HOCHSCHILD COHOMOLOGY AND REPRESENTATION- FINITE ALGEBRAS · HOCHSCHILD COHOMOLOGY AND...

HOCHSCHILD COHOMOLOGY AND REPRESENTATION-FINITE ALGEBRAS

RAGNAR-OLAF BUCHWEITZ AND SHIPING LIU

Dedicated to Idun Reiten to mark her sixtieth birthday

Introduction

Hochschild cohomology is a subtle invariant of associative algebras; see, forexample, [13, 15, 21]. The lower-dimensional Hochschild cohomology groups havewell-known interpretations. Indeed, the 3rst, the second, and the third Hochschildcohomology groups control the in3nitesimal deformation theory; see, for example,[15]. Recall that the 3rst Hochschild cohomology group can be interpreted as thegroup of classes of outer derivations, and here we are mainly concerned with thisgroup, in particular, we wish to understand when and why it may vanish. In moreconcrete terms, the content of the paper is as follows.

We 3rst approach the vanishing of Hochschild cohomology via semicontinuity.As recently pointed out by Hartshorne [20], Grothendieck’s classical semiconti-nuity result [16] on the variation of cohomological functors in a 5at family admitsa very simple and elegant extension to half-exact coherent functors, a frameworkintroduced by M. Auslander in his fundamental paper [3]. Using these tools wederive two semicontinuity results: the 3rst one applies to all Hochschildcohomology groups of an algebra that is 3nitely generated projective over itsbase ring, whereas the second one pertains to homogeneous components of the3rst Hochschild cohomology group of graded algebras and is tailored towards itsapplication to mesh algebras.

Indeed, we will show that a translation quiver is simply connected if andonly if its mesh algebra over a domain admits no outer derivation, whichholds if and only if its mesh algebra over any commutative ring admits noouter derivation.

We then move on to investigate how the Hochschild cohomology of an algebrarelates to that of the endomorphism algebra of a module over it. In the mostgeneral context we obtain an exact sequence relating 3rst Hochschild cohomologygroups. This enables us, in particular, to show that the 3rst Hochschildcohomology group of the Auslander algebra of a representation-3nite artinalgebra always embeds into that of the algebra. In the more restrictive situationwhere both algebras are projective over the base ring, we will deduce from aclassical pair of spectral sequences the invariance of Hochschild cohomologyunder pseudo-tilting, a notion that includes tilting, co-tilting, and thus,Morita equivalence.

Finally, we apply the results obtained so far to investigate the vanishing of the3rst Hochschild cohomology group of a 3nite-dimensional algebra. Our main result

Received 8 May 2002; revised 27 January 2003.

2000 Mathematics Subject Classi�cation 16E30, 16G30.

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Proc. London Math. Soc. (3) 88 (2004) 355--380 q 2004 London Mathematical SocietyDOI: 10.1112/S0024611503014394

in this direction establishes the equivalence of the following conditions for a 3nite-dimensional algebra A of 3nite representation type over an algebraically closed3eld and G its Auslander algebra:

(1) A admits no outer derivation;(2) G admits no outer derivation;(3) A is simply connected;(4) G is strongly simply connected.Note that the representation theory of a simply connected algebra is well

understood, and in most cases, for example, if the base 3eld is of characteristicdiHerent from 2, one can use covering techniques to reduce the representationtheory of an algebra of 3nite representation type to that of a simply connectedalgebra [9].

To prove the equivalences just stated, we 3rst reduce to standard algebras andthen apply our result on general mesh algebras. Some of the implications inquestion were already known and some have at least been claimed to be trueunder varying additional hypotheses. To be more speci3c, let us recall brie5y someof the history. First, Happel proved in [17, (5.5)] the equivalence of (1) and (3) forA of directed representation type. Further, the equivalence of (3) and (4) is due toAssem and Brown [1]. Moreover, as pointed out by SkowroInnski, in case A isstandard, one deduces that (1) implies (3) from [22, Theorem 1; 11, (1.2); 9, (4.7)].Finally, the equivalence of (2) and (3) is claimed in [18, x 4] for a base 3eld ofcharacteristic zero. However, in the proof given there, it is assumed implicitly thatA is tilted. Unfortunately, not only is the given argument thus incomplete, but ithas also been widely misquoted; see, for example, [1, 14].

To conclude this introduction, we draw attention to the long standing questionas to whether vanishing of the 3rst Hochschild cohomology of an algebra precludesthe existence of an oriented cycle in its ordinary quiver. We showed earlier thatthe answer is negative in general [12]. However, it would still be interesting toknow for which classes of algebras the answer is aKrmative. Our results show thatalgebras of 3nite representation type, Auslander algebras, and mesh algebras eachform such a class.

1. Coherent functors and semicontinuity

The main objective of this section is to derive two semicontinuity results onHochschild cohomology. The 3rst one is a direct application of Grothendieck’ssemicontinuity theorem [16, (7.6.9)] for homological functors arising fromcomplexes of 3nitely generated projective modules, whereas the second oneneeds more care and is crucial for our characterization of (strongly) simplyconnected (Auslander) algebras.We useHartshorne’s reworking of the semicontinuitytheorem as it saves some work and makes the proof more transparent.

All rings and algebras in this paper are associative with unit and all modulesare unital. Throughout, R denotes a commutative ring and unadorned tensorproducts are taken over R. Let ModR denote the category of R-modules andmodR its full subcategory of 3nitely generated modules.

Let F be an endofunctor on ModR, that is, an R-linear covariant functor fromModR to itself. Recall that F is half-exact on modR if for any exact sequence

0�!M �!N �!Q�! 0

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RAGNAR-OLAF BUCHWEITZ AND SHIPING LIU356

in modR, the sequence

F ðMÞ �! F ðNÞ �! F ðQÞ

is exact. Following Auslander [3], we say that F is coherent on modR if thereexists an exact sequence of functors

HomRðM;�Þ �!HomRðN;�Þ �! F �! 0;

with M and N in modR. Such an exact sequence is called a coherent presentationof F . If F is coherent on modR, then F clearly commutes with inductive limitsin modR. In particular, for any 3eld L that is an R-algebra, F ðLÞ is a3nite-dimensional vector space over L. We 3rst reformulate a result of Hartshorne[20, (4.6)] that characterizes half-exact coherent functors. For convenience, we calla sequence

M: M0 �!fM1 �!

gM2

of morphisms in an abelian category with gf ¼ 0 a short complex and denote byHðMÞ the unique homology group of M.

1.1. PROPOSITION (Hartshorne). Let R be a noetherian commutative ring.(1) An endofunctor F of ModR admits a coherent presentation

HomRðP;�Þ �! HomRðN;�Þ �! F �! 0

with P projective if and only if there exists a short complex P of �nitely generatedprojective R-modules such that F ffi HðPR �Þ.

(2) An endofunctor of ModR is half-exact coherent on modR if and only if it isa direct summand of an endofunctor satisfying the conditions stated in (1).

In [16, x 7], Grothendieck discusses the behaviour under base change ofcohomological functors that satisfy the conditions stated in Proposition 1.1(1). AsHartshorne observed, the key semicontinuity theorem [16, (7.6.9)] can beformulated just as well for half-exact coherent functors. To do so, we denote asusual by kðpÞ the residue 3eld of the localization Rp of a commutative ring R at aprime ideal p.

1.2. THEOREM (Grothendieck). Let R be a noetherian commutative ring andF an endofunctor on ModR that is half-exact and coherent on modR. Thedimension function

p 7�! dimkðpÞF ðkðpÞÞ

is then upper semicontinuous on SpecðRÞ and takes on only �nitely many values.

To apply this result to the Hochschild cohomology of algebras, let us brie5yrecall the de3nition; see [13] or [21] for more details. From now on, A denotes anR-algebra. Let A� ¼ fa� j a 2 Ag be the opposite algebra and Ae ¼ A� A theenveloping algebra of A over R. An A-bimodule X will always be assumed to besymmetric as R-module, whence it becomes a right Ae-module via x � ða� bÞ ¼ axb.This action does not interfere with the left R-module structure on X, whence forany R-module M, the tensor product M X over R inherits a right Ae-module

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HOCHSCHILD COHOMOLOGY 357

structure through that on X, as well as a compatible left R-module structurethrough that on M.

The bar resolution of A over R is given by the complex

. . .�!Aðiþ2Þ �!bi Aðiþ1Þ �! . . .�!A3�!b1 A2�!�A

A�! 0;

where �A is the multiplication map on A, and bi is the map given by

biða0 a1 . . . aiþ1Þ ¼Xij¼0

ð�1Þja0 . . . ajajþ1 . . . aiþ1:

Each term in the bar resolution of A over R is naturally an A-bimodule and thediHerentials respect that structure, whence the bar resolution can be viewed as acomplex of right Ae-modules. Let B denote the truncated bar resolution of Aover R. The ith cohomology group of the complex HomAeðB; XÞ is called theith Hochschild cohomology group of A over R with coeKcients in X and isdenoted by HHi

RðA;XÞ. In case X ¼ A, we write HHiRðAÞ ¼ HHi

RðA;AÞ. If Ris understood, we shall simply write HHiðA;XÞ for HHi

RðA;XÞ and HHiðAÞfor HHi

RðAÞ.The following semicontinuity result on Hochschild cohomology is a straight-

forward application of Theorem 1.2.

1.3. PROPOSITION. Let R be a noetherian commutative ring. Let A be anR-algebra and X an A-bimodule, and assume that both A and X are �nitelygenerated projective as R-modules. For each i> 0, the dimension function

p 7! dimkðpÞ HHikðpÞðkðpÞ R A; kðpÞ R XÞ

is then upper semicontinuous on SpecðRÞ and takes on only �nitely many values.

Proof. As A is 3nitely generated projective over R, so is Ai for each i> 0.There is thus an isomorphism of functors on ModR as follows:

�X HomRðAi; RÞ ffi HomAeðAðiþ2Þ;�XÞ;

where the right Ae-module structure on �X is inherited from the one on X.Consequently, for each R-module M, the complex HomAeðB;M XÞ is isomorphicto a complex P� of the following form:

0�!M X HomRðR;RÞ �! . . .�!M X HomRðAi; RÞ �! . . . ;

and so HHiRðA;M XÞ ffi HiðP�Þ. As X is assumed to be 3nitely generated

projective over R, the same holds for X HomRðAi; RÞ, whence the functorHHi

RðA;�XÞ satis3es the assumptions of (1.2) for each i> 0.To conclude, let S be a commutative R-algebra and consider the S-algebra

AS ¼ S A. With B the truncated bar resolution of A over R, the truncated barresolution of AS over S is isomorphic to BS ¼ S B. Moreover, XS ¼ S X isnaturally an AS-bimodule. Now adjunction gives rise to the following isomorphismof complexes:

HomðASÞeðBS;XSÞ ffi HomAeðB; XSÞ;

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whence HHiSðAS;XSÞ ffi HHi

RðA;XSÞ for each i> 0. Applying this isomorphism toS ¼ kðpÞ for p 2 SpecðRÞ completes the proof. �

Now we turn our attention to the 3rst Hochschild cohomology group. Note thatHH1

RðA;XÞ ffi Ext1AeðA;XÞ for any R-algebra A and any A-bimodule X. As usual,it is useful to interpret this group through (classes of) outer derivations. To thisend, recall that an R-derivation on A with values in X is an R-linear map � : A! Xsuch that �ðabÞ ¼ �ðaÞbþ a�ðbÞ for all a; b 2 A. For each x 2 X, the map

adðxÞ ¼ ½x;�� : A�!X : a 7�! ½x; a� ¼ xa� ax

is an R-derivation. A derivation of this form is called inner, whereas the others arecalled outer. Denoting by DerRðA;XÞ the R-module of R-derivations on A withvalues in X, and by InnRðA;XÞ its submodule of inner derivations, one can identifythe 3rst Hochschild cohomology group as HH1

RðA;XÞ ffi DerRðA;XÞ=InnRðA;XÞ,whence it can be de3ned through the exact sequence of R-modules

X��!ad DerRðA;XÞ ��!HH1RðA;XÞ ��! 0: ð1Þ

In case X ¼ A, we shall simply write DerRðAÞ ¼ DerRðA;AÞ and InnRðAÞ ¼InnRðA;AÞ.

Assume now that we are given a second R-algebra B and a homomorphismB! A of R-algebras. For each A-bimodule X we have then an R-linearrestriction map DerRðA;XÞ ! DerRðB;XÞ whose kernel we denote by DerBRðA;XÞand call the B-normalized derivations on A. Recall that an R-algebra B isseparable if B is projective as (right) B e-module. In particular, for everyB-bimodule Y , any R-derivation on B with values in Y is inner. One may exploitthis as follows.

1.4. LEMMA. Assume that the structure map of the R-algebra A factorsthrough a separable R-algebra B. For an A-bimodule X, set

XB ¼ fx 2 X jxb ¼ bx for each b 2 Bg;the B invariants of X. The following sequence of R-modules is then exact:

XB��!ad DerBRðA;XÞ ��!HH1RðA;XÞ ��! 0; ð2Þ

equivalently, every derivation on A is the sum of a B-normalized derivation andan inner one. Moreover, XB is a direct summand of X as R-module.

Proof. For the 3rst statement, just apply the Ker-Coker-Lemma to the maps

X��!ad DerRðA;XÞ ��!DerRðB;XÞand observe that the composition is surjective as B is separable. For the 3nalstatement, note that XB ¼ Xe, where e 2 B e is a separating idempotent for theseparable algebra B. The lemma is thus established. �

We will use the following standard application of this result. Let U be acomplete set of pairwise orthogonal idempotents of the R-algebra A. ThenB ¼

Le2U Re is an R-subalgebra of A that is separable over R. An R-derivation

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� : A! X vanishes on B if and only if it vanishes on each idempotent in U,whence such a derivation is also called U-normalized, or simply normalized in caseU is understood. Let DerURðA;XÞ denote the R-module of U-normalized derivationsand InnURðA;XÞ its submodule of U-normalized inner derivations. For eachA-bimodule X, its B-invariants are easy to describe:

XB ¼Me2U

eXe: ð3Þ

Let now A ¼L

i> 0Ai be a positively graded R-algebra and X ¼L

j2Z Xj agraded A-bimodule. An R-derivation � : A! X is of degree d if �ðAiÞ � Xiþd forall i> 0. Let DerURðA;XÞd denote the R-module of U-normalized derivations of

degree d and InnURðA;XÞd that of U-normalized inner derivations of degree d. TheR-module HH1

RðA;XÞ contains then

HH1RðA;XÞd ¼ DerURðA;XÞd=InnURðA;XÞd;

the group of outer derivations of degree d, as a submodule. Repeating thearguments above in the graded context yields the following result.

1.5. LEMMA. Let A ¼L

i> 0Ai be a graded R-algebra and X ¼L

j2ZXj agraded A-bimodule. Let U be a complete set of pairwise orthogonal idempotents ofA. There exists an exact sequence of R-modules as follows:M

e2UeXde��!

adDerURðA;XÞd ��!HH1

R ðA;XÞd ��! 0: ð4Þ

Proof. We need to show that any normalized inner derivation of degree dis of the form ½x;�� with x 2

Le2U eXde. Clearly ½x;�� 2 lnnUR ðA;XÞd if

x 2P

e2U eXde. Conversely, let x ¼P

j xj with xj 2 Xj be such that ½x;�� isnormalized of degree d. For each i> 0 and any j, one has ½xj; Ai� � Xjþi. Hence½xj; Ai� ¼ 0 for all i> 0 and each j 6¼ d, whence ½x;�� ¼ ½xd;��. As ½xd;�� is thennormalized, it is an element of degree d in

Le2U eXe, that is, in

Le2U eXde as

each idempotent is of degree zero. The proof of the lemma is complete. �

The graded R-algebra A is called �nitely generated in degrees 0 and 1 if thereexist a naturally graded tensor algebra T ¼

Li> 0 Ti, where T0 and T1 are 3nitely

presented over R and Ti with i> 2 is the i-fold tensor product of T1 with itselfover T0, and a homogeneous ideal I contained in

Li> 2 Ti such that A ffi T=I.

Note that the homogeneous components of T are 3nitely presented over R andthose of A are 3nitely generated over R.

The following is the semicontinuity result on the 3rst Hochschild cohomologygroup that we alluded to before.

1.6. THEOREM. Let R be a commutative ring, A a graded R-algebra �nitelygenerated in degrees 0 and 1, and X ¼

Lj2ZXj a graded A-bimodule. Let A ffi

T=I be a presentation as above such that I is �nitely generated as ideal andT0 ¼

Le2U Re with U a complete set of pairwise orthogonal idempotents of T .

Denote by MðIÞ the set of degrees of a �nite set of homogeneous generators of I.(1) For any commutative R-algebra S and any integer d, there is a natural

isomorphism HH1RðA; S R XÞd ffi HH1

SðS R A; S R XÞd:

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(2) If the commutative R-algebra S is -at over R, then for any integer d thereis an isomorphism, natural in X, as follows:

S R HH1RðA;XÞd ffi HH1

SðS R A; S R XÞd:

(3) Let R be noetherian. If d is an integer such thatL

e2U eXde is a �nitelygenerated R-module and Xiþd is �nitely generated projective over R for eachi 2 f1g [MðIÞ, then the dimension function

p 7�! dimkðpÞ HH1kðpÞðkðpÞ R A; kðpÞ R XÞd

is upper semicontinuous on Spec ðRÞ.

Proof. The set U is 3nite, say U ¼ fe1; . . . ; eng. Then V ¼ fe ¼ eþ I j e 2 Ugis a complete set of pairwise orthogonal idempotents of A.

Since T is freely generated as an R-algebra by T0 and T1, every R-derivation isuniquely determined by its values on these R-modules. Each � 2 DerURðT;XÞdvanishes on T0 and determines R-linear maps from eT1e

0 to eXdþ1e0 for each pair

e; e0 2 U. As T1 generates T freely over T0, each such family of maps determinesconversely such a derivation. Thus, DerURðT;XÞd ffi

Li; jHomRðeiT1ej; eiXdþ1ejÞ.

Moreover, a derivation � in DerURðT;XÞd induces a derivation in DerVRðA;XÞd ifand only if �ðIÞ ¼ 0, equivalently, �ðIiÞ ¼ 0 for all i 2 MðIÞ. Write Tij ¼ eiT1ej,Imij ¼ eiImej and Xmij ¼ eiXdþmej, for 16 i; j6n and m 2 MðIÞ. Thus, we obtainan exact sequence of R-modules

0�!DerVRðA;XÞd �!Mi; j

HomRðTij;X1ijÞ �!Mm;i; j

HomRðImij;XmijÞ; ð5Þ

where i and j range over f1; . . . ; ng and m over MðIÞ.Now let S be a commutative R-algebra and set TS ¼ S T , IS ¼ S I and

AS ¼ S A ¼ TS=IS. Clearly, VS ¼ f1S ðei þ IÞ j 16 i6ng is a complete set ofpairwise orthogonal idempotents in AS and XS ¼ S X ¼

Lj2ZðS XiÞ is a

graded AS-bimodule. Repeating the above argument, we obtain the sequence (5)as well for the corresponding tensored objects.

(1) Apply the exact sequence (5) twice to XS, 3rst as A-bimodule and then asAS-bimodule. Using adjunction in the Hom-terms, we conclude that

DerVSS ðAS;XSÞd ffi DerVRðA; S XÞd:

As furthermore S ðL

e eXdeÞ ffiL

e eðS XÞde, the exact sequence (4) yields

HH1RðA; S XÞd ffi HH1

SðS A; S XÞd:

(2) Assume that S is 5at as R-module. We use again the exact sequence (5)twice: 3rst, we tensor it with S, and secondly apply it to XS. By assumption, themodule T1 is 3nitely presented over R and so then are the direct summands Tij.Moreover, as I is a 3nitely generated ideal, the R-modules Imij are 3nitelygenerated over R. This implies that the natural map

S Mi; j

HomRðTij;X1ijÞ �!Mi; j

HomRðTij; S X1ijÞ

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is an isomorphism, whereas the natural map

S Mm;i; j

HomRðImij;XmijÞ �!Mm;i; j

HomRðImij; S XmijÞ

is at least injective. We infer that S DerVRðA;XÞd ffi DerVRðA; S XÞd for any d.Applying the same argument as in (1) to the exact sequence (4), we obtainS HH1

RðA;XÞd ffi HH1RðA; S XÞd. Combining this isomorphism with the one

from part (1) yields the claim.(3) By assumption now, Xdþm is 3nitely generated projective over R for each

m 2 f1g [MðIÞ. As direct R-summands of Xdþm, the Xmij are all 3nitelygenerated projective R-modules too. Write ð�Þ� ¼ HomRð�; RÞ for the R-dual. IfP is 3nitely generated projective over R, then so are P � and P �� ffi P . Hence forany R-modules M and N, we obtain 3rst M Xmij ffi HomRðX�

mij;MÞ, and thenHomRðN;M XmijÞ ffi HomRðX�

mij N;MÞ by adjunction. Employing this lastisomorphism in the exact sequence (5), we deduce that the functor DerVRðA;�XÞdis the kernel of a morphism of functors

HomR

Mi; j

X�1ij Tij;�

!�!HomR

Mm;i; j

X�mij Imij;�

!:

Therefore, DerVRðA;�XÞd ¼ HomRðC;�Þ, whereC is the cokernel of someR-linearmap from

Lm;i; j X

�mij Imij to

Li; j X

�1ij Tij. As

Li; j X

�1ij Tij is a 3nitely

generated R-module, so is C.To conclude the argument, consider the functor GðMÞ ¼

Le eðM XÞde on

ModR. The operations of applying ð Þd and multiplying withP

e e e 2 Ae areexact functors. Thus, Gð�Þ is right exact and so isomorphic to �GðRÞ.Now GðRÞ ¼

Le eXde is 3nitely generated over R and we may choose a

surjection from a 3nitely generated projective R-module Q onto it. But then�Q! �

Le eXde is an epimorphism of functors on ModR. Finally, we

may identify �Q ffi HomRðQ�;�Þ, to obtain from the above and the exactsequence (4) the coherent presentation

HomRðQ�;�Þ �!HomRðC;�Þ �!HH1RðA;�XÞd �! 0:

Thus, HH1ðA;�XÞd is half-exact and coherent on modR by Hartshorne’s result.Hence, dimkðpÞHH

1RðA; kðpÞ XÞd, which equals dimkð pÞHH

1kð pÞðkðpÞ A; kðpÞ XÞd

by part (1), varies upper semicontinuously with p on Spec ðRÞ. This 3nishes theproof of the theorem. �

2. Mesh algebras without outer derivations

The main objective of this section is to show that a 3nite translation quiver issimply connected if and only if its mesh algebra over a domain admits no outerderivation, and if this is the case, its mesh algebra over any commutative ringadmits no outer derivation.

We begin with some combinatorial considerations on quivers and theirunderlying graphs. A sequence

a0e1 a1 . . . ar�1

er ar

of edges of a graph such that ei 6¼ eiþ1 for all 16 i < r is called a reduced walk,

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and the sequence is a cycle if ar ¼ a0 and er 6¼ e1. A vertex of a graph isconsidered as a trivial reduced walk.

Now let Q be a quiver, that is, an oriented graph. A reduced walk or a cycle ofQ is in fact a reduced walk or a cycle, respectively, of its underlying graph of edges.A sequence of arrows a0 ! a1 ! . . . ! an with n> 1 is called a path of length nfrom a0 to an, and is an oriented cycle if a0 ¼ an. Such an oriented cycle is calledsimple if the ai with 06 i < n are pairwise distinct. A vertex a is considered as apath of length 0 from a to a. The set of paths of length n is denoted by Qn. Inparticular, Q0 is the set of vertices and Q1 is that of arrows.

A vertex a of Q is called a source if no arrow ends in a, a sink if no arrow startsfrom a. Let ( be a possibly empty subquiver of Q. We say that ( is convex in Qif a path of Q lies entirely in ( as soon as its end-points lie in (. A convexsubquiver in our terminology is thus in particular a full subquiver. We say that Qis a one-point extension of ( by a vertex a, if a is a source of Q and is the onlyvertex of Q that is not in (. In the dual situation, we say that Q is a one-pointco-extension of (. Note that ( is convex in Q in either situation. The followingsimple lemma is a reformulation of [2, (2.3)]; see also [19, (7.6)].

2.1. LEMMA. Let Q be a �nite connected quiver. If Q contains no oriented cycle,then Q is a one-point extension or co-extension of one of its connected subquivers.

Now 3x a commutative ring R and a 3nite quiver Q. For each i> 0, let RQi bethe free R-module with basis Qi. Then RQ ¼

Li> 0RQi is a positively graded

R-algebra, called the path algebra of Q over R, with respect to the multiplicationthat is induced from the composition of paths. Note that we use the convention tocompose paths from the left to the right. The set Q0 yields a complete set ofpairwise orthogonal idempotents of RQ and for a; b 2 Q0, the Peirce componentaðRQÞb is the free R-module spanned by the paths from a to b.

Two or more paths ofQ are parallel if they have the same start-point and the sameend-point. A relation on Q over R is an element ) ¼

Pri¼1 *ipi 2 RQ, where the

*i 2 R are all non-zero and the pi are parallel paths of length at least 2. In this case,we say that p1; . . . ; pr are the paths forming ) and that a path appears in ) if it is asubpath of one of the pi. The relation ) is called polynomial if r> 2; and homogeneousif the pi are of the same length. Moreover, a relation onQ over Z with only coeKcients1 or �1 is called universal. The point is that a universal relation on Q remains arelation involving the same paths over any commutative ring.

Let N be a set of relations on Q over R. The pair ðQ;NÞ is called a boundquiver, while the quotient RðQ;NÞ of RQ modulo the ideal generated by N iscalled the algebra of the bound quiver ðQ;NÞ. If N contains only homogeneousrelations, then RðQ;NÞ is a graded R-algebra with grading induced from that ofRQ. Moreover, RðQ;NÞ is called monomial if N contains no polynomial relation.The following easy lemma generalizes slightly a result of Bardzell and Marcos[5, (2.2)] which states that Q is a tree if RðQ;NÞ is monomial withoutouter derivations.

2.2. LEMMA. Let R be a commutative ring, and let Q be a �nite quiver with Na set of relations on Q over R. If there exists a cycle in Q containing an arrowthat appears in no polynomial relation in N, then HH1

RðRðQ;NÞÞ does not vanish.

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Proof. Let + be an arrow of Q. There exists a normalized derivation � of RQsuch that �ð+Þ ¼ + and �ð,Þ ¼ 0 for any other arrow ,. Assume that + appears inno polynomial relation in N. Then � preserves the ideal I generated by N. Thus �induces a derivation � of RðQ;NÞ. Suppose that � is inner. Then there existsu ¼

Pa2Qa

*aaþ w 2 RQwith *a 2 R andw 2L

i> 1RQi such that �ðvÞ � ½u; v� 2 I,for any v 2 RQ. Suppose further that + is on a cycle

a0+1 a1 . . . ar�1

+r ar ¼ a0;

where +i : ai aiþ1 denotes an arrow in either direction. We may assume that+ ¼ +1 and +i 6¼ + for all 1 < i6 r. Since I �

Li> 2RQi, evaluating � on the +i

yields *a0 ¼ *a1 � 1 and *ai ¼ *aiþ1 for all 16 i < r. This contradiction establishesthe lemma. �

Applying our semicontinuity result obtained in the previous section for the 3rstHochschild cohomology group of a graded algebra, we are able to exclude orientedcycles from Q if RðQ;NÞ is well graded and admits no outer derivation.

2.3. PROPOSITION. Let Q be a �nite quiver and n> 2 an integer. Assume thatN is a set of universal relations formed by paths of Qn such that every path of Qn

appears in at most one relation. Let O be a subset of Qn obtained by removing,for each relation ) 2 N, one path from those forming ).

(1) For any commutative ring R, the component of degree n of RðQ;NÞ is afree R-module having as basis the set of the classes of the paths of O.

(2) If R is a domain such that HH1ðRðQ;NÞÞ vanishes, then Q contains nooriented cycle and any two parallel paths have the same length.

Proof. (1) Let R be a commutative ring. By assumption, the ideal I of RQgenerated by N is generated by elements of degree n. Write RðQ;NÞ ¼

Li> 0Ai

with Ai ¼ ðRQi þ IÞ=I. Statement (1) is obvious.(2) To simplify the notation, we write RðQÞ ¼ RðQ;NÞ. For any commutative

R-algebra S, one has clearly

SðQÞ ffi S R RðQÞ:The general assumptions in Theorem 1.6 are thus satis3ed for A ¼ X ¼ RðQÞ. Nowlet R be a domain and L its 3eld of fractions. Assume that HH1

RðRðQÞÞ ¼ 0. Inparticular, HH1

RðRðQÞ; RðQÞÞ0 ¼ 0. As L is 5at over R, Theorem 1.6(2) yields

0 ¼ L HH1RðRðQÞ; RðQÞÞ0 ffi HH1

LðLðQÞ; LðQÞÞ0:

Suppose that L is of prime characteristic. Let 0 : Z ! L be the canonical ringhomomorphism and let p be its kernel so that Zp ¼ Z=p becomes a sub3eld of L.Using Theorem 1.6(2) again, we get

LZp HH1ZpðZpðQÞ;ZpðQÞÞ0 ffi HH1

LðLðQÞ; LðQÞÞ0 ¼ 0:

As a consequence, HH1ZpðZpðQÞ;ZpðQÞÞ0 ¼ 0. Now we apply Theorem 1.6(3) with

R ¼ Z and d ¼ 0. The conditions there are satis3ed as A0 and A1 are free of 3niterank by de3nition, and so is An by part (1). Semicontinuity then shows thatHH1

QðQðQÞ;QðQÞÞ0 ¼ 0. Thus we may assume that L is of characteristic zero.Note that the Euler derivation E of LðQÞ is normalized of degree 0, and hence isinner. As already observed by Happel [17] evaluating E on an oriented cycle of Q

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would then give a contradiction. Evaluating it on two parallel paths shows thatthey are of the same length. The proof of the proposition is completed. �

We now turn our attention to translation quivers. Let P be a translation quiverwith translation 1 , that is, P is a quiver containing neither loops nor multiplearrows and 1 is a bijection from a subset of P0 to another one such that, for eacha 2 P0 with 1ðaÞ de3ned, there exists at least one arrow + : b! a and any sucharrow determines a unique arrow 2ð+Þ : 1ðaÞ ! b; see [9, (1.1)]. One de3nes theorbit graph OðPÞ of P as follows: the 1-orbit of a vertex a is the set oðaÞ ofvertices of the form 1nðaÞ with n 2 Z; the vertices of OðPÞ are the 1-orbits of P,and there exists an edge oðaÞ oðbÞ in OðPÞ if P contains an arrow x! y ory! x with x 2 oðaÞ and y 2 oðbÞ. Note that OðPÞ contains no multiple edge byde3nition. If P contains no oriented cycle, then OðPÞ is the graph GP de3ned in [9,(4.2)]. Now we say that P is simply connected if P contains no oriented cycle andOðPÞ is a tree; see [7, 8]. By [9, (1.6), (4.1), (4.2)], this de3nition is equivalent tothat in [9, (1.6)]. Finally, if ( is a convex subquiver of P, then ( is a translationquiver with respect to the translation induced from that of P. In this case, Oð(Þis clearly a subgraph of OðPÞ. Thus, if P is simply connected, then so is everyconnected convex translation subquiver of P.

We shall study some properties of simply connected translation quivers. If a is avertex of a quiver, we shall denote by a� the set of immediate predecessors and byaþ that of immediate successors of a.

2.4. LEMMA. Let P be a translation quiver, containing no oriented cycle.(1) Let a be a vertex of P and a1; a2 2 a� or a1; a2 2 aþ. Then oða1Þ ¼ oða2Þ if

and only if a1 ¼ a2.(2) If P is simply connected, then an arrow + : a! b is the only path of P from

a to b.

Proof. (1) It suKces to consider the case where a1; a2 2 a�. Suppose thatoða1Þ ¼ oða2Þ. We may assume that a2 ¼ 1ra1 for some r> 0. If r > 0, thena! 1r�1a1 ! . . . ! a1 ! a would be an oriented cycle in P. Hence r ¼ 0, thatis, a1 ¼ a2.

(2) Let + : a! b be an arrow and

p : a�!+1a1 �! . . .�! as�1 �! as ¼ b

a diHerent path from a to b. Then b 6¼ a1 since P contains neither multiplearrows nor oriented cycle. Therefore oðbÞ 6¼ oða1Þ by (1). In particular, the edgesoðaÞ oðbÞ and oðaÞ oðb1Þ of OðPÞ are distinct. Now p induces a walk

wðpÞ : oðaÞ oða1Þ . . . oðasÞ

in OðPÞ, and wðpÞ, in turn, determines a unique reduced walk wredðpÞ from oðaÞ tooðbÞ. We claim that wredðpÞ starts with the edge oðaÞ oða1Þ. This implies thatOðPÞ is not a tree, that is, P is not simply connected.

To prove our claim, it suKces to show that oðaÞ 6¼ oðaiÞ for all 16 i6 s. If thisis not the case, then at ¼ 1�ma for some 16 t6 s and m 2 Z. Now m > 0 as Pcontains no oriented cycle, and consequently there exists a path from 1�a to1�ma. Further the arrow a! b gives rise to an arrow b! 1�a. Thus we obtain an

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oriented cycle

b1 �! 1�a�! . . .�! 1�ma ¼ at �! atþ1 �! . . .�! as�1 �! b

in P. This contradiction completes the proof of the lemma. �

Recall that a vertex a of P is projective or injective if, respectively, 1ðaÞ or1�ðaÞ is not de3ned. The following result demonstrates how to constructinductively simply connected translation quivers.

2.5. LEMMA. Let P be a connected translation quiver that is a one-pointextension of a simply connected translation subquiver ( by a vertex a. Then P isnot simply connected if and only if a is injective and is the start-point of at leasttwo distinct arrows.

Proof. Suppose that P is not simply connected. Then OðPÞ contains a cycleoða0Þ oða1Þ . . . oðar�1Þ oða0Þ. This cycle contains oðaÞ, since Oð(Þ isa tree by assumption. We may assume that oða0Þ ¼ oðaÞ. If a is not injective, thenb ¼ 1�ðaÞ 2 (, and hence oða0Þ ¼ oðbÞ. Therefore the preceding cycle gives rise toa cycle in Oð(Þ. This contradiction shows that a is injective, and hence the onlyvertex in oðaÞ. Therefore, P contains arrows + : a! b and , : a! c with b 2 oða1Þand c 2 oðar�1Þ. Since OðPÞ contains no multiple edge, we have oða1Þ 6¼ oðar�1Þ. Inparticular, b 6¼ c. This shows that the conditions on a are necessary.

Conversely suppose that a is injective and that there exist two distinct arrows+ : a! b and , : a! c. By Lemma 2.4(1), oðbÞ 6¼ oðcÞ. In particular, oðaÞ oðbÞand oðaÞ oðcÞ are two distinct edges of OðPÞ. Being connected, Oð(Þ containsa non-trivial reduced walk oðbÞ oðc1Þ . . . oðcs�1Þ oðcÞ that can beconsidered as a reduced walk in OðPÞ. This gives rise to a cycle

oðaÞ oðbÞ oðc1Þ . . . oðcs�1Þ oðcÞ oðaÞ

in OðPÞ, that is, P is not simply connected. The proof of the lemmais complete. �

Let us now recall the de3nition of a mesh algebra. Let P be a 3nite translationquiver. A non-projective vertex a of P determines a universal relation on P, calleda mesh relation, mðaÞ ¼

P2ð+iÞ+i, where the sum is taken over the arrows

ending in a. It is clear that every path of length 2 appears in at most one meshrelation on P. Let R be a commutative ring. The ideal of the path algebra RPgenerated by the mesh relations is called the mesh ideal, whereas the quotientRðPÞ of RP modulo the mesh ideal is called the mesh algebra of P over R.

2.6. PROPOSITION. Let R be a commutative ring and P a �nite translationquiver. If P is simply connected, then RðPÞ admits no outer derivation.

Proof. Assume that P is simply connected. Then an arrow + : x! y is theonly path of P from x to y by Lemma 2.4(2). Hence every normalized derivation ofRðPÞ is of degree zero. We shall use induction on the number n of vertices of P toprove the result. If n ¼ 1, then RðPÞ ffi R and the result holds trivially. Assumethat n > 1 and the result holds for n� 1. By Lemma 2.1, we may assume that P

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is a one-point extension of a connected translation subquiver ( by a vertex a.Note that a is projective and ( is simply connected.

Let � be a normalized derivation of RðPÞ. For w 2 RP, write w ¼ wþ IP, whereIP is the mesh ideal of RP. Since � is of degree zero, for each arrow + 2 P,�ð+Þ ¼ *++ for some *+ 2 R. In particular, �ðRð(ÞÞ � Rð(Þ. Thus the restriction�( of � to Rð(Þ is a normalized derivation of Rð(Þ. Now �( is inner by theinductive hypothesis and is of degree zero. Therefore there exists u ¼

Px2(0

�xx,with �x 2 R, such that �( ¼ ½u;��. Since P is connected, there exist arrows startingfrom a. Let +i : a! bi with 16 i6 r be all such arrows. Set �a ¼ �b1 þ *+1

andv ¼ �aaþ u. Then �ð+1Þ ¼ ½v; +1�. We claim that �ð+iÞ ¼ ½v; +i�, for all 16 i6 r.This is trivial if r ¼ 1. Assume now that r > 1. Then a is not injective by Lemma2.5. Therefore, c ¼ 1�ðaÞ exists and lies in (. Hence ( admits arrows ,i : bi ! c,for i ¼ 1; . . . ; r. Evaluating � on the equality

Pri¼1 +i,i ¼ 0 gives rise toPr

i¼2ð*+i þ *,i � *+1� *,1Þ+i,i ¼ 0. Note that +2,2; . . . ; +r,r are linearly inde-

pendent over R by Proposition 2.3(1). Hence *+i þ *,i ¼ *+1þ *,1 for all 26 i6 r.

Moreover, evaluating �( ¼ ½u;�� on ,i results in �c ¼ �bi � *,i for all 16 i6 r, thatis �bi � *,i ¼ �b1 � *,1 , for all 16 i6 r. Now one can deduce that �ð+iÞ ¼ ½v; +i�, forall 16 i6 r. This shows that �ð,Þ ¼ ½v; ,� for any arrow of P, and consequently,� ¼ ½v;��. The proof of the proposition is complete. �

We 3nally reach the main result of this section.

2.7. THEOREM. Let R be a domain and P a �nite connected translationquiver. The following are equivalent:

(1) HH1ðRðPÞÞ ¼ 0;(2) P is simply connected;(3) HH1ðRð(ÞÞ ¼ 0 for every connected convex translation subquiver ( of P.

Proof. It is trivial that (3) implies (1). Assume that P is simply connected.Then every connected convex translation subquiver of P is simply connected, andhence its mesh algebra admits no outer derivation by Proposition 2.6. This provesthat (2) implies (3).

We shall show by induction on the number n of vertices of P that (1) implies (2).This is trivial if n ¼ 1. Assume that n > 1 and this implication holds for n� 1.Suppose that HH1ðRðPÞÞ ¼ 0. By Proposition 2.3(2), P contains no oriented cycleand any two parallel paths are of the same length. In particular, a normalizedderivation of RðPÞ is of degree zero. By Lemma 2.1, we may assume that P is aone-point extension of a connected translation subquiver ( by a vertex a.

Suppose on the contrary that P is not simply connected. First we consider thecase where ( is simply connected. By Lemma 2.5, a is injective and there existtwo distinct arrows + : a! b and , : a! c with b; c 2 (0. Since ( is connected,b and c are connected by a reduced walk in (. Therefore, + lies on a cycle of Pand it appears in no mesh relation on P since a is injective. This contradictsLemma 2.2.

We now turn to the case where ( is not simply connected. By the inductivehypothesis, there exists a normalized outer derivation � of Rð(Þ that is of degreezero. Thus for each arrow + 2 (, we have �ð+Þ ¼ *++ with *+ 2 R, where wdenotes the class of w 2 R( modulo the mesh ideal I( of R(. Let � be the

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normalized derivation of R( such that �ð+Þ ¼ *++ for all + 2 (. Then�ðI(Þ � I(. Being connected, P contains arrows starting with a. Let +i : a! biwith 16 i6 s be all such arrows. If a is injective, let @ be the normalizedderivation of RP such that @ coincides with � on R( and @ð+iÞ ¼ 0 for all16 i6 s. Then @ preserves the mesh ideal IP of P, since IP is generated by themesh relations on ( in this case. Suppose now that a is not injective and let ,i bethe arrow from bi to 1�ðaÞ for 16 i6 s. Then m ¼

Psi¼1 +i,i is the only mesh

relation on P that is not in I(. Note that ,i 2 (, and in particular, *,i is de3nedfor all 16 i6 s. Let @ be the normalized derivation of RP such that @ coincideswith � on R( and @ð+iÞ ¼ ð1� *,iÞ+i for all 16 i6 s. It is easy to see that in thiscase @ preserves IP as well. Now @ induces a normalized derivation @ of RðPÞ, thatcoincides with � on Rð(Þ and is of degree zero. However, @ is inner sinceHH1ðRðPÞÞ ¼ 0. Hence its restriction to Rð(Þ, that is �, is easily seen to be inner aswell. This contradiction shows that (1) implies (2). The proof of the theorem isnow complete. �

REMARK. In case P contains no oriented cycle and R is an algebraically closed3eld, Coelho and Vargas proved in [14] the equivalence of (2) and (3) in terms ofstrongly simply connected algebras. They claimed as well the equivalence of (1)and (2) when in addition the base 3eld is of characteristic zero. However, theproof for this equivalence given there misquoted the incomplete argument given inthe proof of the second theorem in [18, x 4].

3. Hochschild cohomology of endomorphism algebras

The objective of this section is to compare Hochschild cohomology of an algebraand that of the endomorphism algebra of a module. We shall 3rst studythe relation between the 3rst Hochschild cohomology groups in the mostgeneral situation and then prove the invariance of Hochschild cohomology incase the algebras involved are projective over the base ring and the moduleis pseudo-tilting.

As before, let A be an R-algebra and Ae its enveloping algebra. Recall that allunadorned tensor products are taken over R. We 3x the following extension ofA-bimodules:

!A: 0�! NA�!jA

A A�!�A

A�! 0;

where �A is the multiplication map of A and jA is the inclusion map. Each mapf 2 HomAeðNA;XÞ determines an R-derivation �f : A! X : a 7! fða 1� 1 aÞ.The following well-known facts will be used extensively in our investigationbelow, whence we state them explicitly for the convenience of the reader. We referto [10, (AIII.132)] for a proof of the 3rst part.

3.1. LEMMA. Let A be an R-algebra and X an A-bimodule.(1) The following map is an isomorphism of R-modules:

D : HomAeðNA;XÞ �!DerRðA;XÞ : f 7! �f :

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(2) There exists a commutative diagram of R-modules:

Xad

DerRðA;XÞ

ffi???y ffi

???yD�1

HomAeðA A;XÞðjA;XÞ

HomAeðNA;XÞ

Now let B be another R-algebra. Any B-A-bimodule X becomes a rightB� A-module through x � ðbo aÞ ¼ bxa. Let M and N be B-A-bimodules.Forgetting the right A-module structure or the left B-module structure on anextension of B-A-bimodules yields respectively the following maps:

fA : Ext1B �AðM;NÞ �! Ext1BðM;NÞ;fB : Ext1B �AðM;NÞ �! Ext1AðM;NÞ:

The kernels of these maps are identi3ed in the following result.

3.2. PROPOSITION. Let M and N be B-A-bimodules. The forgetful mapsintroduced above �t into the following exact sequences:

(1) 0 ! HH1ðA;HomBðM;NÞÞ�!gA

Ext1B �AðM;NÞ�!fA

Ext1BðM;NÞ,

(2) 0 ! HH1ðB;HomAðM;NÞÞ�!gB

Ext1B �AðM;NÞ�!fB

Ext1AðM;NÞ.

Proof. It suKces to show the 3rst part of the proposition. To this end, we shallde3ne explicitly the map gA. First applying Lemma 3.1 to the A-bimoduleHomBðM;NÞ and then using adjunction, we get the following commutative diagram:

HomBðM;NÞ adDerRðA;HomBðM;NÞÞ

ffi???y ffi

???yHomAeðAA;HomBðM;NÞÞ

ðjA; ðM;NÞÞHomAeðNA;HomBðM;NÞÞ

ffi???y ffi

???yHomB �AðM A;NÞ

ðM jA;NÞHomB �AðM A NA;NÞ

ð6Þ

Since !A splits as an extension of left A-modules, the sequence

M A !A: 0 M A NAM A jA

M AM A �A

M 0

is an extension of B-A-bimodules that splits as an extension of left B-modules. For� 2 DerRðA;HomBðM;NÞÞ, let e�� 2 HomB �AðM A NA; NÞ be the correspondingmap under the isomorphisms given in the above diagram (6). Pushing outM A !Aalong the map e��, we get an exact commutative diagram


M AM A �A

M 0

�

???y ???y e��ðM A !AÞ: 0 N E M 0

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where the rows are extensions of B-A-bimodules that split as extensions of leftB-modules. Now the lower row splits as an extension of B-A-bimodules if and onlyif e�� factors through M A jA which is so if and only if � 2 InnRðA;HomBðM;NÞÞ.This shows that associated to the class ½�� 2 HH1ðA;HomBðM;NÞÞ of a derivation�, the class

gAð½��Þ ¼ ½e��ðM A !AÞ� 2 Ext1B �AðM;NÞ

yields a well-de3ned injective map gA with image contained in the kernel of fA.It remains to show that the kernel of fA is contained in the image of gA. For

this purpose, let

>: 0�!N �! E�!pM �! 0

be an extension of B-A-bimodules that splits as an extension of left B-modules.Let q :M ! E be a B-linear map such that pq ¼ 1M , and let 2 :M A! E bethe B-A-bilinear map such that 2ðm aÞ ¼ qðmÞa. These data de3ne an exactcommutative diagram of B-A-bimodules,


M AM A �A

M 0

@

???y 2

???y >: 0 N E

pM 0

where @ is induced by 2. In view of the isomorphisms given in (6), there exists aderivation � 2 DerRðA; HomBðM;NÞÞ whose image is @. Hence gAð½��Þ ¼ ½>�. Thiscompletes the proof of the proposition. �

Now we specialize the preceding proposition to the case M ¼ N. The canonicalalgebra anti-homomorphism ) : A! EndBðMÞ is a homomorphism of A-bimodules.Thus ) induces an R-linear map

HH1ðA; )Þ : HH1ðA;AÞ �!HH1ðA;EndBðMÞÞ;

which in turn allows us to de3ne a map 0M : HH1ðAÞ ! Ext1AðM;MÞ through thefollowing commutative diagram:

HH1ðAÞ0M

Ext1AðM;MÞ

HH1ðA; )Þ???y fB

x???HH1ðA;EndBðMÞÞ

gAExt1B �AðM;MÞ

To give an explicit description of 0M , consider the commutative diagram

DerRðA;AÞ) � �

DerRðA;EndBðMÞÞ

ffi???y ffi

???yHomAeðNA;AÞ

M A �HomB �AðM A NA;MÞ

where ) � denotes composition with ), the isomorphism on the left comes fromLemma 3.1(2) while that on the right comes from the diagram (6) in the proof ofProposition 3.2. Let � 2 DerRðA;AÞ be a derivation and let f 2 HomAeðNA;AÞ be

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the corresponding map. Applying MA -- to the extension !A and pushing outM A !A along the map M A f yields a self-extension ðM A fÞ�ðM A !AÞ ofthe right A-module M. Then

0Mð½��Þ ¼ ½ðM A fÞ�ðM A !AÞ�

by our earlier de3nition of gA and fB.To study the map 0M further, we give an alternative and explicit description.

To this end, we de3ne the diHerentiation of a morphism between projectivemodules of the following form: let

A :Mnj¼1

ujA�!Mmi¼1

viA

be an A-linear map with uj and vi some idempotents of A. Let � 2 DerRðAÞ besuch that �ðviAujÞ � viAuj for all i; j. If the matrix of A with respect to the givendecomposition is ðaijÞm�n with aij 2 viAuj, then we call the A-linear map

�ðAÞ :Mnj¼1

ujA�!Mmi¼1

viA

given by the matrix ð�ðaijÞÞm�n the derivative of A along �.

3.3. LEMMA. Let M be a B-A-bimodule that is �nitely presented as rightA-module. Let

P2 Mn

j¼1ujA

A Mmi¼1

viA"M 0

be an exact sequence of right A-modules with P2 projective and uj and viidempotents in A. Let � 2 DerRðAÞ be such that �ðviAujÞ � viAuj for all i; j.Then the image of ½�� 2 DerRðAÞ=InnRðAÞ under 0M is the class of the self-extension of M that corresponds to the class ½"�ðAÞ� in Ext1AðM;MÞ.

Proof. Let f 2 HomAeðNA;AÞ be such that �ðaÞ ¼ fða 1� 1 aÞ for alla 2 A; see Lemma 3.1(1). We have seen that 0Mð½��Þ ¼ ½ðM A fÞ�ðM A !AÞ�.We claim that there exist A-linear maps + and , rendering the followingdiagram commutative:Mn

i¼1ujA

A Mmj¼1

viA"

M 0

�ðAÞ????y

��!,

��!+

===========Mmj¼1

viA M A NAM jA

M AM �A

M 0��!

" M A f

M

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In fact, since "ðviÞ vi ¼ ð"ðviÞ viÞvi 2 ðM AÞvi, there exists a uniqueA-linearmap + :

Lmi¼1 viA!M A such that +ðviÞ ¼ "ðviÞ vi for all 16 i6m. Moreover,

assume that the matrix of A is ðaijÞm�n with aij 2 viAuj. Then AðujÞ ¼Pm

i¼1 viaijand �ðAÞðujÞ ¼

Pmi¼1 vi�ðaijÞ for all 16 j6n. Let

xj ¼Xmi¼1

"ðviÞ A ðaij 1� 1 aijÞ 2M A NA:

Note that aij ¼ aijuj andPm

i¼1 "ðviÞaij ¼ "ðAðujÞÞ ¼ 0. This implies thatxj ¼ xjuj 2 ðM A NAÞuj. Therefore, there exists a unique A-linear map

, :Mnj¼1

ujA�!M A NA

such that ,ðujÞ ¼ xj for all 16 j6n. It is now easy to verify that the abovediagram is commutative. Consider now the following exact commutative diagram:

0 NMj Mm

i¼1viA

"M 0

D

???y +

???y M A !A: 0 M A NA

M A jAM A

M A �AM 0

M A f

???y ???y >: 0 M E M 0

where j is the inclusion map and D is induced by +. Let eAA :Ln

j¼1 ujA! NM be

such that A ¼ jeAA. Then , ¼ D eAA, and hence "�ðAÞ ¼ ðM A fÞD eAA. Note thateAA ¼ 0. This implies "�ðAÞ 2 Kerð ;MÞ and exhibits > as the self-extension of Mcorresponding to "�ðAÞ. The proof of the lemma is complete. �

Next, we look more carefully at the image of the map 0M .

3.4. LEMMA. Let M be a B-A-bimodule with M ¼Ln

i¼1Mi a decompositionof right A-modules. Then 0M factors as follows:

HH1ðAÞ 0MExt1AðM;MÞX

i

0Mi &Mni¼1

Ext1AðMi;MiÞ

Proof. Let � 2 DerRðAÞ be a derivation and f 2 HomAeðNA;AÞ the corre-sponding map; see Lemma 3.1(1). For each i, the map f de3nes an exactcommutative diagram:

Mi A !A: 0 Mi A NAMi A jA

Mi AMi A �A

Mi 0

Mi A f??y 2i

??y >i : 0 Mi Ei Mi 0

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Summing up yields the following exact commutative diagram:


M AM A �A

M 0

M A f

???y Mi

2i

???y Mi

>i: 0 MMi

Ei M 0

Therefore, 0Mð½��Þ ¼ ½L

i @i� 2Ln

i¼1 Ext1AðMi;MiÞ. This completes the proof of

the lemma. �

In our main application, we shall consider the case where M is a right A-moduleand B ¼ EndAðMÞ, whence M is endowed with the canonical B-A-bimodulestructure. Moreover, there exists a morphism of B-A-bimodules ) : A! EndBðMÞ,where the map )ðaÞ for a 2 A is given by )ðaÞðxÞ ¼ xa, for all x 2M. Recall thatM is a faithfully balanced B-A-bimodule if ) is an isomorphism.

3.5. THEOREM. Let A be an algebra over a commutative ring R. Let M be aright A-module and set B ¼ EndAðMÞ. Assume that M is a faithfully balancedB-A-bimodule with Ext1BðM;MÞ ¼ 0. Then there exists an exact sequence

0 HH1ðBÞ HH1ðAÞ0M

Ext1AðM;MÞ

of R-modules. Moreover, if M ¼Ln

i¼1Mi is a decomposition of right A-modules,then the image of 0M lies in the diagonal part

Lni¼1 Ext

1AðMi;MiÞ.

Proof. By hypothesis, HH1ðA; )Þ : HH1ðAÞ ! HH1ðA;EndBðMÞÞ is an iso-morphism. Furthermore, the map gA in Proposition 3.2(1) is an isomorphism sinceExt1BðM;MÞ ¼ 0. Thus D ¼ gA �HH1ðA; )Þ is an isomorphism. Set c ¼ D�1 � gB. Itfollows from the de3nition of 0M that the diagram

0 HH1ðBÞ cHH1ðAÞ

0MExt1AðM;MÞ D

???y ffi

0 HH1ðBÞ gBExt1B �AðM;MÞ fB

Ext1AðM;MÞ

is commutative. By Proposition 3.2(2), the lower row is exact, and hence theupper row is an exact sequence as desired. The last part of the theorem followsfrom Lemma 3.4. This completes the proof. �

For a right A-module, denote by addðMÞ the full subcategory of the category ofright A-modules generated by the direct summands of direct sums of 3nitely manycopies of M. We call M an A-generator if A lies in addðMÞ. We shall now studythe behaviour of Hochschild cohomology under (r-)pseudo-tilting, a notion wede3ne as follows.

3.6. DEFINITION. Let A be an algebra over a commutative ring R and letr> 0 be an integer. A right A-module M is called r-pseudo-tilting if it satis3es thefollowing conditions:

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(1) ExtiAðM;MÞ ¼ 0 for 16 i6 r;(2) there exists an exact sequence

0�!A�!M0 �!M1 �! . . .�!Mr �! 0

of right A-modules, where M0; . . . ;Mr 2 addðMÞ.Moreover, the module M is called pseudo-tilting if ExtiAðM;MÞ ¼ 0 for all i> 1

and (2) holds for some r> 0; in other words, M is r-pseudo-tilting for all suKcientlylarge r.

Note that a 0-pseudo-tilting A-module is nothing but an A-generator. Moreover,a tilting or co-tilting module of projective or injective dimension at most r,respectively, in the sense of [23] is an r-pseudo-tilting module, and consequently, apseudo-tilting module. We recall now the following result from [23, (1.4)].

3.7. LEMMA (Miyashita). Let A be an R-algebra and r> 0 an integer. Let Mbe an r-pseudo-tilting right A-module and set B ¼ EndAðMÞ. Then M is afaithfully balanced B-A-bimodule and ExtiBðM;MÞ ¼ 0 for all i> 1.

As an immediate consequence of Lemma 3.7 and Theorem 3.5, we get thefollowing result.

3.8. PROPOSITION. Let A be an algebra over a commutative ring R. Let M bean r-pseudo-tilting right A-module and set B ¼ EndAðMÞ.

(1) There exists an exact sequence of R-modules

0��!HH1ðBÞ ��!HH1ðAÞ��!0M

Ext1AðM;MÞ:(2) If r> 1, then HH1ðAÞ ffi HH1ðBÞ.

So far we have not put any restriction on the R-algebra structure. As usual inHochschild theory, much sharper results can be obtained if the algebras involvedare projective as R-modules, for example, when R is a 3eld. First we recall thefollowing result from [13, p. 346].

3.9. PROPOSITION (Cartan--Eilenberg). Let A and B be algebras over acommutative ring R, and let M and N be B-A-bimodules. If A and B areprojective as R-modules, then there exist spectral sequences

HHiðA;ExtjBðM;NÞÞ¼)ExtiþjB �AðM;NÞand

HHiðB;ExtjAðM;NÞÞ¼)ExtiþjB �AðM;NÞ:

REMARK. The 3rst spectral sequence yields the following exact sequence oflower terms that extends the exact sequence from Proposition 3.2(1):

0�!HH1ðA;HomBðM;NÞÞ�!g1A

Ext1B �AðM;NÞ�!f1A

HH0ðA;Ext1BðM;NÞÞ

�!HH2ðA;HomBðM;NÞÞ�!g2A

Ext2B �AðM;NÞ:

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Applying the preceding proposition to the case where M ¼ N is a (r-)pseudo-tilting A-module and B ¼ EndAðMÞ, we obtain the following result.

3.10. THEOREM. Let R be a commutative ring, and let A be an R-algebrathat is projective as R-module. Let M be a right A-module such thatB ¼ EndAðMÞ is projective as R-module as well.

(1) If M is r-pseudo-tilting, then HHiðBÞ ffi HHiðAÞ for 06 i6 r.(2) If M is pseudo-tilting, then HHiðBÞ ffi HHiðAÞ for all i> 0.

Proof. First we note that the edge homomorphisms of the 3rst spectralsequence in Proposition 3.9 become giA : HHiðAÞ ! ExtiB�AðM;MÞ and

f iA : ExtiB �AðM;MÞ �!HH0ðA;ExtiBðM;MÞÞ; for all i> 0;

whereas those of the second one become giB : HHiðBÞ ! ExtiB �AðM;MÞ and

f iB : ExtiB �AðM;MÞ �!HH0ðA;ExtiAðM;MÞÞ; for all i> 0:

Assume now that M is r-pseudo-tilting. By Lemma 3.7, ExtiBðM;MÞ ¼ 0 for alli> 1. So each giB with i> 0 is an isomorphism. Moreover, since ExtiAðM;MÞ ¼ 0for 16 i6 r, the giA are isomorphisms for 06 i6 r. These give rise to the desiredisomorphisms

ci ¼ ðgiAÞ�1giB : HHiðBÞ �!HHiðAÞ for i ¼ 0; 1; . . . ; r:

If M is pseudo-tilting, then the map ci is de3ned and an isomorphism for eachi> 0. This completes the proof of the theorem. �

REMARK. In case A is a 3nite-dimensional algebra over an algebraically closed3eld and M is a 3nite-dimensional tilting A-module, Happel showed in [17, (4.2)]that HHiðAÞ ffi HHiðBÞ for all i> 0.

4. Representation-�nite algebras without outer derivations

The objective of this section is to investigate when an algebra of 3niterepresentation type admits no outer derivation. To begin with, let A be an artinalgebra. Denote by modA the category of 3nitely generated right A-modules andby indA its full subcategory generated by a chosen complete set of representativesof isoclasses of the indecomposable modules. Let PA denote the Auslander--Reitenquiver of A, which is a translation quiver with respect to the Auslander--Reitentranslation DTr. We refer to [4] for general results on almost split sequences andirreducible maps as they pertain to the structure of Auslander--Reiten quivers.

Assume that A is of 3nite representation type, that is, indA contains only3nitely many objects, say M1; . . . ;Mn. Then M ¼

Lni¼1 Mi is an A-generator,

called a minimal representation generator for A. One calls G ¼ EndAðMÞ theAuslander algebra of A. Applying Lemma 3.7 and Theorem 3.5, one obtainsimmediately the following exact sequence:

0�! HH1ðGÞ �!HH1ðAÞ�!0 Mn

i¼1Ext1A ðMi;MiÞ:

Thus, HH1ðGÞ is always embedded into HH1ðAÞ. To investigate when this

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embedding is an isomorphism, recall that a module in indA is a brick if itsendomorphism algebra is a division ring. It is shown in [25, (4.6)] that if indAcontains only bricks, then A is of 3nite representation type. Conversely, it is wellknown that if A is of 3nite representation type and PA contains no oriented cycle,then indA contains only bricks.

4.1. PROPOSITION. Let A be an artin algebra such that indA contains onlybricks, and let G be the Auslander algebra of A. Then HH1ðGÞ ffi HH1ðAÞ.

Proof. In view of the exact sequence stated above, we need only show thatExt1A ðN;NÞ ¼ 0 for every module N in indA. Assume on the contrary that this failsfor somemoduleM in indA. ThenM is non-projective andHomAðM;DTrMÞ 6¼ 0; see,for example, [4, p. 131]. Let

0�!DTrM �! E �!M �! 0

be an almost split sequence in modA. Using the fact that an irreducible map iseither a monomorphism or an epimorphism, one 3nds easily that either M or anindecomposable direct summand of E is not a brick. This contradiction completes theproof of the proposition. �

From now on, let k be an algebraically closed 3eld and A a 3nite-dimensionalk-algebra. It is well known (see [9, (2.1)]) that there exist a unique 3nite quiverQA (called the ordinary quiver of A) and an admissible ideal IA in kQA such thatthe basic algebra B of A is isomorphic to kQA=IA. Such an isomorphism B ffi kQA=IAis called a presentation of B. Moreover, one says that A is connected or triangularif QA is connected or contains no oriented cycle, respectively. In case A is of 3niterepresentation type, one says that A is standard [9, (5.1)] if its Auslander algebrais isomorphic to the mesh algebra kðPAÞ of PA over k. It follows from [11, (3.1)]that this de3nition is equivalent to that given in [6, (1.11)].

4.2. LEMMA. Let A be of �nite representation type, and let G be its Auslanderalgebra. If HH1ðAÞ ¼ 0 or HH1ðGÞ ¼ 0, then A is standard.

Proof. It is well known that HH1ðAÞ is invariant under Morita equivalence;see also Proposition 3.8. We may hence assume that A is basic. Suppose that A isnot standard. It follows from [6, (9.6)] that there exists a presentation A ffi kQA=IAsuch that the bound quiver ðQA; IAÞ contains a Riedtmann contour C as a fullbound subquiver. This means that the ordinary quiver QC of C consists of avertex a, a loop ) at a, and a simple oriented cycle

a ¼ a0 �!+1

a1�! . . .�!an�1 �!+n

an ¼ a;

bound by the following relations:

)2 � +1 . . .+n ¼ +n+1 � +n)+1

¼ +iþ1 . . .+n)+1 . . .+fðiÞ ¼ 0; for i ¼ 1; . . . ; n� 1;

where f : f1; . . . ; n� 1g ! f1; . . . ; n� 1; ng is a non-decreasing function that is notconstant with value 1. Furthermore, )3 62 IA since )2 is a non-deep path [6, (2.5),(2.7), (6.4), (9.2)]. Using [6, (7.7)] and its dual, we have the following fact.

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(1) If p is a path of QA that does not lie completely in QC but contains ) as asubpath, then p 2 IA.

Next, write x ¼ xþ IA 2 A for x 2 kQA. For simplicity, we identify thevertex set of QA with a complete set of orthogonal primitive idempotents of A.Since A is 3nite dimensional, there exists an integer m> 4 such that )m ¼ 0 and)m�1 6¼ 0. Now

+n)2 ¼ +n+1 . . .+n ¼ +n)+1 . . .+n ¼ +n))

2 ¼ +n)3 ¼ . . . ¼ +n)

m ¼ 0:

Thus, +n)2 2 IA and analogously )2+1 2 IA. We now show the following.

(2) If p is a non-trivial path of QA di.erent from ), then )m�2p; p)m�2 2 IA. Asa consequence, )m�1 lies in the socle of A.

In fact, write p ¼ p1, with , an arrow. If , 6¼ ), using (1) together with+n)

2 2 IA and m> 4, we infer that ,)m�2 2 IA. If , ¼ ), then p1 is non-trivial andwe can write p1 ¼ p2D, with D an arrow. Then D,)m�2 ¼ D)m�1 2 IA, and thereforep)m�2 2 IA. Similarly, )m�2p 2 IA, and we have established statement (2).

Now let @ be the unique normalized derivation of kQA with @ð)Þ ¼ )m�1 and@ð+Þ ¼ 0 for any arrow + 6¼ ). Our next claim is as follows.

(3) If p is a path of QA that is di.erent from ), then @ðpÞ 2 IA.

This is trivially true if p is of length at most 1. Assume thus that p isnon-trivial and that (3) holds for paths shorter than p. If ) does not appear inp, then @ðpÞ ¼ 0. Otherwise p ¼ u)v, where u and v are paths shorter than p withu or v non-trivial. Now @ðpÞ ¼ @ðuÞ)vþ u)m�1vþ u)@ðvÞ. It follows from (2) thatu)m�1v 2 IA. Moreover, if u 6¼ ), then @ðuÞ 2 IA by the inductive hypothesis.Otherwise, @ðuÞ)v ¼ )mv 2 IA. Similarly, u)@ðvÞ 2 IA. Therefore, @ðpÞ 2 IA,which proves statement (3).

Since IA is generated by linear combinations of paths of length at least 2,@ðIAÞ � IA by (3). Hence @ induces a normalized derivation � of A with �ðyÞ ¼ @ðyÞfor all y 2 kQA. We now wish to show that � is an outer derivation of A. Assume onthe contrary that � ¼ ½x;�� for some x 2 A. Write x ¼ x1 þ x2, where x1 2 Aaþ aAand x2 2 a0Aa0 with a0 ¼ 1� a. Then �ð)Þ ¼ ½x1; )� ¼ )m�1 6¼ 0: Thus x1 62 aAasince ) lies in the center of aAa. Therefore x1 ¼ z1 þ z2 þ z, where z1 2 a0Aa,z2 2 aAa0, z 2 aAa and z1 þ z2 6¼ 0. However, this would imply that �ðaÞ ¼z1 � z2 6¼ 0, contrary to � being normalized. Hence [�] is a non-zero elementof HH1ðAÞ.

Let M be a minimal representation generator of A such that G ¼ EndAðMÞ. ByProposition 3.8, there exists an exact sequence

0�!HH1ðGÞ �!HH1ðAÞ0M

Ext1AðM;MÞ:We want to show that ½�� 2 Kerð0MÞ. Let

P1�!AP0 �!

"M �! 0

be a minimal projective presentation of M. Let b1; . . . ; bn and c1; . . . ; cm be verticesof QA such that P1 ¼

Lnj¼1 bjA and P0 ¼

Lmi¼1 ciA. The matrix of A is then

ðxijÞm�n with xij 2 ci radðAÞ bj. Since � is normalized, �ðciAbjÞ � ciAbj. Hence the

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matrix of �ðAÞ : P1 ! P0 is ð�ðxijÞÞm�n. For each 3xed 16 i6m, we consider theith row ðxi1; xi2; . . . ; xinÞ of the matrix of A. If c1 6¼ a, then �ðxijÞ ¼ 0 for all16 j6n by (3). Thus,

ð�ðxi1Þ; �ðxi2Þ; . . . ; �ðxinÞÞ ¼ 0 � ðxi1; xi2; . . . ; xinÞ:

Assume that ci ¼ a. If bj 6¼ a, then �ðxijÞ ¼ 0 by (3) and )m�2xij ¼ 0 by (2),since xij 2 cirad ðAÞbj. If bj ¼ a, then xij ¼

Pm�1s¼1 *s)

s with *s 2 k sinceaAa ¼ k½)�=ð)mÞ. Now �ð)Þ ¼ )m�2 � ), whereas for s> 2, both �ð)sÞ ¼ 0 and)m�2 � )s ¼ 0. This shows that

ð�ðxi1Þ; �ðxi2Þ; . . . ; �ðxinÞÞ ¼ )m�2 � ðxi1; xi2; . . . ; xinÞ:For 16 i; j6m, let yij ¼ 0 if i 6¼ j, yii ¼ 0 if ci 6¼ a, and yii ¼ )m�2 if ci ¼ a.Then ð�ðxijÞÞ ¼ ðyijÞðxijÞ. Let @ : P0 ! P0 be the A-linear map given by thematrix ðyijÞ. Then �ðAÞ ¼ @A, and hence "�ðAÞ ¼ ð"@ÞA. This shows that the class½"�ðAÞ� 2 Ext1AðM;MÞ is zero. Now Lemma 3.3 implies that the class ½�� 2 HH1ðAÞlies in the kernel of 0M . Therefore, HH1ðGÞ 6¼ 0. This completes the proof ofthe lemma. �

If A is of 3nite representation type, we say that A is simply connected if PA is asimply connected translation quiver. In general, one says that A is strongly simplyconnected if A is connected and triangular, and its basic algebra B admits apresentation B ffi kQA=IA such that the 3rst Hochschild cohomology groupvanishes for every algebra de3ned by a convex bound subquiver of ðQA; IAÞ. Itfollows from [24, (4.1)] that this de3nition coincides with the original one given in[24, (2.2)]. Moreover, if A is of 3nite representation type, then A is simplyconnected if and only if it is strongly simply connected. We are now ready toestablish the main result of this section.

4.3. THEOREM. Let A be a connected �nite-dimensional algebra over analgebraically closed �eld k. Assume that A is of �nite representation type and letG be its Auslander algebra. The following statements are equivalent:

(1) HH1ðAÞ ¼ 0;(2) HH1ðGÞ ¼ 0;(3) A is simply connected;(4) G is strongly simply connected.

Proof. First of all, we claim that each of the conditions stated in the theoremimplies that A is standard. Indeed, for (1) or (2), this follows from Lemma 4.2.Now each of (3) or (4) implies that PA contains no oriented cycle. This in turnimplies that the ordinary quiver of A contains no oriented cycle, as A is of 3niterepresentation type. Therefore, A is standard by [6, (9.6)].

Thus we may assume that A is standard, that is, G ffi kðPA). Now theequivalence of (2), (3), and (4) follows immediately from Theorem 2.7. Moreover,(1) implies (2) since HH1ðGÞ embeds into HH1ðAÞ. Finally, if (2) holds, then (3)holds as well. In particular, indA contains only bricks. By Proposition 4.1,HH1ðAÞ ¼ HH1ðGÞ ¼ 0. This completes the proof of the theorem. �

We conclude this paper with a consequence of Theorem 4.3. Recall that A is anAuslander algebra if its global dimension is less than or equal to 2 while its

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dominant dimension is greater than or equal to 2. Note that A is an Auslanderalgebra if and only if it is Morita equivalent to the Auslander algebra of analgebra of 3nite representation type; see, for example, [4].

4.4. COROLLARY. Let A be of �nite representation type or an Auslanderalgebra. If HH1ðAÞ ¼ 0, then HHiðAÞ ¼ 0 for all i> 1. In particular, A is rigid inthis case.

Proof. For the 3rst part, we apply Theorem 4.3 and the two results stated,respectively, in [17, (5.4)] and [18, x 5]. For the last part, observe that HH2ðAÞ ¼ 0implies rigidity, as originally proved by Gerstenhaber [15]. �

Acknowledgements. Both authors gratefully acknowledge partial supportfrom the Natural Sciences and Engineering Research Council of Canada. Theauthors were also supported in part by respectively an Ontario--QuIeebec ExchangeGrant of the Ministry of Education and Training of Ontario and a grant ofCoopIeeration QuIeebec-Provinces canadiennes en Enseignement supIeerieur etRecherche of the Ministry of Education of QuIeebec.

References

1. I. ASSEM and P. BROWN, ‘Strongly simply connected Auslander algebras’, Glasgow Math. J.39 (1997) 21--27.

2. I. ASSEM and S. LIU, ‘Strongly simply connected algebras’, J. Algebra 207 (1998) 449--477.3. M. AUSLANDER, ‘Coherent functors’, Proceedings of the Conference on Categorical Algebra

(ed. S. Eilenberg, D. K. Harrison, S. MacLane and H. Rohrl), La Jolla, 1965 (Springer,New York, 1966) 189--231.

4. M. AUSLANDER, I. REITEN and S. O. SMALU, Representation theory of Artin algebras,Cambridge Studies in Advanced Mathematics 36 (Cambridge University Press, 1995).

5. M. J. BARDZELL and E. N. MARCOS, ‘Induced boundary maps for the cohomology ofmonomial and Auslander algebras’, Algebras and modules II (ed. I. Reiten, S. O. SmalVand U. Solberg), Canadian Mathematical Society Conference Proceedings 24 (AmericanMathematical Society, Providence, RI, 1998) 47--54.

6. R. BAUTISTA, P. GABRIEL, A. V. ROITER and L. SALMERIOON, ‘Representation-3nite algebrasand multiplicative basis’, Invent. Math. 81 (1985) 217--285.

7. R. BAUTISTA and F. LARRI IOON, ‘Auslander--Reiten quivers for certain algebras of 3niterepresentation type’, J. London Math. Soc. (2) 26 (1982) 43--52.

8. R. BAUTISTA, F. LARRI IOON and L. SALMERIOON, ‘On simply connected algebras’, J. LondonMath. Soc. (2) 27 (1983) 212--220.

9. K. BONGARTZ and P. GABRIEL, ‘Covering spaces in representation theory’, Invent. Math. 65(1982) 331--378.

10. N. BOURBAKI, IEElIeements de MathIeematiques: algWeebre, Chapitres 1 Waa 3 (Hermann, Paris,1970).

11. O. BRETSCHER and P. GABRIEL, ‘The standard form of a representation-3nite algebra’, Bull.Soc. Math. France 111 (1983) 21--40.

12. R.-O. BUCHWEITZ and S. LIU, ‘Artin algebras with loops but no outer derivations’, Algebr.Represent. Theory 5 (2002) 149--162.

13. H. CARTAN and S. EILENBERG, Homological algebra (Princeton University Press, 1956).14. F. U. COEHLO and R. R. S. VARGAS, ‘Strongly simply connected mesh algebras’, Preprint,

University of S~aao Paulo, 1999.15. M. GERSTENHABER, ‘On the deformations of rings and algebras’, Ann. of Math. 79 (1964)

59--103.16. A. GROTHENDIECK, ‘IEElIeements de gIeeomIeetrie algIeebrique III: IEEtude cohomologique des

faisceaux cohIeerents II’, Inst. Hautes IEEtudes Sci. Publ. Math. 17 (1963).17. D. HAPPEL, ‘Hochschild cohomology of 3nite-dimensional algebras’, SIeeminaire d’algWeebra

Paul Dubreil et Marie-Paul Malliavin (ed. M.-P. Malliavin), Lecture Notes inMathematics 1404 (Springer, Berlin, 1989) 108--126.

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18. D. HAPPEL, ‘Hochschild cohomology of Auslander algebras’, Topics in algebra (ed.S. Balcerzyk et al.), Banach Center Publications 26 (Banach Center, Warsaw, 1990)303--310.

19. D. HAPPEL, ‘Quasitilted algebras’, Algebras and modules I (ed. I. Reiten, S. O. SmalV andU. Solberg), Canadian Mathematical Society Conference Proceedings 23 (AmericanMathematical Society, Providence, RI, 1998) 55--83.

20. R. HARTSHORNE, ‘Coherent functors’, Adv. in Math. 140 (1998) 44--94.21. J.-L. LODAY, Cyclic homology, Grundlehren der mathematischen Wissenschaften 301

(Springer, New York, 1992).22. MA. I. R. MARTINS and J. A. DE LA PE~NNA, ‘Comparing the simplicial and the Hochschild

cohomologies of a 3nite dimensional algebra’, J. Pure Appl. Algebra 138 (1999) 45--58.23. Y. MIYASHITA, ‘Tilting modules of 3nite projective dimension’, Math. Z. 193 (1986)

113--146.24. A. SKOWROINNSKI, ‘Simply connected algebras and Hochschild cohomologies’, Representations

of algebras (ed. V. Dlab and H. Lenzing), Canadian Mathematical Society ConferenceProceedings 14 (American Mathematical Society, Providence, RI, 1993) 431--447.

25. A. SKOWROINNSKI, ‘Minimal representation-in3nite artin algebras’, Math. Proc. CambridgePhilos. Soc. 116 (1994) 229--243.

Ragnar-Olaf BuchweitzDepartment of MathematicsUniversity of TorontoTorontoOntarioCanada M5S 3G3

[email protected]

Shiping LiuD �eepartement de math �eematiquesUniversit �ee de SherbrookeSherbrookeQu �eebecCanada J1K 2R1

[email protected]

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