − ( B, X ) I → . . . δ → H nu ( B, X ) I → H nu ( A, X ) K → H nu ( A | B, X ) . roof. For degrees ≥ the terms at page of the spectral sequence whichconverges to the gap are E p,q = Tor B e p + q ( X, ( A/B ) ⊗ B p ) for p, q > and elsewhere.Since A/B is projective on one side, we have that ( A/B ) ⊗ B p is of projectivedimension at most pu (see [4, Chapter IX, Proposition 2.6]).If p + q ≥ nu , then p ≥ n or p + q > pu . Hence if p + q ≥ nu then E p,q = 0 .Consequently the gap is in degrees ≥ nu . ⋄ References [1] Andr´e, M. Homologie des alg`ebres commutatives. Die Grundlehren dermathematischen Wissenschaften (In Einzeldarstellungen mit besondererBer¨ucksichtigung der Anwendungsgebiete), vol 206. Springer, Berlin, Heidel-berg, 1974.[2] Barr, M. Harrison homology, Hochschild homology and triples. J. Algebra 8(1968) 314–323.[3] B¨uhler, T. Exact categories, Expo. Math. 28 (2010), 1–69.[4] Cartan, H.; Eilenberg, S. Homological algebra. Princeton University Press,Princeton, N. J., 1956.[5] Cibils, C. Rigidity of truncated quiver algebras. Adv. Math. 79 (1990), 18–42.[6] Cibils, C. Tensor Hochschild homology and cohomology. Interactions betweenring theory and representations of algebras (Murcia), 35–51, Lecture Notes inPure and Appl. Math., 210, Dekker, New York, 2000.[7] Cibils, C.; Lanzilotta, M.; Marcos, M.; Solotar, A. Split bounded extensionalgebras and Han’s conjecture. Pacific J. Math. 307 (2020), 63–77.https://doi.org/10.2140/pjm.2020.307.63[8] Cibils, C.; Lanzilotta, M.; Marcos, M.; Schroll, S.; Solotar, A. The firstHochschild (co)homology when adding arrows to a bound quiver algebra J.Algebra 540 (2019) 63–77.[9] Cibils, C.; Lanzilotta, M.; Marcos, M.; Solotar, A. Adding or deleting arrowsof a bound quiver algebra and Hochschild (co)homology. Proc. Amer. Math.Soc. 148 (2020), 2421–2432.https://doi.org/10.1090/proc/14936[10] Gerstenhaber, M.; Schack, S. D. Relative Hochschild cohomology, rigid alge-bras, and the Bockstein. J. Pure Appl. Algebra 43 (1986), 53–74.[11] Ginot, G. On the Hochschild and Harrison (co)homology of C ∞ -algebras andapplications to string topology. Deformation spaces, 1–51, Aspects Math., E40,Vieweg + Teubner, Wiesbaden, 2010.
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C.C.:Institut Montpelli´erain Alexander Grothendieck, CNRS, Univ. Montpellier, France.
M.L.:Instituto de Matem´atica y Estad´ıstica “Rafael Laguardia”, Facultad de Ingenier´ıa, Universidad dela Rep´ublica, Uruguay. [email protected]
E.N.M.:Departamento de Matem´atica, IME-USP, Universidade de S˜ao Paulo, Brazil. [email protected]
A.S.:IMAS-CONICET y Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Argentina. [email protected]@dm.uba.ar