hochschild cohomology and representation- finite algebras - TSpace

PLMS 1439---10/2/2004---SRUMBAL---79431 Proc. London Math. Soc. (3) 88 (2004) 355--380 q 2004 London Mathematical Society DOI: 10.1112/S0024611503014394

HOCHSCHILD COHOMOLOGY AND REPRESENTATIONFINITE 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, for example, [13, 15, 21]. The lower-dimensional Hochschild cohomology groups have well-known interpretations. Indeed, the 3rst, the second, and the third Hochschild cohomology groups control the in3nitesimal deformation theory; see, for example, [15]. Recall that the 3rst Hochschild cohomology group can be interpreted as the group of classes of outer derivations, and here we are mainly concerned with this group, in particular, we wish to understand when and why it may vanish. In more concrete 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 semicontinuity result [16] on the variation of cohomological functors in a 5at family admits a very simple and elegant extension to half-exact coherent functors, a framework introduced by M. Auslander in his fundamental paper [3]. Using these tools we derive two semicontinuity results: the 3rst one applies to all Hochschild cohomology groups of an algebra that is 3nitely generated projective over its base ring, whereas the second one pertains to homogeneous components of the 3rst Hochschild cohomology group of graded algebras and is tailored towards its application to mesh algebras. Indeed, we will show that a translation quiver is simply connected if and only if its mesh algebra over a domain admits no outer derivation, which holds if and only if its mesh algebra over any commutative ring admits no outer derivation. We then move on to investigate how the Hochschild cohomology of an algebra relates to that of the endomorphism algebra of a module over it. In the most general context we obtain an exact sequence relating 3rst Hochschild cohomology groups. This enables us, in particular, to show that the 3rst Hochschild cohomology group of the Auslander algebra of a representation-3nite artin algebra always embeds into that of the algebra. In the more restrictive situation where both algebras are projective over the base ring, we will deduce from a classical pair of spectral sequences the invariance of Hochschild cohomology under 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 the 3rst Hochschild cohomology group of a 3nite-dimensional algebra. Our main result Received 8 May 2002; revised 27 January 2003. 2000 Mathematics Subject Classication 16E30, 16G30.

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RAGNAR-OLAF BUCHWEITZ AND SHIPING LIU

in this direction establishes the equivalence of the following conditions for a 3nitedimensional algebra A of 3nite representation type over an algebraically closed 3eld 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 characteristic diHerent from 2, one can use covering techniques to reduce the representation theory of an algebra of 3nite representation type to that of a simply connected algebra [9]. To prove the equivalences just stated, we 3rst reduce to standard algebras and then apply our result on general mesh algebras. Some of the implications in question were already known and some have at least been claimed to be true under varying additional hypotheses. To be more speci3c, let us recall brie5y some of the history. First, Happel proved in [17, (5.5)] the equivalence of (1) and (3) for A of directed representation type. Further, the equivalence of (3) and (4) is due to Assem and Brown [1]. Moreover, as pointed out by SkowroI nski, in case A is standard, 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 of characteristic zero. However, in the proof given there, it is assumed implicitly that A is tilted. Unfortunately, not only is the given argument thus incomplete, but it has also been widely misquoted; see, for example, [1, 14]. To conclude this introduction, we draw attention to the long standing question as to whether vanishing of the 3rst Hochschild cohomology of an algebra precludes the existence of an oriented cycle in its ordinary quiver. We showed earlier that the answer is negative in general [12]. However, it would still be interesting to know for which classes of algebras the answer is aKrmative. Our results show that algebras of 3nite representation type, Auslander algebras, and mesh algebras each form such a class.

1. Coherent functors and semicontinuity The main objective of this section is to derive two semicontinuity results on Hochschild cohomology. The 3rst one is a direct application of Grothendieck’s semicontinuity theorem [16, (7.6.9)] for homological functors arising from complexes of 3nitely generated projective modules, whereas the second one needs more care and is crucial for our characterization of (strongly) simply connected (Auslander) algebras. We use Hartshorne’s reworking of the semicontinuity theorem as it saves some work and makes the proof more transparent. All rings and algebras in this paper are associative with unit and all modules are unital. Throughout, R denotes a commutative ring and unadorned tensor products are taken over R. Let Mod R denote the category of R-modules and mod R its full subcategory of 3nitely generated modules. Let F be an endofunctor on Mod R, that is, an R-linear covariant functor from Mod R to itself. Recall that F is half-exact on mod R if for any exact sequence 0 ! M ! N ! Q ! 0

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in mod R, the sequence F ðMÞ ! F ðNÞ ! F ðQÞ is exact. Following Auslander [3], we say that F is coherent on mod R if there exists an exact sequence of functors HomR ðM; Þ ! HomR ðN; Þ ! F ! 0; with M and N in mod R. Such an exact sequence is called a coherent presentation of F . If F is coherent on mod R, then F clearly commutes with inductive limits in mod R. In particular, for any 3eld L that is an R-algebra, F ðLÞ is a 3nite-dimensional vector space over L. We 3rst reformulate a result of Hartshorne [20, (4.6)] that characterizes half-exact coherent functors. For convenience, we call a sequence f g M: M0 ! M1 ! M2 of morphisms in an abelian category with gf ¼ 0 a short complex and denote by HðMÞ the unique homology group of M.

1.1. PROPOSITION (Hartshorne). Let R be a noetherian commutative ring. (1) An endofunctor F of Mod R admits a coherent presentation HomR ðP; Þ ! HomR ðN; Þ ! F ! 0 with P projective if and only if there exists a short complex P of nitely generated projective R-modules such that F ffi HðP R Þ. (2) An endofunctor of Mod R is half-exact coherent on mod R if and only if it is a direct summand of an endofunctor satisfying the conditions stated in (1). In [16, x 7], Grothendieck discusses the behaviour under base change of cohomological functors that satisfy the conditions stated in Proposition 1.1(1). As Hartshorne observed, the key semicontinuity theorem [16, (7.6.9)] can be formulated just as well for half-exact coherent functors. To do so, we denote as usual by kðpÞ the residue 3eld of the localization Rp of a commutative ring R at a prime ideal p.

1.2. THEOREM (Grothendieck). Let R be a noetherian commutative ring and F an endofunctor on Mod R that is half-exact and coherent on mod R. The dimension function p 7! dimkð pÞ F 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 brie5y recall the de3nition; see [13] or [21] for more details. From now on, A denotes an R-algebra. Let A ¼ fa j a 2 Ag be the opposite algebra and Ae ¼ A A the enveloping algebra of A over R. An A-bimodule X will always be assumed to be symmetric 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 for any R-module M, the tensor product M X over R inherits a right Ae -module

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structure through that on X, as well as a compatible left R-module structure through that on M. The bar resolution of A over R is given by the complex A bi b1 . . . ! A ðiþ2Þ ! A ðiþ1Þ ! . . . ! A 3 ! A 2 ! A ! 0; where A is the multiplication map on A, and bi is the map given by bi ða0 a1 . . . aiþ1 Þ ¼

i X ð1Þj a0 . . . aj ajþ1 . . . aiþ1 : j¼0

Each term in the bar resolution of A over R is naturally an A-bimodule and the diHerentials respect that structure, whence the bar resolution can be viewed as a complex of right Ae -modules. Let B denote the truncated bar resolution of A over R. The ith cohomology group of the complex HomAe ðB; XÞ is called the ith Hochschild cohomology group of A over R with coeKcients in X and is denoted by HHiR ðA; XÞ. In case X ¼ A, we write HHiR ðAÞ ¼ HHiR ðA; AÞ. If R is understood, we shall simply write HHi ðA; XÞ for HHiR ðA; XÞ and HHi ðAÞ for HHiR ðAÞ. The following semicontinuity result on Hochschild cohomology is a straightforward application of Theorem 1.2.

1.3. PROPOSITION . Let R be a noetherian commutative ring. Let A be an R-algebra and X an A-bimodule, and assume that both A and X are nitely generated projective as R-modules. For each i > 0, the dimension function p 7! dimkð pÞ HHikð pÞ ðkð pÞ 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 A i for each i > 0. There is thus an isomorphism of functors on Mod R as follows: X HomR ðA i ; RÞ ffi HomAe ðA ðiþ2Þ ; XÞ; where the right Ae -module structure on X is inherited from the one on X. Consequently, for each R-module M, the complex HomAe ðB; M XÞ is isomorphic to a complex P of the following form: 0 ! M X HomR ðR; RÞ ! . . . ! M X HomR ðA i ; RÞ ! . . . ; and so HHiR ðA; M XÞ ffi Hi ðP Þ. As X is assumed to be 3nitely generated projective over R, the same holds for X HomR ðA i ; RÞ, whence the functor HHiR ðA; 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 bar resolution of AS over S is isomorphic to BS ¼ S B. Moreover, XS ¼ S X is naturally an AS -bimodule. Now adjunction gives rise to the following isomorphism of complexes: HomðAS Þe ðBS ; XS Þ ffi HomAe ðB; XS Þ;


HHiS ðAS ; XS Þ

359

HHiR ðA; XS Þ

whence ffi for each i > 0. Applying this isomorphism to S ¼ kðpÞ for p 2 SpecðRÞ completes the proof. Now we turn our attention to the 3rst Hochschild cohomology group. Note that HH1R ðA; XÞ ffi Ext1Ae ðA; 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 this end, recall that an R-derivation on A with values in X is an R-linear map : A ! X such that ðabÞ ¼ ðaÞb þ aðbÞ 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 are called outer. Denoting by DerR ðA; XÞ the R-module of R-derivations on A with values in X, and by InnR ðA; XÞ its submodule of inner derivations, one can identify the 3rst Hochschild cohomology group as HH1R ðA; XÞ ffi DerR ðA; XÞ=InnR ðA; XÞ, whence it can be de3ned through the exact sequence of R-modules ad ! HH1R ðA; XÞ ! 0: X ! DerR ðA; XÞ

ð1Þ

In case X ¼ A, we shall simply write DerR ðAÞ ¼ DerR ðA; AÞ and InnR ðAÞ ¼ InnR ðA; AÞ. Assume now that we are given a second R-algebra B and a homomorphism B ! A of R-algebras. For each A-bimodule X we have then an R-linear restriction map DerR ðA; XÞ ! DerR ðB; XÞ whose kernel we denote by DerB R ðA; XÞ and call the B-normalized derivations on A. Recall that an R-algebra B is separable if B is projective as (right) B e -module. In particular, for every B-bimodule Y , any R-derivation on B with values in Y is inner. One may exploit this as follows.

1.4. LEMMA . Assume that the structure map of the R-algebra A factors through a separable R-algebra B. For an A-bimodule X, set X B ¼ fx 2 X j xb ¼ bx for each b 2 Bg; the B invariants of X. The following sequence of R-modules is then exact: ad ! DerB ! HH1R ðA; XÞ ! 0; X B R ðA; XÞ

ð2Þ

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

Proof. For the 3rst statement, just apply the Ker-Coker-Lemma to the maps ad ! DerR ðB; XÞ X ! DerR ðA; XÞ and observe that the composition is surjective as B is separable. For the 3nal statement, note that X B ¼ Xe, where e 2 B e is a separating idempotent for the separable algebra B. The lemma is thus established. We will use the following standard application of this result. Let U be a complete L set of pairwise orthogonal idempotents of the R-algebra A. Then B ¼ e2U 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 case U is understood. Let DerUR ðA; XÞ denote the R-module of U-normalized derivations and InnUR ðA; XÞ its submodule of U-normalized inner derivations. For each A-bimodule X, its B-invariants are easy to describe: M eXe: ð3Þ XB ¼ e2U

L Let now A ¼ i > 0 Ai be a positively graded R-algebra and X ¼ j2Z Xj a graded A-bimodule. An R-derivation : A ! X is of degree d if ðAi Þ Xiþd for all i > 0. Let DerUR ðA; XÞd denote the R-module of U-normalized derivations of degree d and InnUR ðA; XÞd that of U-normalized inner derivations of degree d. The R-module HH1R ðA; XÞ contains then L

HH1R ðA; XÞd ¼ DerUR ðA; XÞd =InnUR ðA; XÞd ; the group of outer derivations of degree d, as a submodule. Repeating the arguments above in the graded context yields the following result. L L 1.5. LEMMA . Let A ¼ i > 0 Ai be a graded R-algebra and X ¼ j2Z Xj a graded A-bimodule. Let U be a complete set of pairwise orthogonal idempotents of A. There exists an exact sequence of R-modules as follows: M ad eXd e ! DerUR ðA; XÞd ! HH1R ðA; XÞd ! 0: ð4Þ e2U

Proof. We need to show that any L normalized inner derivation Uof degree d is of the form ½x; with x 2 e2U eXd e. Clearly ½x; 2 lnnR ðA; XÞd if P P x 2 e2U eXd e. Conversely, let x ¼ j xj with xj 2 Xj be such that ½x; is normalized of degree d. For each i > 0 and any j, one has ½xj ; Ai Xjþi . Hence ½xj ; Ai ¼ 0 for all i > 0 and each j 6¼ d, whence ½xd ; is then L ½x; ¼ ½xd ; . As L normalized, it is an element of degree d in eXe, that is, in e2U e2U eXd e as each idempotent is of degree zero. The proof of the lemma is complete. The graded R-algebra A is called nitely L generated in degrees 0 and 1 if there exist a naturally graded tensor algebra T ¼ i > 0 Ti , where T0 and T1 are 3nitely presented over R and Ti with i > 2 is the i-fold tensor L product of T1 with itself over T0 , and a homogeneous ideal I contained in i > 2 Ti such that A ffi T =I. Note that the homogeneous components of T are 3nitely presented over R and those of A are 3nitely generated over R. The following is the semicontinuity result on the 3rst Hochschild cohomology group that we alluded to before.

1.6. THEOREM . Let R be a commutative L ring, A a graded R-algebra nitely generated in degrees 0 and 1, and X ¼ j2Z Xj a graded A-bimodule. Let A ffi T =I be L a presentation as above such that I is nitely generated as ideal and T0 ¼ e2U 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 HH1R ðA; S R XÞd ffi HH1S ðS 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 there is an isomorphism, natural in X, as follows: S R HH1R ðA; XÞd ffi HH1S ðS R A; S R XÞd : L (3) Let R be noetherian. If d is an integer such that e2U eXd e is a nitely generated R-module and Xiþd is nitely generated projective over R for each i 2 f1g [ MðIÞ, then the dimension function p 7! dimkð pÞ HH1kð pÞ ðkðpÞ R A; kðpÞ R XÞd is upper semicontinuous on Spec ðRÞ.

Proof. The set U is 3nite, say U ¼ fe1 ; . . . ; en g. Then V ¼ fe ¼ e þ I j e 2 Ug is 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 is uniquely determined by its values on these R-modules. Each 2 DerUR ðT; XÞd vanishes on T0 and determines R-linear maps from eT1 e0 to eXdþ1 e0 for each pair e; e0 2 U. As T1 generates T freely over T0 , each suchLfamily of maps determines conversely such a derivation. Thus, DerUR ðT; XÞd ffi i; j HomR ðei T1 ej ; ei Xdþ1 ej Þ. Moreover, a derivation in DerUR ðT; XÞd induces a derivation in DerVR ðA; XÞd if and only if ðIÞ ¼ 0, equivalently, ðIi Þ ¼ 0 for all i 2 MðIÞ. Write Tij ¼ ei T1 ej , Imij ¼ ei Im ej and Xmij ¼ ei Xdþm ej , for 1 6 i; j 6 n and m 2 MðIÞ. Thus, we obtain an exact sequence of R-modules M M 0 ! DerVR ðA; XÞd ! HomR ðTij ; X1ij Þ ! HomR ðImij ; Xmij Þ; ð5Þ i; j

m;i; j

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 1 6 i 6 ng isLa complete set of pairwise orthogonal idempotents in AS and XS ¼ S X ¼ j2Z ðS Xi Þ 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 as AS -bimodule. Using adjunction in the Hom-terms, we conclude that DerVSS ðAS ; XS Þd ffi DerVR ðA; S XÞd : L L As furthermore S ð e eXd eÞ ffi e eðS XÞd e, the exact sequence (4) yields HH1R ðA; S XÞd ffi HH1S ðS 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, the module 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 3nitely generated over R. This implies that the natural map M M S HomR ðTij ; X1ij Þ ! HomR ðTij ; S X1ij Þ i; j

i; j

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is an isomorphism, whereas the natural map M M S HomR ðImij ; Xmij Þ ! HomR ðImij ; S Xmij Þ m;i; j

m;i; j

DerVR ðA; XÞd

is at least injective. We infer that S ffi DerVR ðA; S XÞd for any d. Applying the same argument as in (1) to the exact sequence (4), we obtain S HH1R ðA; XÞd ffi HH1R ðA; 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 3nitely generated projective R-modules too. Write ðÞ ¼ HomR ð; RÞ for the R-dual. If P is 3nitely generated projective over R, then so are P and P ffi P . Hence for any R-modules M and N, we obtain 3rst M Xmij ffi HomR ðXmij ; MÞ, and then HomR ðN; M Xmij Þ ffi HomR ðXmij N; MÞ by adjunction. Employing this last isomorphism in the exact sequence (5), we deduce that the functor DerVR ðA; XÞd is the kernel of a morphism of functors ! ! M M HomR X1ij Tij ; ! HomR Xmij Imij ; : i; j

m;i; j

V Therefore, Der L R ðA; XÞd ¼ map from m;i; j Xmij Imij

HomL of some R-linear R ðC; Þ, where C is the L cokernel to X T . As X T ij ij is a 3nitely i; j 1ij i; j 1ij

generated R-module, so is C. L To conclude the argument, consider the functor GðMÞ ¼ P e eðM XÞd e on e Mod R. The operations of applying ð Þd and multiplying with e e e 2 A are exact functors. L Thus, GðÞ is right exact and so isomorphic to GðRÞ. Now GðRÞ ¼ e eXd e is 3nitely generated over R and we may choose a surjection fromLa 3nitely generated projective R-module Q onto it. But then Q ! e eXd e is an epimorphism of functors on Mod R. Finally, we may identify Q ffi HomR ðQ ; Þ, to obtain from the above and the exact sequence (4) the coherent presentation HomR ðQ ; Þ ! HomR ðC; Þ ! HH1R ðA; XÞd ! 0: Thus, HH1 ðA; XÞd is half-exact and coherent on mod R by Hartshorne’s result. Hence, dimkð pÞ HH1R ðA; kðpÞ XÞd , which equals dimkð pÞ HH1kð pÞ ðkð pÞ A; kð pÞ XÞd by part (1), varies upper semicontinuously with p on Spec ðRÞ. This 3nishes the proof of the theorem.

2. Mesh algebras without outer derivations The main objective of this section is to show that a 3nite translation quiver is simply connected if and only if its mesh algebra over a domain admits no outer derivation, and if this is the case, its mesh algebra over any commutative ring admits no outer derivation. We begin with some combinatorial considerations on quivers and their underlying graphs. A sequence a0

e1

a1

...

ar1

er

ar

of edges of a graph such that ei 6¼ eiþ1 for all 1 6 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 is considered as a trivial reduced walk. Now let Q be a quiver, that is, an oriented graph. A reduced walk or a cycle of Q 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 n from a0 to an , and is an oriented cycle if a0 ¼ an . Such an oriented cycle is called simple if the ai with 0 6 i < n are pairwise distinct. A vertex a is considered as a path of length 0 from a to a. The set of paths of length n is denoted by Qn . In particular, 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 starts from a. Let ( be a possibly empty subquiver of Q. We say that ( is convex in Q if a path of Q lies entirely in ( as soon as its end-points lie in (. A convex subquiver in our terminology is thus in particular a full subquiver. We say that Q is a one-point extension of ( by a vertex a, if a is a source of Q and is the only vertex of Q that is not in (. In the dual situation, we say that Q is a one-point co-extension of (. Note that ( is convex in Q in either situation. The following simple 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 L Q. For each i > 0, let RQi be the free R-module with basis Qi . Then RQ ¼ i > 0 RQi is a positively graded R-algebra, called the path algebra of Q over R, with respect to the multiplication that is induced from the composition of paths. Note that we use the convention to compose paths from the left to the right. The set Q0 yields a complete set of pairwise orthogonal idempotents of RQ and for a; b 2 Q0 , the Peirce component aðRQÞb is the free R-module spanned by the paths from a to b. Two or more paths of Q are parallel if they have the same P start-point and the same end-point. A relation on Q over R is an element ) ¼ ri¼1 *i pi 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 a subpath of one of the pi . The relation ) is called polynomial if r > 2; and homogeneous if the pi are of the same length. Moreover, a relation on Q over Z with only coeKcients 1 or 1 is called universal. The point is that a universal relation on Q remains a relation 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 bound quiver, while the quotient RðQ; NÞ of RQ modulo the ideal generated by N is called the algebra of the bound quiver ðQ; NÞ. If N contains only homogeneous relations, then RðQ; NÞ is a graded R-algebra with grading induced from that of RQ. 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 without outer derivations.

2.2. LEMMA . Let R be a commutative ring, and let Q be a nite quiver with N a set of relations on Q over R. If there exists a cycle in Q containing an arrow that appears in no polynomial relation in N, then HH1R ðRðQ; NÞÞ does not vanish.

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Proof. Let + be an arrow of Q. There exists a normalized derivation of RQ such that ð+Þ ¼ + and ð,Þ ¼ 0 for any other arrow ,. Assume that + appears in no polynomial relation in N. Then preserves the ideal I generated by N. Thus induces that is inner. Then there exists P a derivation of RðQ; NÞ. Suppose L u ¼ a2Qa *a a þ w 2 RQ with *a 2 R and w 2 i > 1 RQi such that ðvÞ ½u; v 2 I, for any v 2 RQ. Suppose further that + is on a cycle + + a0 1 a1 ... ar1 r ar ¼ a0 ; aiþ1 denotes an arrow in either where +i : ai Ldirection. We may assume that + ¼ +1 and +i 6¼ + for all 1 < i 6 r. Since I i > 2 RQi , evaluating on the +i yields *a0 ¼ *a1 1 and *ai ¼ *aiþ1 for all 1 6 i < r. This contradiction establishes the lemma. Applying our semicontinuity result obtained in the previous section for the 3rst Hochschild cohomology group of a graded algebra, we are able to exclude oriented cycles 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 that N 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 a free R-module having as basis the set of the classes of the paths of O. (2) If R is a domain such that HH1 ðRðQ; NÞÞ vanishes, then Q contains no oriented cycle and any two parallel paths have the same length. Proof. (1) Let R be a commutative ring. By assumption, the ideal ILof RQ generated by N is generated by elements of degree n. Write RðQ; NÞ ¼ i > 0 Ai 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. Now let R be a domain and L its 3eld of fractions. Assume that HH1R ðRðQÞÞ ¼ 0. In particular, HH1R ðRðQÞ; RðQÞÞ0 ¼ 0. As L is 5at over R, Theorem 1.6(2) yields 0 ¼ L HH1R ðRðQÞ; RðQÞÞ0 ffi HH1L ðLðQÞ; LðQÞÞ0 : Suppose that L is of prime characteristic. Let 0 : Z ! L be the canonical ring homomorphism 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 L Zp HH1Zp ðZp ðQÞ; Zp ðQÞÞ0 ffi HH1L ðLðQÞ; LðQÞÞ0 ¼ 0: As a consequence, HH1Zp ðZp ðQÞ; Zp ðQÞÞ0 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 3nite rank by de3nition, and so is An by part (1). Semicontinuity then shows that HH1Q ðQðQÞ; 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 is inner. As already observed by Happel [17] evaluating E on an oriented cycle of Q


365

would then give a contradiction. Evaluating it on two parallel paths shows that they 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 quiver with translation 1, that is, P is a quiver containing neither loops nor multiple arrows and 1 is a bijection from a subset of P0 to another one such that, for each a 2 P0 with 1ðaÞ de3ned, there exists at least one arrow + : b ! a and any such arrow determines a unique arrow 2ð+Þ : 1ðaÞ ! b; see [9, (1.1)]. One de3nes the orbit graph OðPÞ of P as follows: the 1-orbit of a vertex a is the set oðaÞ of vertices of the form 1 n ðaÞ 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 or y ! x with x 2 oðaÞ and y 2 oðbÞ. Note that OðPÞ contains no multiple edge by de3nition. 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 and OðPÞ is a tree; see [7, 8]. By [9, (1.6), (4.1), (4.2)], this de3nition is equivalent to that in [9, (1.6)]. Finally, if ( is a convex subquiver of P, then ( is a translation quiver with respect to the translation induced from that of P. In this case, Oð(Þ is clearly a subgraph of OðPÞ. Thus, if P is simply connected, then so is every connected convex translation subquiver of P. We shall study some properties of simply connected translation quivers. If a is a vertex of a quiver, we shall denote by a the set of immediate predecessors and by aþ 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 oða1 Þ ¼ oða2 Þ 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 that oða1 Þ ¼ oða2 Þ. We may assume that a2 ¼ 1 r a1 for some r > 0. If r > 0, then a ! 1 r1 a1 ! . . . ! a1 ! a would be an oriented cycle in P. Hence r ¼ 0, that is, a1 ¼ a2 . (2) Let + : a ! b be an arrow and +1 p : a ! a1 ! . . . ! as1 ! as ¼ b a diHerent path from a to b. Then b 6¼ a1 since P contains neither multiple arrows nor oriented cycle. Therefore oðbÞ 6¼ oða1 Þ by (1). In particular, the edges oðaÞ oðbÞ and oðaÞ oðb1 Þ of OðPÞ are distinct. Now p induces a walk wð pÞ : oðaÞ

oða1 Þ

...

oðas Þ

in OðPÞ, and wðpÞ, in turn, determines a unique reduced walk wred ðpÞ from oðaÞ to oðbÞ. We claim that wred ðpÞ starts with the edge oðaÞ oða1 Þ. This implies that OðPÞ is not a tree, that is, P is not simply connected. To prove our claim, it suKces to show that oðaÞ 6¼ oðai Þ for all 1 6 i 6 s. If this is not the case, then at ¼ 1 m a for some 1 6 t 6 s and m 2 Z. Now m > 0 as P contains no oriented cycle, and consequently there exists a path from 1 a to 1 m a. 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 m a ¼ at ! atþ1 ! . . . ! as1 ! 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 or 1 ðaÞ is not de3ned. The following result demonstrates how to construct inductively simply connected translation quivers.

2.5. LEMMA . Let P be a connected translation quiver that is a one-point extension of a simply connected translation subquiver ( by a vertex a. Then P is not simply connected if and only if a is injective and is the start-point of at least two distinct arrows. Proof. Suppose that P is not simply connected. Then OðPÞ contains a cycle oða0 Þ oða1 Þ ... oðar1 Þ oða0 Þ. This cycle contains oðaÞ, since Oð(Þ is a tree by assumption. We may assume that oða0 Þ ¼ oðaÞ. If a is not injective, then b ¼ 1 ðaÞ 2 (, and hence oða0 Þ ¼ oðbÞ. Therefore the preceding cycle gives rise to a cycle in Oð(Þ. This contradiction shows that a is injective, and hence the only vertex in oðaÞ. Therefore, P contains arrows + : a ! b and , : a ! c with b 2 oða1 Þ and c 2 oðar1 Þ. Since OðPÞ contains no multiple edge, we have oða1 Þ 6¼ oðar1 Þ. In particular, 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 oðbÞ and oðaÞ oðcÞ are two distinct edges of OðPÞ. Being connected, Oð(Þ contains a non-trivial reduced walk oðbÞ oðc1 Þ ... oðcs1 Þ oðcÞ that can be considered as a reduced walk in OðPÞ. This gives rise to a cycle oðaÞ

oðbÞ

oðc1 Þ

...

oðcs1 Þ

oðcÞ

oðaÞ

in OðPÞ, that is, P is not simply connected. The proof of the lemma is complete. Let us now recall the de3nition of a mesh algebra. Let P be a 3nite translation quiver. A non-projective vertex P a of P determines a universal relation on P, called a mesh relation, mðaÞ ¼ 2ð+i Þ+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 mesh relation on P. Let R be a commutative ring. The ideal of the path algebra RP generated by the mesh relations is called the mesh ideal, whereas the quotient RðPÞ 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 translation quiver. If P is simply connected, then RðPÞ admits no outer derivation. Proof. Assume that P is simply connected. Then an arrow + : x ! y is the only path of P from x to y by Lemma 2.4(2). Hence every normalized derivation of RðPÞ is of degree zero. We shall use induction on the number n of vertices of P to prove the result. If n ¼ 1, then RðPÞ ffi R and the result holds trivially. Assume that n > 1 and the result holds for n 1. By Lemma 2.1, we may assume that P


367

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 w þ IP , where IP 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 P by the inductive hypothesis and is of degree zero. Therefore there exists u ¼ x2(0 x x, with x 2 R, such that ( ¼ ½u; . Since P is connected, there exist arrows starting from a. Let +i : a ! bi with 1 6 i 6 r be all such arrows. Set a ¼ b1 þ *+1 and v ¼ a a þ u. Then ð+1 Þ ¼ ½v; +1 . We claim that ð+i Þ ¼ ½v; +i , for all 1 6 i 6 r. This is trivial if r ¼ 1. Assume now that r > 1. Then a is not injective by Lemma 2.5. Therefore, c ¼ 1 ðaÞ exists and lies in (. HenceP ( admits arrows ,i : bi ! c, r for i ¼ 1; . . . ; r. Evaluating on the equality i¼1 +i , i ¼ 0 gives rise to Pr ð* þ * * * Þ+ , ¼ 0. Note that + , ; . ,i +1 ,1 i i 2 2 . . ; +r , r are linearly indei¼2 +i pendent over R by Proposition 2.3(1). Hence *+i þ *,i ¼ *+1 þ *,1 for all 2 6 i 6 r. Moreover, evaluating ( ¼ ½u; on ,i results in c ¼ bi *,i for all 1 6 i 6 r, that is bi *,i ¼ b1 *,1 , for all 1 6 i 6 r. Now one can deduce that ð+i Þ ¼ ½v; +i , for all 1 6 i 6 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 translation quiver. The following are equivalent: (1) HH1 ðRðPÞÞ ¼ 0; (2) P is simply connected; (3) HH1 ðRð(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, and hence its mesh algebra admits no outer derivation by Proposition 2.6. This proves that (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 HH1 ðRðPÞÞ ¼ 0. By Proposition 2.3(2), P contains no oriented cycle and any two parallel paths are of the same length. In particular, a normalized derivation of RðPÞ is of degree zero. By Lemma 2.1, we may assume that P is a one-point extension of a connected translation subquiver ( by a vertex a. Suppose on the contrary that P is not simply connected. First we consider the case where ( is simply connected. By Lemma 2.5, a is injective and there exist two 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 P and it appears in no mesh relation on P since a is injective. This contradicts Lemma 2.2. We now turn to the case where ( is not simply connected. By the inductive hypothesis, there exists a normalized outer derivation of Rð(Þ that is of degree zero. Thus for each arrow + 2 (, we have ð+Þ ¼ *+ + with *+ 2 R, where w denotes 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 ! bi with 1 6 i 6 s be all such arrows. If a is injective, let @ be the normalized derivation of RP such that @ coincides with on R( and @ð+i Þ ¼ 0 for all 1 6 i 6 s. Then @ preserves the mesh ideal IP of P, since IP is generated by the mesh relations on ( in this case. Suppose now that a is Pnot injective and let ,i be the arrow from bi to 1 ðaÞ for 1 6 i 6 s. Then m ¼ si¼1 +i ,i is the only mesh relation on P that is not in I( . Note that ,i 2 (, and in particular, *,i is de3ned for all 1 6 i 6 s. Let @ be the normalized derivation of RP such that @ coincides with on R( and @ð+i Þ ¼ ð1 *,i Þ+i for all 1 6 i 6 s. It is easy to see that in this case @ preserves IP as well. Now @ induces a normalized derivation @ of RðPÞ, that coincides with on Rð(Þ and is of degree zero. However, @ is inner since HH1 ðRðPÞÞ ¼ 0. Hence its restriction to Rð(Þ, that is , is easily seen to be inner as well. This contradiction shows that (1) implies (2). The proof of the theorem is now complete.

REMARK . In case P contains no oriented cycle and R is an algebraically closed 3eld, Coelho and Vargas proved in [14] the equivalence of (2) and (3) in terms of strongly simply connected algebras. They claimed as well the equivalence of (1) and (2) when in addition the base 3eld is of characteristic zero. However, the proof for this equivalence given there misquoted the incomplete argument given in the 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 algebra and that of the endomorphism algebra of a module. We shall 3rst study the relation between the 3rst Hochschild cohomology groups in the most general situation and then prove the invariance of Hochschild cohomology in case the algebras involved are projective over the base ring and the module is pseudo-tilting. As before, let A be an R-algebra and Ae its enveloping algebra. Recall that all unadorned tensor products are taken over R. We 3x the following extension of A-bimodules: A jA !A : 0 ! NA ! A A ! A ! 0; where A is the multiplication map of A and jA is the inclusion map. Each map f 2 HomAe ðNA ; 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 investigation below, whence we state them explicitly for the convenience of the reader. We refer to [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 : HomAe ðNA ; XÞ ! DerR ðA; XÞ : f 7! f :

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HOCHSCHILD COHOMOLOGY

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

X ? ? ffi? y HomAe ðA A; XÞ

ðjA ; XÞ

DerR ðA; XÞ ? ? 1 ffi? yD HomAe ðNA ; XÞ

Now let B be another R-algebra. Any B-A-bimodule X becomes a right B 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 an extension of B-A-bimodules yields respectively the following maps: fA : Ext1B A ðM; NÞ ! Ext1B ðM; NÞ; fB : Ext1B A ðM; NÞ ! Ext1A ðM; 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 maps introduced above t into the following exact sequences: gA fA (1) 0 ! HH1 ðA; HomB ðM; NÞÞ ! Ext1B A ðM; NÞ ! Ext1B ðM; NÞ, gB fB (2) 0 ! HH1 ðB; HomA ðM; NÞÞ ! Ext1B A ðM; NÞ ! Ext1A ðM; NÞ. Proof. It suKces to show the 3rst part of the proposition. To this end, we shall de3ne explicitly the map gA . First applying Lemma 3.1 to the A-bimodule HomB ðM; NÞ and then using adjunction, we get the following commutative diagram: ad

HomB ðM; NÞ ? ? ffi? y

DerR ðA; HomB ðM; NÞÞ ? ? ffi? y

ð jA ; ðM; NÞÞ HomAe ðNA ; HomB ðM; NÞÞ HomAe ðA A; HomB ðM; NÞÞ ? ? ? ? ? ffiy ffi? y ðM jA ; NÞ HomB A ðM A NA ; NÞ HomB A ðM A; NÞ

ð6Þ

Since !A splits as an extension of left A-modules, the sequence M A !A : 0

M A NA

M A jA

M A

M A A

0

M

is an extension of B-A-bimodules that splits as an extension of left B-modules. For 2 DerR ðA; HomB ðM; NÞÞ, let e 2 HomB A ðM A NA ; NÞ be the corresponding map under the isomorphisms given in the above diagram (6). Pushing out M A !A along the map e, we get an exact commutative diagram M A !A : 0

e ðM A !A Þ: 0

M ?A NA ? ? y N

M A jA

M ?A ? ? y E

M A A

M

0

M

0

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where the rows are extensions of B-A-bimodules that split as extensions of left B-modules. Now the lower row splits as an extension of B-A-bimodules if and only if e factors through M A jA which is so if and only if 2 InnR ðA; HomB ðM; NÞÞ. This shows that associated to the class ½ 2 HH1 ðA; HomB ðM; NÞÞ of a derivation , the class gA ð½Þ ¼ ½e ðM A !A Þ 2 Ext1 ðM; NÞ B A

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 p >: 0 ! N ! E ! M ! 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 be the B-A-bilinear map such that 2ðm aÞ ¼ qðmÞa. These data de3ne an exact commutative diagram of B-A-bimodules, M A !A : 0

M ? A NA ? @? y

>: 0

M A jA

N

M? A ? 2? y E

M A A

p

M

0

M

0

where @ is induced by 2. In view of the isomorphisms given in (6), there exists a derivation 2 DerR ðA; HomB ðM; NÞÞ whose image is @. Hence gA ð½Þ ¼ ½>. This completes the proof of the proposition. Now we specialize the preceding proposition to the case M ¼ N. The canonical algebra anti-homomorphism ) : A ! EndB ðMÞ is a homomorphism of A-bimodules. Thus ) induces an R-linear map HH1 ðA; )Þ : HH1 ðA; AÞ ! HH1 ðA; EndB ðMÞÞ; which in turn allows us to de3ne a map 0M : HH1 ðAÞ ! Ext1A ðM; MÞ through the following commutative diagram: HH?1 ðAÞ ? HH1 ðA; )Þ? y

0M

HH1 ðA; EndB ðMÞÞ

gA

Ext1A ðM; x MÞ ? fB ? ? Ext1B A ðM; MÞ

To give an explicit description of 0M , consider the commutative diagram DerR ðA; AÞ ? ? ffi? y HomAe ðNA ; AÞ

)

M A

DerR ðA; EndB ðMÞÞ ? ? ffi? y HomB A ðM A NA ; MÞ

where ) denotes composition with ), the isomorphism on the left comes from Lemma 3.1(2) while that on the right comes from the diagram (6) in the proof of Proposition 3.2. Let 2 DerR ðA; AÞ be a derivation and let f 2 HomAe ðNA ; AÞ be

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the corresponding map. Applying M A -- to the extension !A and pushing out M A !A along the map M A f yields a self-extension ðM A fÞ ðM A !A Þ of the right A-module M. Then 0M ð½Þ ¼ ½ðM A fÞ ðM A !A Þ 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 projective modules of the following form: let n m M M A: uj A ! vi A j¼1

i¼1

be an A-linear map with uj and vi some idempotents of A. Let 2 DerR ðAÞ be such that ðvi Auj Þ vi Auj for all i; j. If the matrix of A with respect to the given decomposition is ðaij Þmn with aij 2 vi Auj , then we call the A-linear map ðAÞ :

n M

uj A !

j¼1

m M

vi A

i¼1

given by the matrix ððaij ÞÞmn the derivative of A along .

3.3. LEMMA . Let M be a B-A-bimodule that is nitely presented as right A-module. Let n m M A M " P2 uj A vi A 0 M j¼1

i¼1

be an exact sequence of right A-modules with P2 projective and uj and vi idempotents in A. Let 2 DerR ðAÞ be such that ðvi Auj Þ vi Auj for all i; j. Then the image of ½ 2 DerR ðAÞ=InnR ðAÞ under 0M is the class of the selfextension of M that corresponds to the class ½"ðAÞ in Ext1A ðM; MÞ.

Proof. Let f 2 HomAe ðNA ; AÞ be such that ðaÞ ¼ fða 1 1 aÞ for all a 2 A; see Lemma 3.1(1). We have seen that 0M ð½Þ ¼ ½ðM A fÞ ðM A !A Þ. We claim that there exist A-linear maps + and , rendering the following diagram commutative: n M

m M A " uj A vi A 0 M= == i¼1 ? j¼1 == ? , + == ? == ðAÞ? ! == ! y m M M A M jA vi A M A NA M A M j¼1 " M A f !

M

0

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In fact,Lsince "ðvi Þ vi ¼ ð"ðvi Þ vi Þvi 2 ðM AÞvi , there exists a unique A-linear map + : m i¼1 vi A ! M A such that +ðvi Þ ¼ "ðvi Þ vi for all 1 6 i 6 m. Moreover, P assume that thePmatrix of A is ðaij Þmn with aij 2 vi Auj . Then Aðuj Þ ¼ m i¼1 vi aij m and ðAÞðuj Þ ¼ i¼1 vi ðaij Þ for all 1 6 j 6 n. Let xj ¼

m X

"ðvi Þ A ðaij 1 1 aij Þ 2 M A NA :

i¼1

Pm Note that aij ¼ aij uj and i¼1 "ðvi Þaij ¼ "ðAðuj ÞÞ ¼ 0. This implies that xj ¼ xj uj 2 ðM A NA Þuj . Therefore, there exists a unique A-linear map ,:

n M

uj A ! M A NA

j¼1

such that ,ðuj Þ ¼ xj for all 1 6 j 6 n. It is now easy to verify that the above diagram is commutative. Consider now the following exact commutative diagram: 0

M A ! A : 0

>: 0

NM ? ? D? y M A NA ? ? M A f ? y

m M

j

M A jA

vA ?i i¼1 ? +? y

M A ? ? ? y

"

M A A

M

0

M

0

M 0 L where j is the inclusion map and D is induced by +. Let Ae : nj¼1 uj A ! NM be such that A ¼ jAe. Then , ¼ D Ae, and hence "ðAÞ ¼ ðM A fÞD Ae. Note that Ae ¼ 0. This implies "ðAÞ 2 Kerð ; MÞ and exhibits > as the self-extension of M corresponding to "ðAÞ. The proof of the lemma is complete. M

E

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

3.4. LEMMA . Let M be a B-A-bimodule with M ¼ of right A-modules. Then 0M factors as follows: 0M Ext1A ðM; MÞ HH1 ðAÞ X

0M i

i

&

n M

Ln

i¼1

Mi a decomposition

Ext1A ðMi ; Mi Þ

i¼1

Proof. Let 2 DerR ðAÞ be a derivation and f 2 HomAe ðNA ; AÞ the corresponding map; see Lemma 3.1(1). For each i, the map f de3nes an exact commutative diagram: Mi A !A : 0

>i : 0

Mi A NA ? ? Mi A f y Mi

Mi A jA

Mi A ? ? 2i y Ei

Mi A A

Mi

0

Mi

0

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Summing up yields the following exact commutative diagram: M A !A : 0 M

M A NA ? ? M A f ? y

>i : 0

i

Therefore, 0M ð½Þ ¼ ½ the lemma.

M L

i @i

2

Ln

i¼1

M A A M A M

? M ?

? 2i y

i M Ei M

M A jA

0

0

i

Ext1A ðMi ; Mi Þ. This completes the proof of

In our main application, we shall consider the case where M is a right A-module and B ¼ EndA ðMÞ, whence M is endowed with the canonical B-A-bimodule structure. Moreover, there exists a morphism of B-A-bimodules ) : A ! EndB ðMÞ, where the map )ðaÞ for a 2 A is given by )ðaÞðxÞ ¼ xa, for all x 2 M. Recall that M 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 a right A-module and set B ¼ EndA ðMÞ. Assume that M is a faithfully balanced B-A-bimodule with Ext1B ðM; MÞ ¼ 0. Then there exists an exact sequence 0

HH1 ðBÞ

HH1 ðAÞ

0M

Ext1A ðM; MÞ

L of R-modules. Moreover, if M ¼ ni¼1 Mi is a decomposition of right A-modules, Ln 1 then the image of 0M lies in the diagonal part i¼1 ExtA ðMi ; Mi Þ.

Proof. By hypothesis, HH1 ðA; )Þ : HH1 ðAÞ ! HH1 ðA; EndB ðMÞÞ is an isomorphism. Furthermore, the map gA in Proposition 3.2(1) is an isomorphism since Ext1B ðM; MÞ ¼ 0. Thus D ¼ gA HH1 ðA; )Þ is an isomorphism. Set c ¼ D 1 gB . It follows from the de3nition of 0M that the diagram 0M c 0 HH1 ðBÞ Ext1A ðM; MÞ HH1 ðAÞ

?

?

? Dy ffi

g f B B 0 HH1 ðBÞ Ext1B A ðM; MÞ Ext1A ðM; MÞ is commutative. By Proposition 3.2(2), the lower row is exact, and hence the upper row is an exact sequence as desired. The last part of the theorem follows from Lemma 3.4. This completes the proof. For a right A-module, denote by addðMÞ the full subcategory of the category of right A-modules generated by the direct summands of direct sums of 3nitely many copies of M. We call M an A-generator if A lies in addðMÞ. We shall now study the behaviour of Hochschild cohomology under (r-)pseudo-tilting, a notion we de3ne as follows.

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

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ExtiA ðM; MÞ

(1) ¼ 0 for 1 6 i 6 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 ExtiA ðM; MÞ ¼ 0 for all i > 1 and (2) holds for some r > 0; in other words, M is r-pseudo-tilting for all suKciently large 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, a pseudo-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 M be an r-pseudo-tilting right A-module and set B ¼ EndA ðMÞ. Then M is a faithfully balanced B-A-bimodule and ExtiB ðM; MÞ ¼ 0 for all i > 1. As an immediate consequence of Lemma 3.7 and Theorem 3.5, we get the following result.

3.8. PROPOSITION . Let A be an algebra over a commutative ring R. Let M be an r-pseudo-tilting right A-module and set B ¼ EndA ðMÞ. (1) There exists an exact sequence of R-modules 0M 0 ! HH1 ðBÞ ! HH1 ðAÞ ! Ext1A ðM; MÞ: (2) If r > 1, then HH1 ðAÞ ffi HH1 ðBÞ. So far we have not put any restriction on the R-algebra structure. As usual in Hochschild theory, much sharper results can be obtained if the algebras involved are projective as R-modules, for example, when R is a 3eld. First we recall the following result from [13, p. 346].

3.9. PROPOSITION (Cartan--Eilenberg). Let A and B be algebras over a commutative ring R, and let M and N be B-A-bimodules. If A and B are projective as R-modules, then there exist spectral sequences HHi ðA; ExtjB ðM; NÞÞ ¼) Extiþj B A ðM; NÞ and HHi ðB; ExtjA ðM; NÞÞ ¼) Extiþj B A ðM; NÞ:

REMARK . The 3rst spectral sequence yields the following exact sequence of lower terms that extends the exact sequence from Proposition 3.2(1): g1A fA1 0 ! HH1 ðA; HomB ðM; NÞÞ ! Ext1B A ðM; NÞ ! HH0 ðA; Ext1B ðM; NÞÞ gA2 ! HH2 ðA; HomB ðM; NÞÞ ! Ext2B A ðM; NÞ:

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

3.10. THEOREM . Let R be a commutative ring, and let A be an R-algebra that is projective as R-module. Let M be a right A-module such that B ¼ EndA ðMÞ is projective as R-module as well. (1) If M is r-pseudo-tilting, then HHi ðBÞ ffi HHi ðAÞ for 0 6 i 6 r. (2) If M is pseudo-tilting, then HHi ðBÞ ffi HHi ðAÞ for all i > 0. Proof. First we note that the edge homomorphisms of the 3rst spectral sequence in Proposition 3.9 become gAi : HHi ðAÞ ! ExtiB A ðM; MÞ and fAi : ExtiB A ðM; MÞ ! HH0 ðA; ExtiB ðM; MÞÞ;

for all i > 0;

whereas those of the second one become gBi : HHi ðBÞ ! ExtiB A ðM; MÞ and fBi : ExtiB A ðM; MÞ ! HH0 ðA; ExtiA ðM; MÞÞ;

for all i > 0:

Assume now that M is r-pseudo-tilting. By Lemma 3.7, ExtiB ðM; MÞ ¼ 0 for all i > 1. So each gBi with i > 0 is an isomorphism. Moreover, since ExtiA ðM; MÞ ¼ 0 for 1 6 i 6 r, the gAi are isomorphisms for 0 6 i 6 r. These give rise to the desired isomorphisms c i ¼ ðgAi Þ1 gBi : HHi ðBÞ ! HHi ðAÞ for i ¼ 0; 1; . . . ; r: If M is pseudo-tilting, then the map c i is de3ned and an isomorphism for each i > 0. This completes the proof of the theorem.

REMARK . In case A is a 3nite-dimensional algebra over an algebraically closed 3eld and M is a 3nite-dimensional tilting A-module, Happel showed in [17, (4.2)] that HHi ðAÞ ffi HHi ðBÞ for all i > 0. 4. Representation-nite algebras without outer derivations The objective of this section is to investigate when an algebra of 3nite representation type admits no outer derivation. To begin with, let A be an artin algebra. Denote by mod A the category of 3nitely generated right A-modules and by ind A its full subcategory generated by a chosen complete set of representatives of isoclasses of the indecomposable modules. Let PA denote the Auslander --Reiten quiver of A, which is a translation quiver with respect to the Auslander --Reiten translation DTr. We refer to [4] for general results on almost split sequences and irreducible maps as they pertain to the structure of Auslander --Reiten quivers. Assume that A is of 3nite representation type, L that is, ind A contains only 3nitely many objects, say M1 ; . . . ; Mn . Then M ¼ ni¼1 Mi is an A-generator, called a minimal representation generator for A. One calls G ¼ EndA ðMÞ the Auslander algebra of A. Applying Lemma 3.7 and Theorem 3.5, one obtains immediately the following exact sequence: n 0 M Ext1A ðMi ; Mi Þ: 0 ! HH1 ðGÞ ! HH1 ðAÞ ! i¼1 1

Thus, HH ðGÞ is always embedded into HH1 ðAÞ. To investigate when this

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

4.1. PROPOSITION . Let A be an artin algebra such that ind A contains only bricks, and let G be the Auslander algebra of A. Then HH1 ðGÞ ffi HH1 ðAÞ. Proof. In view of the exact sequence stated above, we need only show that Ext1A ðN; NÞ ¼ 0 for every module N in ind A. Assume on the contrary that this fails for some module M in ind A. Then M is non-projective and HomA ðM; DTr MÞ 6¼ 0; see, for example, [4, p. 131]. Let 0 ! DTr M ! E ! M ! 0 be an almost split sequence in mod A. Using the fact that an irreducible map is either a monomorphism or an epimorphism, one 3nds easily that either M or an indecomposable direct summand of E is not a brick. This contradiction completes the proof of the proposition. From now on, let k be an algebraically closed 3eld and A a 3nite-dimensional k-algebra. It is well known (see [9, (2.1)]) that there exist a unique 3nite quiver QA (called the ordinary quiver of A) and an admissible ideal IA in kQA such that the basic algebra B of A is isomorphic to kQA =IA . Such an isomorphism B ffi kQA =IA is called a presentation of B. Moreover, one says that A is connected or triangular if QA is connected or contains no oriented cycle, respectively. In case A is of 3nite representation type, one says that A is standard [9, (5.1)] if its Auslander algebra is isomorphic to the mesh algebra kðPA Þ 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 Auslander algebra. If HH1 ðAÞ ¼ 0 or HH1 ðGÞ ¼ 0, then A is standard. Proof. It is well known that HH1 ðAÞ is invariant under Morita equivalence; see also Proposition 3.8. We may hence assume that A is basic. Suppose that A is not standard. It follows from [6, (9.6)] that there exists a presentation A ffi kQA =IA such that the bound quiver ðQA ; IA Þ contains a Riedtmann contour C as a full bound subquiver. This means that the ordinary quiver QC of C consists of a vertex a, a loop ) at a, and a simple oriented cycle +n +1 a ¼ a0 ! a1 ! . . . !an1 ! 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 not constant 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.


377

(1) If p is a path of QA that does not lie completely in QC but contains ) as a subpath, then p 2 IA . Next, write x ¼ x þ IA 2 A for x 2 kQA . For simplicity, we identify the vertex 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 )m1 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 )m2 p; p)m2 2 IA . As a consequence, ) m1 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 ,)m2 2 IA . If , ¼ ), then p1 is non-trivial and we can write p1 ¼ p2 D, with D an arrow. Then D,)m2 ¼ D)m1 2 IA , and therefore p)m2 2 IA . Similarly, )m2 p 2 IA , and we have established statement (2). Now let @ be the unique normalized derivation of kQA with @ð)Þ ¼ )m1 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 is non-trivial and that (3) holds for paths shorter than p. If ) does not appear in p, then @ð pÞ ¼ 0. Otherwise p ¼ u)v, where u and v are paths shorter than p with u or v non-trivial. Now @ð pÞ ¼ @ðuÞ)v þ u)m1 v þ u)@ðvÞ. It follows from (2) that u)m1 v 2 IA . Moreover, if u 6¼ ), then @ðuÞ 2 IA by the inductive hypothesis. Otherwise, @ðuÞ)v ¼ )m v 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, @ðIA Þ IA by (3). Hence @ induces a normalized derivation of A with ðyÞ ¼ @ðyÞ for all y 2 kQA . We now wish to show that is an outer derivation of A. Assume on the contrary that ¼ ½x; for some x 2 A. Write x ¼ x1 þ x2 , where x1 2 Aa þ aA and x2 2 a0 Aa0 with a0 ¼ 1 a. Then ð)Þ ¼ ½x1 ; ) ¼ ) m1 6¼ 0: Thus x1 62 aAa since ) lies in the center of aAa. Therefore x1 ¼ z1 þ z2 þ z, where z1 2 a0 Aa, z2 2 aAa0 , z 2 aAa and z1 þ z2 6¼ 0. However, this would imply that ðaÞ ¼ z1 z2 6¼ 0, contrary to being normalized. Hence [] is a non-zero element of HH1 ðAÞ. Let M be a minimal representation generator of A such that G ¼ EndA ðMÞ. By Proposition 3.8, there exists an exact sequence 0M 0 ! HH1 ðGÞ ! HH1 ðAÞ Ext1A ðM; MÞ: We want to show that ½ 2 Kerð0M Þ. Let A " P1 ! P0 ! M ! 0 be a minimal projective presentation of M. Let L Lb1 ; . . . ; bn and c1 ; . . . ; cm be vertices of QA such that P1 ¼ nj¼1 bj A and P0 ¼ m i¼1 c i A. The matrix of A is then ðxij Þmn with xij 2 c i radðAÞ bj . Since is normalized, ðc i Abj Þ c i Abj . Hence the

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matrix of ðAÞ : P1 ! P0 is ððxij ÞÞmn . For each 3xed 1 6 i 6 m, we consider the ith row ðxi1 ; xi2 ; . . . ; xin Þ of the matrix of A. If c1 6¼ a, then ðxij Þ ¼ 0 for all 1 6 j 6 n by (3). Thus, ððxi1 Þ; ðxi2 Þ; . . . ; ðxin ÞÞ ¼ 0 ðxi1 ; xi2 ; . . . ; xin Þ: by (3) and ) m2 xij ¼ 0 by (2), Assume that c i ¼ a. If bj 6¼ a, then ðxij Þ ¼ 0 P s since xij 2 c i rad ðAÞbj . If bj ¼ a, then xij ¼ m1 with *s 2 k since s¼1 *s ) m m2 aAa ¼ k½)=ð) Þ. Now ð)Þ ¼ ) ), whereas for s > 2, both ð) s Þ ¼ 0 and ) m2 ) s ¼ 0. This shows that ððxi1 Þ; ðxi2 Þ; . . . ; ðxin ÞÞ ¼ ) m2 ðxi1 ; xi2 ; . . . ; xin Þ: For 1 6 i; j 6 m, let yij ¼ 0 if i 6¼ j, yii ¼ 0 if c i 6¼ a, and yii ¼ ) m2 if c i ¼ a. Then ððxij ÞÞ ¼ ðyij Þðxij Þ. Let @ : P0 ! P0 be the A-linear map given by the matrix ðyij Þ. Then ðAÞ ¼ @A, and hence "ðAÞ ¼ ð"@ÞA. This shows that the class ½"ðAÞ 2 Ext1A ðM; MÞ is zero. Now Lemma 3.3 implies that the class ½ 2 HH1 ðAÞ lies in the kernel of 0M . Therefore, HH1 ðGÞ 6¼ 0. This completes the proof of the lemma. If A is of 3nite representation type, we say that A is simply connected if PA is a simply connected translation quiver. In general, one says that A is strongly simply connected if A is connected and triangular, and its basic algebra B admits a presentation B ffi kQA =IA such that the 3rst Hochschild cohomology group vanishes for every algebra de3ned by a convex bound subquiver of ðQA ; IA Þ. It follows 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 simply connected if and only if it is strongly simply connected. We are now ready to establish the main result of this section.

4.3. THEOREM . Let A be a connected nite-dimensional algebra over an algebraically closed eld k. Assume that A is of nite representation type and let G be its Auslander algebra. The following statements are equivalent: (1) HH1 ðAÞ ¼ 0; (2) HH1 ðGÞ 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 theorem implies 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 turn implies that the ordinary quiver of A contains no oriented cycle, as A is of 3nite representation type. Therefore, A is standard by [6, (9.6)]. Thus we may assume that A is standard, that is, G ffi kðPA ). Now the equivalence of (2), (3), and (4) follows immediately from Theorem 2.7. Moreover, (1) implies (2) since HH1 ðGÞ embeds into HH1 ðAÞ. Finally, if (2) holds, then (3) holds as well. In particular, ind A contains only bricks. By Proposition 4.1, HH1 ðAÞ ¼ HH1 ðGÞ ¼ 0. This completes the proof of the theorem. We conclude this paper with a consequence of Theorem 4.3. Recall that A is an Auslander algebra if its global dimension is less than or equal to 2 while its


379

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

4.4. COROLLARY . Let A be of nite representation type or an Auslander algebra. If HH1 ðAÞ ¼ 0, then HHi ðAÞ ¼ 0 for all i > 1. In particular, A is rigid in this 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 HH2 ðAÞ ¼ 0 implies rigidity, as originally proved by Gerstenhaber [15]. Acknowledgements. Both authors gratefully acknowledge partial support from the Natural Sciences and Engineering Research Council of Canada. The authors were also supported in part by respectively an Ontario --QuIebec Exchange Grant of the Ministry of Education and Training of Ontario and a grant of CoopIeration QuIebec-Provinces canadiennes en Enseignement supIerieur et Recherche of the Ministry of Education of QuIebec.

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Ragnar-Olaf Buchweitz Department of Mathematics University of Toronto Toronto Ontario Canada M5S 3G3 [email protected]

Shiping Liu De partement de mathe matiques Universite de Sherbrooke Sherbrooke Quebec Canada J1K 2R1 [email protected]