Front. Math. China 2012, 7(3): 415–426DOI 10.1007/s11464-012-0200-y

Module-relative-Hochschild (co)homology

of tensor products

Yuan CHEN

School of Mathematics and Computer Science, Hubei University, Wuhan 430062, ChinaSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

c© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012

Abstract In this paper, we consider the module-relative-Hochschild homologyand cohomology of tensor products of algebras and relate them to those of thefactor algebras. Moreover, we show that the tensor product is formally smoothif and only if one of its factor algebras is formally smooth and the other isseparable.

Keywords Tensor product, module-relative-Hochschild (co)homology, formalsmoothnessMSC 16E10, 16E40, 18G10, 18G25

1 Introduction

Module-relative-Hochschild (co)homology was introduced in [2] by Ardizzoniet al. when they studied the formal smoothness. It plays an important role innon-commutative algebraic geometry and provides a natural characterizationof the separable bimodules and formally smooth bimodules. One can viewthe separable bimodules as (non-commutative, relative) “bundles of points”,that is, the objects with relative-Hochschild cohomology dimension zero; andthe formally smooth bimodules can be understood as (non-commutative,relative) “bundles of curves” or “line bundles”, that is, the objects with relative-Hochschild cohomology dimension at most one.

The notion of formal smoothness has attracted much attention in recentliterature (see [1–3,5,6,8,11–13,16,17]). A convenient description and conceptualinterpretation of formal smoothness is provided by E -relative derived functors(see [7,8,10]). Ardizzoni et al. have introduced in [3] the Hochschild cohomologyin monoidal abelian categories in this way, instead of generalizing ordinaryHochschild’s construction [9] or by using the (co)simplicial approach explained

Received June 3, 2011; accepted February 20, 2012E-mail: cy2010@hubu.edu.cn

416 Yuan CHEN

in [14]. This general algebraic approach to formal smoothness in monoidalabelian categories, including the cohomological aspects, was also proposed in [1].These gave rise to the introduction of module-relative-Hochschild cohomology[2].

Let A and B be k-algebras with k a field. Given a bimodule BMA suchthat M is a generator in BM , we consider the following projective class ofepimorphisms:

EM,B := {f ∈ BMB | HomB(M,f) is split epimorphic in AMB}.

Based on the theory of EM,B-relative derived functor, the nth M -Hochschildcohomology and homology of B over A are defined by

HnEM,B

(B) := ExtnEM,B(B,B)

andHEM,B

n (B) := TorEM,Bn (B,B),

respectively. In particular, taking BMA=BBk, we get the ordinary Hochschild(co)homology of B; moreover, if there is an algebra homomorphism μ : A →B, then by taking BMA=BBA, we get the relative Hochschild (co)homologyof B with respect to μ. Thus, the concept of module-relative-Hochschild(co)homology is in fact a generalization of the notion of ordinary (relative)Hochschild (co)homology.

In this paper, we will consider the module-relative-Hochschild homologyand cohomology of tensor products of algebras and show that they can beobtained by those of the factor algebras. This generalizes the known result onthe ordinary Hochschild (co)homology. It is shown in [15] that the computationof the ordinary Hochschild (co)homology of the tensor product can be relatedto the (co)homology of each tensor factor. Moreover, we prove that the tensorproduct is formally smooth if and only if one of its factor algebras is formallysmooth and the other is separable.

2 Module-relative-Hochschild (co)homology

Throughout this paper, for an algebra (resp. a ring), we mean a unitalassociative algebra (resp. a ring). Let BM , MA, and BMA denote categoriesof (unital) left B-modules, right A-modules, and B-A-bimodules, respectively.The notation BMA means that M is a B-A-bimodule.

Let A and B be k-algebras with k a field. Let Ae = A ⊗k Aop denote theenveloping algebra of A. Given a bimodule BMA, we consider the followingadjunction:

LB := M ⊗A − : AMB → BMB,

RB := HomB(M,−) : BMB → AMB .

Module-relative-Hochschild (co)homology of tensor products 417

Let

EM,B := {f ∈ BMB | HomB(M,f) is split epimorphic in AMB}.EM,B is always a projective class (see [1, Theorem 1.4]), and if M is a generatorin BM , then EM,B is a projective class of epimorphisms (see [2, Proposition3.1]). Then every object in BMB has an EM,B-projective resolution, which isunique up to a homotopy. The reader is referred to [8] for further informationon relatively projective object and projective class of epimorphisms. Note thatBPB is EM,B-projective if and only if HomBe(P,−) is EM,B-exact. Anothercondition is that BPB is EM,B-projective if and only if there is a splitepimorphism π : LB(X) → P for a suitable X ∈ AMB . Therefore, it is easy tosee that all projective B-B-bimodules and B-B-bimodules of the form LB(X),X ∈ AMB , are EM,B-projective.

Recall first from [2] some definitions. Let M be a B-A-bimodule which is agenerator as a left B-module. The nth M -Hochschild cohomology of B over Awith coefficients in a B-B-bimodule Y is defined to be

HnEM,B

(B,Y ) := ExtnEM,B(B,Y ).

In particular, if Y = B, then

HnEM,B

(B) := HnEM,B

(B,B)

is called the nth M -Hochschild cohomology of B over A. The number

min{n ∈ N | Hn+1EM,B

(B,Y ) = 0, ∀ Y ∈ BMB}

is called the M -Hochschild cohomology dimension of B (if it exists), anddenoted by hch.dimM (B). If such an n does not exist, we will say that theM -Hochschild cohomology dimension of B is infinite.

Using the relative-Tor-functor, we propose the following definition.

Definition 2.1 Consider a B-A-bimodule M such that M is a generator inBM . The nth M -Hochschild homology of B over A with coefficients in BYB isdefined by

HEM,Bn (B,Y ) := TorEM,B

n (B,Y ).

In particular, if Y = B, then

HEM,Bn (B) := HEM,B

n (B,B)

is called the nth M -Hochschild homology of B over A. The number

min{n ∈ N | HEM,B

n+1 (B,Y ) = 0, ∀ Y ∈ BMB}is called the M -Hochschild homology dimension of B (if it exists), and denotedby hh.dimM (B). If such an n does not exist, we will say that the M -Hochschildhomology dimension of B is infinite.

418 Yuan CHEN

Similar to the non-relative case, M -Hochschild (co)homology can beequivalently described as the (co)homology of a complex associated with thestandard resolution. Let εB : LBRB → Id

BMBbe the counit of the adjunction

(LB , RB), and let M be a B-A-bimodule which is a generator in BM . Then,for every B-B-bimodule X, the associated augmented chain complex (PX , d∗)of BXB :

· · · −→ (LBRB)2(X) d1−→ LBRB(X) d0−→ (LBRB)0(X) =: X −→ 0,

where

dn =n∑

i=0

(−1)i(LBRB)i(εB((LBRB)n−i(B))),

is an EM,B-projective resolution of BXB , called the standard EM,B-projectiveresolution of BXB .

3 Module-relative-Hochschild (co)homology of tensor products

This section is devoted to the module-relative-Hochschild homology andcohomology of the tensor product B ⊗k C of two finite-dimensional k-algebrasB and C. In certain cases, one can get the module-relative-Hochschild(co)homology of B ⊗k C from that of B and C.

Let k be a field, and let A be a k-algebra. Write ⊗ for ⊗k. Consider thetensor product B ⊗C of two finite-dimensional k-algebras B and C. Given twobimodules BXB and CYC , X ⊗ Y is a B ⊗ C-bimodule, with the left operatorsgiven by

(b ⊗ c)(x ⊗ y) = bx ⊗ cy,

and the right operators similarly defined. Set

Λ = B ⊗ C, A = A ⊗ A.

Given bimodules BMA and CNA, the tensor product M ⊗ N becomes aΛ-A-bimodule. Considering the adjunctions

(LB = M ⊗A −, RB = HomB(M,−)),

(LC = N ⊗A −, RC = HomC(N,−)),

and(LΛ = (M ⊗ N) ⊗A −, RΛ = HomΛ(M ⊗ N,−)),

we have the classes EM,B , EN,C , and EM⊗N,Λ.

Lemma 3.1 If BMA and CNA are generators in BM and CM , respectively,then Λ(M ⊗ N)A is a generator in ΛM . Moreover, EM,B , EN,C , and EM⊗N,Λ

are all projective classes of epimorphisms.

Module-relative-Hochschild (co)homology of tensor products 419

Proof Since BM and CN are generators in BM and CM , respectively, B is aquotient of a direct sum of copies of M as a B-module and C is a quotient ofa direct sum of copies of N as a C-module. It follows that B ⊗N is a quotientof a direct sum of copies of M ⊗N as a Λ-module and B ⊗C is a quotient of adirect sum of copies of B ⊗ N as a Λ-module. Hence, B ⊗ C is a quotient of adirect sum of copies of M ⊗ N as a Λ-module. That is, M ⊗ N is a generatorin ΛM . �

From now on, we always assume that bimodules BMA and CNA are finitedimensional and are generators in BM and CM , respectively.

The following lemma about homology products is taken from [15, TheoremVIII.1.1].

Lemma 3.2 For k-algebras B and C, the homology product is an isomorphism⊕

p+q=n

Hp(U ⊗B X) ⊗ Hq(V ⊗C Y) � Hn((U ⊗ V ) ⊗B⊗C (X ⊗ Y))

for any right B-module U, right C-module V, and any complexes X and Y ofleft B-modules and left C-modules, respectively.

If f : X → X′ and g : Y → Y

′ are chain maps, then

(f ⊗ g)(x ⊗ y) = f(x) ⊗ g(y)

gives a chain mapf ⊗ g : X ⊗ Y → X

′ ⊗ Y′.

In this section, the notation f � g : X → X′ means that the chain maps f and

g are chain homotopy equivalent. If

f1 � f2 : X → X′, g1 � g2 : Y → Y

′,

thenf1 ⊗ g1 � f2 ⊗ g2 : X ⊗ Y → X

′ ⊗ Y′,

by [15, Proposition V.9.1].In order to establish the relationship between the M ⊗ N -Hochschild

(co)homology of Λ over A and that of B and C over A, we need to provethe following result about the EM⊗N,Λ-projective resolution of Λ-Λ-bimoduleX⊗Y, where X and Y are any B-B-bimodule and C-C-bimodule, respectively.

Proposition 3.3 Let PXd0→ BXB be the standard EM,B-projective resolution

of BXB , and let PYσ0→ CYC be the standard EN,C-projective resolution of CYC .

Then PX ⊗PY → Λ(X ⊗Y )Λ is an EM⊗N,Λ-projective resolution of Λ(X ⊗Y )Λ.

Proof LetF := LBRB(−), G := LCRC(−).

Then PX and PY are

(PX , d∗) : · · · → FnX → Fn−1X → · · · → F 2X → FX → 0

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and(PY , σ∗) : · · · → GnY → Gn−1Y → · · · → G2Y → GY → 0,

respectively, where FX and GY are of degree 0.First, we need to prove that (PX ⊗PY )n is EM⊗N,Λ-projective for any n � 0.

Note that

(PX ⊗ PY )n

=⊕

p+q=n+2

F pX ⊗ GqY

=⊕

p+q=n+2

F (F p−1X) ⊗ G(Gq−1Y )

=⊕

p+q=n+2

(M ⊗A HomB(M,F p−1X)) ⊗ (N ⊗A HomC(N,Gq−1Y ))

�⊕

p+q=n+2

(M ⊗ N) ⊗A (HomB(M,F p−1X)) ⊗ HomC(N,Gq−1Y ))

�⊕

p+q=n+2

(M ⊗ N) ⊗A HomB⊗C(M ⊗ N,F p−1X ⊗ Gq−1Y ),

where p, q � 1. It is easy to see that (PX ⊗ PY )n is EM⊗N,Λ-projective sinceall Λ-Λ-bimodules of the form (M ⊗ N) ⊗A HomΛ(M ⊗ N,Z), Z ∈ ΛMΛ, areEM⊗N,Λ-projective.

Second, we need to prove that the complex

PX ⊗ PY : PX ⊗ PY → X ⊗ Y → 0

is EM⊗N,Λ-exact. By Lemma 3.2, one can easily check that the complex PX ⊗ PY

is exact. Indeed, choose

U = B, V = C, X = PX , Y = PY .

Since PX and PY are both exact on degree � 1, we get that

Hn(PX ⊗ PY ) =⊕

p+q=n

Hp(PX) ⊗ Hq(PY ) = 0

for n � 1. For n = 0,

H0(PX ⊗ PY ) = H0(PX) ⊗ H0(PY ) = X ⊗ Y.

Then we get the exactness of the complex PX ⊗ PY .

Applying the functor HomΛ(M ⊗ N,−) to PX ⊗ PY , we deduce that it issufficient to prove that the complex

HomΛ(M⊗N, PX ⊗ PY ) : HomΛ(M⊗N, PX⊗PY ) → HomΛ(M⊗N,X⊗Y ) → 0

Module-relative-Hochschild (co)homology of tensor products 421

is split exact in AMΛ. Set

(PX ⊗ PY )−1 = X ⊗ Y.

Note that

HomΛ(M ⊗ N, (PX ⊗ PY )−1) = HomΛ(M ⊗ N,X ⊗ Y )� HomB(M,X) ⊗ HomC(N,Y )

and

HomΛ(M ⊗ N, (PX ⊗ PY )n) = HomΛ

(M ⊗ N,

⊕

p+q=n+2

F pX ⊗ GqY

)

�⊕

p+q=n+2

HomΛ(M ⊗ N,F pX ⊗ GqY )

�⊕

p+q=n+2

HomB(M,F pX) ⊗ HomC(N,GqY )

as A-Λ-bimodules, where p, q � 1. Hence, one can check that the complexHomΛ(M ⊗ N, PX ⊗ PY ) can be identified with the complex

HomB(M, PX) ⊗ HomC(N, PY ) → HomB(M,X) ⊗ HomC(N,Y ) → 0

in the category of A-Λ-bimodule complexes. Let PX and PY denote thecomplexes PX → X and PY → Y, respectively. Since PX → X is EM,B-exact,we conclude that

HomB(M, PX) : HomB(M, PX) → HomB(M,X) → 0

is split exact in the category of A-B-bimodule complexes. We also see that

HomC(N, PY ) : HomC(N, PY ) → HomC(N,Y ) → 0

is split exact in the category of A-C-bimodule complexes, since PY → Y isEN,C-exact. Note that a complex X is spilt exact if and only if id : X → X isnull-homotopic.

Let f = (fi) denote the chain map

HomB(M, PX) → HomB(M, PX)

in the category of A-B-bimodule complexes, where

fi =

⎧⎪⎨

⎪⎩

id, i � 1,

d∗1s0, i = 0,

0, i < 0.

422 Yuan CHEN

Here, s = (si) is the chain homotopy:

id � 0: HomB(M, PX) → HomB(M, PX).

It is easy to see that s = (si) also gives a chain homotopy:

f � 0: HomB(M, PX) → HomB(M, PX).

Noticing thatd∗1s0 = id − s−1d

∗0,

we haveid � f ′ : HomB(M, PX) → HomB(M, PX),

where

f ′ = (f ′i), f ′

i =

{0, i �= 0,

s−1d∗0, i = 0.

Let g = (gi) denote the chain map

HomC(N, PY ) → HomC(N, PY )

in the category of A-C-bimodule complexes, where

gi =

⎧⎪⎨

⎪⎩

id, i � 1,

σ∗1t0, i = 0,

0, i < 0.

Here, t = (ti) is the chain homotopy:

id � 0: HomC(N, PY ) → HomC(N, PY ).

Note that t = (ti) also gives a chain homotopy:

g � 0: HomC(N, PY ) → HomC(N, PY ).

Thus, we have

id � g′ : HomC(N, PY ) → HomC(N, PY ),

where

g′ = (g′i), g′i =

{0, i �= 0,

t−1σ∗0, i = 0.

Then we deduce that

id � f ′⊗ g′ : HomB(M, PX)⊗HomC(N, PY ) → HomB(M, PX)⊗HomC(N, PY )

Module-relative-Hochschild (co)homology of tensor products 423

in the category of A-Λ-bimodule complexes. Let

f ′ ⊗ g′ = (hi),

where

hi =

{0, i �= 0,

s−1d∗0 ⊗ t−1σ

∗0 , i = 0.

Associating with the last term HomB(M,X) ⊗ HomC(N,Y ) of degree −1, onecan easily check that

id � 0: HomΛ(M ⊗ N, PX ⊗ PY ) → HomΛ(M ⊗ N, PX ⊗ PY )

in the category of A-Λ-bimodule complexes. This implies that the complex

PX ⊗ PY : PX ⊗ PY → X ⊗ Y

is EM⊗N,Λ-exact, which completes the proof. �Since Λe � Be ⊗ Ce, we can get the following theorem directly by Lemma

3.2 and Proposition 3.3.

Theorem 3.4 Let A be any k-algebra, and let B and C be finite-dimensionalk-algebras. Consider two finite-dimensional bimodules BMA and CNA whichare generators in BM and CM , respectively. Then, for each n and any BXB ,

BX ′B , CYC , and CY ′

C , we have

TorEM⊗N,Λn (X ′ ⊗ Y ′,X ⊗ Y ) �

⊕

i+j=n

TorEM,B

i (X ′,X) ⊗ TorEN,C

j (Y ′, Y ).

In particular, we have

HEM⊗N,Λn (Λ,X ⊗ Y ) �

⊕

i+j=n

HEM,B

i (B,X) ⊗ HEN,C

j (C, Y ).

Moreover,hh.dimM⊗N (Λ) = hh.dimM (B) + hh.dimN (C).

Proof Note that

TorEM⊗N,Λn (X ′ ⊗ Y ′,X ⊗ Y ) = Hn((X ′ ⊗ Y ′) ⊗B⊗C (PX ⊗ PY ))

�⊕

i+j=n

Hi(X ′ ⊗B PX) ⊗ Hj(Y ′ ⊗C PY )

�⊕

i+j=n

TorEM,B

i (X ′,X) ⊗ TorEN,C

j (Y ′, Y ),

where the first isomorphism follows from Lemma 3.2. Then the result follows.�

424 Yuan CHEN

For the cohomology, additional finite dimension assumptions are needed.

Theorem 3.5 Keep the assumptions in Theorem 3.4. Then, for each n andany BXB , BX ′

B, CYC , and CY ′C , where X and Y are finite dimensional, we

have

ExtnEM⊗N,Λ

(X ⊗ Y,X ′ ⊗ Y ′) �⊕

i+j=n

ExtiEM,B(X,X ′) ⊕ ExtjEN,C

(Y, Y ′).

Proof Suppose that bimodules BMA and CNA are finite dimensional and aregenerators in BM and CM , respectively. First, we should prove that each B-B-bimodule F pX and C-C-bimodule GpY appearing in the standard E -projectiveresolutions are finite dimensional.

Since BMA is finite dimensional, we have the following two exact sequences:

(B ⊗k Aop)r → BMA → 0

andBs → BM → 0

as B-A-bimodules and left B-modules, respectively.Applying the functors − ⊗A HomB(M,X) and HomB(−, BXB) to them,

respectively, we get another two exact sequences:

(B ⊗k Aop)r ⊗A HomB(M,X) → BM ⊗A HomB(M,X) → 0

and0 → HomB(BM, BXB) → HomB(Bs, BXB)

as B-B-bimodules and right B-modules, respectively.Note that HomB(Bs,X) � Xs and thus is finite dimensional. This implies

that HomB(M,X) is also finite dimensional since B is finite dimensional. Hence,we have

(B ⊗k Aop)r ⊗A HomB(M,X) � Br ⊗k HomB(M,X)

is finite dimensional and so is M ⊗A HomB(M,X). Thus, by induction, we canconclude that each F pX is finite dimensional. With the same arguments, wesee that each GpY is also finite dimensional.

By the above arguments and [4, Theorem XI.3.1], we can conclude that

HomBe(PX ,X ′) ⊗ HomCe(PY , Y ′) � HomΛe(PX ⊗ PY ,X ′ ⊗ Y ′).

Then by Lemma 3.2, the result follows. �In particular, we have

Theorem 3.6 Let B and C be finite-dimensional k-algebras. Consider twobimodules BMA and CNA which are finite dimensional and are generators in

Module-relative-Hochschild (co)homology of tensor products 425

BM and CM , respectively. Then, for each n and any B-B-bimodule X andC-C-bimodule Y, we have

HnEM⊗N,Λ

(Λ,X ⊗ Y ) �⊕

i+j=n

HiEM,B

(B,X) ⊗ HjEN,C

(C, Y ).

In particular,

hch.dimM⊗N (Λ) = hch.dimM (B) + hch.dimN (C).

Remark Recall in [2] that B is BMA-separable if and only if

hch.dimM (B) = 0;

and that B is M -smooth if and only if

hch.dimM (B) � 1.

By Theorem 3.6, we directly obtain that Λ = B ⊗C is M ⊗N -separable if andonly if B is M -separable and C is N -separable; Λ = B ⊗ C is M ⊗ N -smoothif and only if either B is M -smooth and C is N -separable or B is M -separableand C is N -smooth.

Acknowledgements The author thanks the referees for some helpful comments and

pointing out a mistake in an earlier version of this paper. She also thank her supervisor

Prof. C. C. Xi and Prof. Y. G. Xu for their helpful suggestions. This work was supported by

the National Natural Science Foundation of China (Grant No. 11126110).

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