A Covering Technique for Derived Equivalence

JOURNAL OF ALGEBRA ARTICLE NO

191, 382]415 Ž1997.

JA976906

A Covering Technique for Derived Equivalence Hideto Asashiba Department of Mathematics, Osaka City Uni¨ ersity, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558, Japan Communicated by Michel Broue´ Received March 4, 1996

Through this paper k denotes an algebraically closed field. The word algebra always means a basic finite-dimensional k-algebra with identity, and all categories and functors are assumed to be k-linear. Consider a Brauer tree with e edges whose exceptional vertex has multiplicity m. Then Rickard proved that the algebra defined by this Brauer tree is derived equivalent to the symmetric Nakayama algebra with e isoclasses of simples and of Loewy length em q 1 w19, Theorem 4.2x. In particular this solves the following problem posed by Broue ´ w3x affirmatively in the cyclic defect case: For a finite group G, is a block of kG with abelian defect derived equivalent to its Brauer correspondent? In this paper we develop a ‘‘covering technique’’ for derived equivalence to give a new approach to this problem Žsee Example 6.2. which extends Rickard’s result above Žsee w2, 5, 6, 8, 21, 22, 24x, etc. for the usual covering techniques in representation theory of algebras.. In fact, as an application, we classify representation-finite self-injective algebras of type A n up to derived equivalence. In the sequel, we consider covering functors to ‘‘algebras,’’ and for an algebra A we deal with its repetition Žsrepetitive ‘‘algebra’’. AZ , which is no longer an algebra with identity but can be regarded as a category. Accordingly, we regard all algebras as categories and work over categories. Our investigation was motivated by looking at quiver presentations of the principal block L of kSLŽ2, 4. and its Brauer correspondent G when k has characteristic 2. As we will see in Sect. 6 we can find algebras A and B 382 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

DERIVED EQUIVALENCE

383

such that there are Galois covering functors AZ ª L, B Z ª G. Further it is not hard to see that A is derived equivalent to B. We want to induce a derived equivalence between L and G from one between A and B. This observation leads us to the following problems, whose answers provide desired tools: Ž1. When does a derived equivalence between algebras A and B yield a derived equivalence between AZ and B Z ? Ž2. When does a derived equivalence between categories R and S yield a derived equivalence between quotient categories RrG and SrH by groups G and H of automorphisms of R and S? In Sect. 1 we answer problem Ž1. to show that if two algebras are derived equivalent, then their repetitions are derived equivalent. Next, in Sect. 2 we show that a Galois covering functor R ª RrG induces a Galois covering functor from a full subcategory of bounded homotopy category of finitely generated projective R-modules consisting of a complete list of indecomposable objects to an essential part of the corresponding full subcategory for RrG with the same group G. We expect that this enables us to calculate the derived category for RrG using that for R. In Sect. 3 we show that a ‘‘push-down’’ of a ‘‘tilting spectroid’’ is also a tilting spectroid using the theorem in Sect. 2, which gives an answer to problem Ž2.. In Sect. 4 we give the first application of our technique, which gives an infinite sequence of derived equivalent pairs of algebras from one such pair. Next, in Sect. 5 we classify self-injective algebras of type A n up to derived equivalence. This classification shows us that stable equivalence and derived equivalence coincide for self-injective algebras of type A n . Notice that the algebras having the form AZrG for some algebra A and for some group G of automorphisms of AZ , to which our technique may be applied, are not very special Žsee, for instance, Pogorzały w17, Theoremx or Hughes and Waschbusch ¨ w11x.. In the last section we give an example to illustrate the proof in the previous section, and we explicitly present our solution of Broue’s ´ problem above for the group SLŽ2, 4..

PRELIMINARIES As stated in the introduction we regard every algebra as a category, more precisely as a finite spectroid. Recall from Gabriel and Roiter w7x that a category A is called s¨ elte Žsskeletally small. if the isoclasses of objects of A form a set; and a svelte category A is called a spectroid Žsa

384

HIDETO ASASHIBA

locally finite-dimensional category in w5x. if the following three conditions are satisfied: Ži.

A is basic, i.e., distinct objects of A are not isomorphic;

Žii. A is semiperfect, i.e., every object of A has a local endomorphism ring; Žiii. A is pointwise finite, i.e., the space AŽ x, y . is finite-dimensional for every x, y g A. For a spectroid A, A is called finite if A has only a finite number of objects; and A is called locally bounded if, for every x g A, there are only finitely many y g A such that AŽ x, y . / 0 or AŽ y, x . / 0. For a spectroid A, we denote by Mod A the category of all Žright. A-modules; by mod A the full subcategory of Mod A consisting of finitely presented objects; by Pro A the full subcategory of Mod A consisting of projective objects; and by pro A the full subcategory of Pro A consisting of finitely generated projective objects. In addition, mod A denotes the Žprojectively. stable category of mod A, and we set D [ Hom k Žy, k .. For an additive category A, we denote by C Ž A . and by C b Ž A . the category of differential complexes and the category of bounded differential complexes in A, respectively; and by H Ž A . and H b Ž A . the corresponding homotopy categories. When A is an abelian category, we denote by D Ž A . and by D b Ž A . the corresponding derived categories. For a triangulated category T , and a class of objects E in T , triŽ E . denotes the smallest full triangulated subcategory of T closed under direct summands and isomorphisms that contains E. Further, if T has infinite direct sums, by TriŽ E . we denote the smallest full triangulated subcategory of T closed under isomorphisms and infinite direct sums that contains E. For a category A, XA denotes the full subcategory of pro A consisting of representable functors AŽy, x . for some x g A. We regard XA : H b Žpro A. by considering each AŽy, x . as a complex concentrated in degree 0. As in the notation AŽy, x ., by Žy. or Ž?. we denote a place for a variable. When we want to stress the difference of variables, we use different symbols like AŽy, ?.. We refer the reader to definitions of S-functors and S-equivalences in Keller and Vossieck w13x.

1. DERIVED EQUIVALENCES OF REPETITIONS We start with a categorical version of Rickard’s theorem w18, Theorem 6.4x.

385

DERIVED EQUIVALENCE

PROPOSITION 1.1. equi¨ alent:

Let A and B be categories. Then the following are

Ž1. There is an S-equi¨ alence D ŽMod B . ª D ŽMod A.. Ž2. There is a full subcategory E of D ŽMod A. such that the following are satisfied: for all T g E, D ŽMod A.ŽT, y. commutes with Ž infinite. direct sums; Žb. for all T, U g E and for all n / 0, D ŽMod A.ŽT, U w n x. s 0; Žc. TriŽ E . s D ŽMod A.; Žd. E is equi¨ alent to B. Ž3. There is a full subcategory E of H b Žpro A. such that the following are satisfied: Ža. for all T, U g E and for all n / 0, H b Žpro A.ŽT, U w n x. s 0; Žb. triŽ E . contains XA ; Žc. E is equi¨ alent to B. Ža.

Proof. Consider A and B as DG-categories concentrated in degree 0. Then the equivalence of Ž1. and Ž2. follows from Keller w12, Corollary 9.2x. Note that, for each T g D ŽMod A., D ŽMod A.ŽT, y. commutes with Žinfinite . direct sums iff T is isomorphic to an object of H b Žpro A. Žsee Rickard w18, 6.3x.. Note also that Ž3.Žb. follows from Ž2.Žc. by Keller w12, 5.3x. Then the equivalence of Ž2. and Ž3. follows. DEFINITION 1.2. We say that categories A and B are deri¨ ed equi¨ alent if one of the equivalent conditions above holds. In this case the triple Ž A, E, B . in Ž3. is called a tilting triple. When both A and B are spectroids, E in Ž3. is called a tilting spectroid for A. LEMMA 1.3. Let A be a spectroid. Then, for each T g H ŽMod A. and for each U g H b Žpro A., we ha¨ e H Ž Mod A . Ž T , U mA DA Ž y, ?. . ( D Ž H Ž Mod A . Ž U, T . . . Proof. For each X s Ž X i, d Xi ., Y s Ž Y i, d Yi . g H ŽMod A., we denote by H o mAŽ X, Y . the Hom-complex, i.e., p

H o mA Ž X , Y . [ Ž H o mA Ž X , Y . , d p . , where H o mAŽ X, Y . p [ Łyiqjsp Hom AŽ X i, Y j . and p

d p : H o mA Ž X , Y . ª H o mA Ž X , Y .

pq1

386

HIDETO ASASHIBA

is defined by

Ž f i:

p

X i ª Y j . ¬ d Yj ( f i y Ž y1 . f iq1 ( d Xi .

Then H 0 H o mAŽ X, Y . s H ŽMod A.Ž X, Y .. Further, for each X as above ˜A Z the and for each Z s Ž Z i, d Zi . g H ŽMod Aop ., we denote by X m p p ˜ ˜ m-complex, i.e., X mA Z [ ŽŽ X mA Z . , d ., where

˜A Z . ŽXm

p

[ [iqjsp X i mA Y j

and

p

˜A Z . ª Ž X m ˜A Z . dp: Ž X m

pq1

is defined by x i m y j ¬ d Xi Ž x i . m y j q Žy1. i x i m d Yj Ž y j .. For each X, Z as above, we set X * [ H o mAŽ X, AŽy, ?.., Z* [ H o mA op Ž Z, Aop Žy, ?... We use the following well-known facts, which are easily verified. Ža. For each X, Y, and Z as above, if X and Z are concentrated in degree zero, then H o mAŽ X, Y . ( Hom AŽ X 0 , Y ., H o mAŽ Y, X . ( ˜A Z ( Y mA Z 0. Hom AŽ Y, X 0 ., and Y m Žb. For each X as above, if X g H b Žpro A., then Ž X *.* ( X. ˜A Y * ( Žc. For each X, Y as above, if Y g H b Žpro A., then X m H o mAŽ Y, X .. Žd. For each X as above, we have DX ( H o mk Ž X, k . ( H o mAŽ X, DŽ AŽy, ?.... Že. For each X, Z as above, we have H o mAŽ X, H o mk Ž Z, k .. ( ˜A Z, k .. H o mk Ž X m Now let T, U be as in the assertion. Then

˜A op Ž U* . * U mA DA Ž y, ?. s DAop Ž ?, y . mA op U( DAop Ž ?, y . m ( H o mA op Ž U*, DAop Ž ?, y . . ( H o mk Ž U*, k . . Hence H o mA Ž T , U mA DA Ž y, ?. . ( H o mA Ž T , H o mk Ž U*, k . .

˜A U*, k . ( H o mk Ž T m ˜A U* . ( D ŽT m H o mA Ž U, T . . ( DH H o mAŽU, T .. Then, since D is exact, Thus H o mAŽT, U mA DAŽy, ?.. ( DH we get the assertion by taking the 0th cohomologies: H Ž Mod A . Ž T , U mA DA Ž y, ?. . ( H 0 D H o mA Ž U, T . ( DH 0 H o mA Ž U, T . H Ž Mod A . Ž U, T . . ( DH

387

DERIVED EQUIVALENCE

We recall the definition of the repetition of a spectroid Žcf. w7x.. Note that we slightly changed the definition. DEFINITION 1.4. Let A be a spectroid. Ž1. We define a spectroid, the repetition AZ of A as follows. Objects are the pairs x w nx [ Ž x, n. with x g A and n g Z. Z

A

Žx

w nx

,y

w mx

¡ f ~ [ . ¢0, w

w nx

[ Ž f , n . N f g A Ž x, y . 4 ,

if m s n,

w nx

[ Ž w , n . N w g DA Ž y, x . 4 ,

if m s n q 1, otherwise.

The composition AZ Ž y w mx , z w l x . = AZ Ž x w nx , y w m x . ª AZ Ž x w nx , z w l x . is given as follows: Ži. If m s n, l s m, then this is the composition of A: A Ž y, z . = A Ž x, y . ª A Ž x, z . . Žii. If m s n, l s m q 1, then this is given by the right Amodule structure of DAŽy, ?.: DA Ž z, y . = A Ž x, y . ª DA Ž z, x . . Žiii. If m s n q 1, l s m, then this is given by the left A-module structure of DAŽy, ?.: A Ž y, z . = DA Ž y, x . ª DA Ž z, x . . Živ. Otherwise the composition is zero. Ž2. For every n g Z, we denote by Aw nx the full subcategory of AZ ; formed by x w nx with x g A, and by 1w nx: A ª Aw nx ¨ AZ , x ¬ x w nx, the embedding functor. Ž3. The Nakayama automorphism n s nA of AZ is defined by w nx x ¬ x w nq1x, f w nx ¬ f w nq1x, w w nx ¬ w w nq1x for all x g A and for all f g AŽ x, y ., w g DAŽ y, x . with x, y g A. Note that if a spectroid A is locally bounded, then so is AZ. THEOREM 1.5. Let A and B be spectroids. If A and B are deri¨ ed equi¨ alent, then their repetitions are deri¨ ed equi¨ alent. Proof. Let Ž A, E, B . be a tilting triple. For every n g Z, AZ Žy, 1w nx?. is an A-AZ -bimodule. For every x g A, we have AŽy, x . mA AZ Žy, 1w nx?. ( AZ Žy, x w nx .. Hence, for every P g H b Žpro A., we see P w nx [ P mA

388

HIDETO ASASHIBA

AZ Žy, 1w nx?. g H b Žpro AZ .. Since E is a subcategory of H b Žpro A., we can define a full subcategory Eˆ of H b Žpro AZ . consisting of T w nx for T g E ˆ B Z . is a tilting triple. and n g Z. We have only to show that Ž AZ , E, Ža. For all T w ix, U w j x g Eˆ and for all n / 0, H b Ž pro AZ . Ž T w ix , U w j x w n x . s 0. In fact, the left-hand side is equal to H b Ž pro AZ . Ž T mA AZ Ž y, 1w ix?. , Ž U mA AZ Ž yy, 1w j x??. . w n x . ( H b Ž pro AZ . Ž T mA AZ Ž y, 1w ix?. , U w n x mA AZ Ž yy, 1w j x??. . ( H Ž Mod AZ . Ž T mA AZ Ž y, 1w ix?. , U w n x mA AZ Ž yy, 1w j x??. . ( H Ž Mod A . Ž T , Hom AZ Ž AZ Ž y, 1w ix?. , U w n x mA AZ Ž yy, 1w j x??. . . ( H Ž Mod A . Ž T , U w n x mA AZ Ž 1w ix?, 1w j x??. .

¡H Ž Mod A. Ž T , U w n x m

(

~

A

A Ž ?, ??. .

( H b Ž pro A . Ž T , U w n x . s 0,

¢

H Ž Mod A . Ž T , U w n x mA DA Ž ??, ?. . , 0,

if j s i , if j s i q 1, otherwise.

Now in the case j s i q 1, we have by Lemma 1.3, H ŽMod A.ŽT, U w n x mA H ŽMod A.ŽU w n x, T . ( DH H b Žpro A.ŽU, T wyn x. s 0. DAŽ??, ?.. ( DH Žb. triŽ Eˆ. = XAZ . In fact, let E be the full subcategory of H b Žpro A. formed by the objects P such that P mA AZ Žy, 1w ix?. g triŽ Eˆ. for all i g Z. Then E s triŽ E . because the functors H b Žpro A. ª H b Žpro AZ ., P ¬ P mA AZ Žy, 1w ix?. are S-functors. Since, for all i g Z and for all T g E, T mA AZ Žy, 1w ix?. s T w ix g Eˆ : triŽ Eˆ., we have E : E . Hence E = triŽ E . = XA . Therefore, for all x g A and for all i g Z, we have tri Ž Eˆ. 2 A Ž y, x . m AZ Ž y, 1w i x?. ( AZ Ž y, x w i x . . Thus triŽ Eˆ. = XAZ . Žc. Eˆ is equivalent to B Z . Let c : E ª B be an equivalence. For T, U g E, let c UT : EŽT, U . ª B Ž c T, c U . be the isomorphism given by c . We construct an equivalence ˆ we define c Z ŽT w nx . [ Ž c T .w nx. Then this gives c Z : Eˆ ª B Z . For T w nx g E,

389

DERIVED EQUIVALENCE

a bijection on objects. Define the value of c Z on morphisms as follows: EˆŽ T w i x , U w j x . ª ˜ H Ž Mod AZ . Ž T mA AZ Ž y, 1w i x?. , U mA AZ Ž yy, 1w j x??. . ª ˜ H Ž Mod A . Ž T , U mA AZ Ž 1w ix?, 1w j x??. .

¡H Ž Mod A. Ž T , U . s E Ž T , U . BŽ c T , c U . ,

6

c UT

ª ˜ ~ DHH Ž Mod A . Ž U, T . s DE Ž U, T .

¢0,

wix

DB Ž c U, c T . ,

6

D ŽŽ c TU .y1 .

ª ˜ BZ Ž Ž c T . , Ž c U .

w jx

if j s i , if j s i q 1, otherwise

. s B Z Ž c Z T w i x , c ZU w j x .

ˆ If we could show that the defined correspondence c Z is for T w i x, U w j x g E. a k-functor, then by construction c Z turns out to be an equivalence. It is obvious that c Z preserves identities and that c Z is k-linear. Hence it only remains to show that c Z commutes with compositions. Let ˆ We have to show the commutativity of the following T w ix, U w j x, V w l x g E. diagram: EˆŽ T w i x , V w l x .

6

EˆŽ U w j x , V w l x . = EˆŽ T w i x , U w j x . Žc Z , c Z .

6

. = BZ Ž Ž c T .w ix , Ž c U .w j x.

6

wlx

6

w x

BZ Ž Ž c U . j , Ž c V .

Ž).

cZ w x

BZ Ž Ž c T . i , Ž c V .

wlx

..

If the upper left corner or the lower right corner is zero, then Ž). is obviously commutative. Hence we may assume that j g  i, i q 14 , l g  j, j q 14 , and l g  i, i q 14 . Ži. Case j s i, l s j. In this case i s j s l and Ž). is isomorphic to the following diagram, whose commutativity follows from the fact that c : E ª B is a functor: 6

Ž c VU , c UT .

c VT

6

6

B Ž c U, c V . = B Ž c T , c U .

EŽ T , V . 6

E Ž U, V . = E Ž T , U .

BŽ c T , c V . .

390

HIDETO ASASHIBA

Žii. Case j s i, l s j q 1. In this case l s i q 1 and Ž). is isomorphic to the following diagram, whose commutatively also follows from c ’s being a functor: 6

Ž D ŽŽ c UV .y1 . , c UT .

D ŽŽ c TV .y1 .

6

6

DB Ž c V , c U . = B Ž c T , c U .

DE Ž V , T . 6

DE Ž V , U . = E Ž T , U .

DB Ž c V , c T . .

Žiii. Case j s i q 1, l s j. In this case l s i q 1 and Ž). is isomorphic to the following commutative diagram: 6

Ž c VU , D ŽŽ c TU .y1 ..

D ŽŽ c TV .y1 .

6

6

B Ž c U, c V . = DB Ž c U, c T .

DE Ž V , T . 6

E Ž U, V . = DE Ž U, T .

DB Ž c V , c T . .

Živ. Case j s i q 1, l s j q 1. In this case l s i q 2 f  i, i q 14 . Hence Ž). is commutative. In any case the diagram Ž). is commutative. Thus c Z : Eˆ ª B Z is a functor that is fully faithful and bijective on objects. As a consequence, ˆ B Z . is a tilting triple. Ž AZ , E,

2. COVERINGS OF SPECTROIDS OF HOMOTOPY CATEGORIES Throughout this section R is a locally bounded spectroid, G is a group of automorphisms of R, and F: R ª RrG is the canonical Galois covering functor. Note in this case that an R-module is finitely presented iff it is finitely generated iff it is finite dimensional. Recall that a category is called an aggregate Žsa Krull]Schmidt category. in case it is additive, svelte, pointwise finite, and each object is a finite direct sum of indecomposables with local endomorphism algebras w7, 3.5x. Note in our case that both H b Žpro R . and H b Žpro RrG. are aggregates because both C b Žpro R . and C b Žpro RrG. are. Recall also that, for an aggregate A, a spectroid of A is a full subcategory of A formed by a complete set of representatives of isoclasses of indecomposable objects of A. In this section we will show that under a suitable condition on G, the covering functor F induces a Galois covering with the group G from a spectroid of H b Žpro R . to a full subcategory of a spectroid of H b Žpro RrG.. The following example of a Galois covering will be used later.

391

DERIVED EQUIVALENCE

EXAMPLE 2.1. Let A be a locally bounded spectroid and n s nA the Nakayama automorphism of AZ . Then AZ r² n : is isomorphic to the trivial extension T Ž A. [ A h DA. Thus we have a Galois covering AZ ª T Ž A. with group ² n :. Recall that the group G acts on Mod R: For each g g G, and for each M g Mod R, M

R op

Mod k.

6

gy1

M [ M ( gy1 : R op

6

g

Since for every g g G, g Žy.: pro R ª pro R is an equivalence, it induces an S-equivalence H b Žpro R . ª H b Žpro R ., which is denoted also by g Žy.. LEMMA 2.2. The action of G on H b Žpro R . is locally bounded, i.e., for e¨ ery pair Ž X, Y . of objects of H b Žpro R ., there are only a finite number of y1 g g G such that H b Žpro R .Ž X, g Y . ( H b Žpro R .Žg X, Y . / 0. Proof. This follows easily from the fact that the G-action on R is locally bounded. See also Remark 2.4Ž2.. Let F : Mod RrG ª Mod R, F Ž M . [ M ( F op be the pull-up functor, and let F : Mod R ª Mod RrG, Ž F Ž M ..Ž a. [ [x g a M Ž x . the push-down functor. Then F is the left adjoint of F , and both F and F are exact functors. Hence F , F induce S-functors, which we denote also by F , F , respectively: v

v

v

v

v

v

v

v

v

v

v

v

6

Fv

6

H Ž Mod R .

H Ž Mod RrG . .

Fv

Then again F is the left adjoint of F . Since F RŽy, x . ( RrGŽy, Fx . for all x g R, F induces a functor v

v

v

v

F : H b Ž pro R . ª H b Ž pro RrG . . v

LEMMA 2.3. Ž1. Ž2. Ž3.

Let X g H b Žpro R .. Then the following hold:

for all g g G, F g X ( F X in H b Žpro RrG., canonically; F F X ( [g g G g X in H ŽMod R ., canonically; for all Y g H b Žpro R ., F induces an isomorphism v

v

v

v

v

[H

b

Ž pro R . Ž X , g Y . ª ˜ H b Ž pro RrG . Ž F X , F Y . . v

v

ggG

Proof. Ž1. and Ž2.. Both isomorphisms are given by Gabriel w5, Lemma 3.2x in the category of differential complexes, and hence in the homotopy category.

392

HIDETO ASASHIBA

Ž3. Since H b ŽPro R .Ž X, y. commutes with direct sums, we have H b Ž pro RrG . Ž F X , F Y . ( H Ž Mod RrG . Ž F X , F Y . v

v

v

v

( H Ž Mod R . Ž X , F F Y . v

v

( H b Ž Pro R . X ,

ž

(

[H

b

[ ggG

g

Y

/

Ž pro R . Ž X , g Y . .

ggG

Remark 2.4. Ž1. In contrast with Ž1. above, we have by the definition of F that F X s F Y implies X s Y for all X, Y g H b Žpro R .. v

v

v

Ž2. In Ž3. above, since H b Žpro RrG.Ž F X, F Y . is finite-dimensional, the subset GX , Y [  g g G N H b Žpro R .Ž X, g Y . / 04 of G is finite. Thus Lemma 2.2 also follows from Ž3., and we have an isomorphism v

H b Ž pro RrG . Ž F X , F Y . ( v

v

[

v

H b Ž pro R . Ž X , g Y . .

ggG X , Y

LEMMA 2.5.

Let X be an indecomposable object of H b Žpro R .. Then

Ž1. F X is indecomposable iff g X ( r X for all 1 / g g G. Ž2. Assume that g X ( r X for all 1 / g g G and let Y g H b Žpro R .. Then F Y ( F X implies that Y ( g X for some g g G. v

v

v

Proof. Ž1. Put H [  g g G N X ( g X 4 2 1. Then H : GX , X . By Remark 2.4Ž2. we have H b Ž pro RrG . Ž F X , F X . ( v

v

[

H b Ž pro R . Ž X , g X . .

ggG X , X

Since rad H b Žpro R .Ž X, g X . s H b Žpro R .Ž X, g X . for all g g G R H, we have H b Ž pro RrG . Ž F X , F X . rrad H b Ž pro RrG . Ž F X , F X . v

(

[H

b

v

v

v

Ž pro R . Ž X , g X . rrad H b Ž pro R . Ž X , g X .

ggH H

(

[ k.

Hence F X is indecomposable iff H s  14 . v

Ž2. Assume that F Y ( F X. Then by Lemma 2.3 we have v

v

[g g G g Y ( [g g G g X ;

393

DERIVED EQUIVALENCE

in particular Y is a direct summand of [g g G g X. We first show that Y ( [g g H g X for some subset H of G. Let Y ªs [g g G g X ªp Y be morphisms in H b ŽPro R . such that ps s 1 Y . Then since Y g H b Žpro R ., we have H b Ž Pro R . Y ,

ž

[ ggG

g

X (

/

[H

b

Ž pro R . Ž Y , g X .

[H

b

Ž pro R . Ž Y , g X .

ggG

(

ggN

( H b Ž pro R . Y ,

ž

[ ggN

g

/

X ,

where N [ G Y , X Ž2.4. is a finite set. Take

s 9 g H b Ž pro R . Ž Y , [g g N g X . that corresponds to s . Then we have s s ts 9, where t : [g g N g X ª [g g G g X is the inclusion. Since Žpt . s 9 s ps s 1 Y , s 9 is a section; thus Y is a direct summand of [g g N g X g H b Žpro R .. Therefore Y ( [g g H g X for some H : N because H b Žpro R . is an aggregate. Now since F X ( F Y ( [g g H F g X is indecomposable by statement Ž1. above, H must be a one-element set, say H s  g 4 with g g G. As a consequence, Y ( g X. v

v

v

DEFINITION 2.6. Ž1. Let I : S be sets or classes and assume that a group H acts on S. Then I is called H-stable if hX g I for all X g I and for all h g H. Ž2. Suppose that a group H acts on a set I. Then we say that the action of H is free Žor that H acts freely . on I if hX / X for all 1 / h g H and for all X g I. Ž3. Let H be a group of automorphisms of an aggregate A. Then an H-action is induced on isoclasses of objects of A defined by hw X x [ w hX x for all h g H and X g A, where w X x denotes the isoclass of X. The H-action of A is said to be free on isoclasses of indecomposable objects of A if the induced H-action on the isoclasses of indecomposable objects of A is free, i.e., if hX ( r X for all 1 / h g H and for all indecomposable objects X g A. Ž4. For an aggregate A choose a spectroid of A and denote it by ind A. We denote by Ind A the full subcategory of A consisting of all indecomposable objects of A. For a full subcategory I of Ind A, a retraction ¨ of I to ind A is a sequence Ž ¨ X . X g I of isomorphisms ¨ X : X ª ¨ Ž X . with ¨ Ž X . g ind A such that ¨ X s 1 X whenever X g ind A. For a : X ª Y in Ž . Ž . I , define ¨ Ž a . [ ¨ Y a ¨ y1 X : ¨ X ª ¨ Y . Then this gives a fully faithful functor ¨ Žy.: I ª ind A.

394

HIDETO ASASHIBA

Ž5. Let L: A ª B be a functor of aggregates. Then a retraction ¨ of  LŽ X . N X g ind A 4 to ind B defines the unique functor L9: ind A ª ind B such that ¨ turns out to be a natural isomorphism from L N ind A to the composite ind A ªL9 B ¨ B. This L9 is denoted by indŽ L, ¨ .. Ž6. For a locally bounded spectroid S, we denote by ¨ S a retraction of Ind H b Žpro S . to ind H b Žpro S .. For each full subcategory I of Ind H b Žpro S ., we set ¨ S N I [ Ž ¨ XS . X g I , which is a retraction of I to ind H b Žpro S .. DEFINITION 2.7. Assume that G acts freely on isoclasses of indecomposable objects of H b Žpro R .. Ž1. Let w X i x N i g I 4 be a complete set of representatives of G-orbits in the set of isoclasses of Ind H b Žpro R .. We choose the object set of ind H b Žpro R . as the union of G-orbits of X i , i g I: objŽind H b Žpro R .. [  g X i N g g G, i g I 4 . Then ind H b Žpro R . is a G-stable spectroid of H b Žpro R ., on which G acts freely. Ž2. We set F [ indŽ F , ¨ R r G .. We choose v

¨ R r G < F

v

Ž X . < X g ind H b Ž pro R . 4

as follows. First, choose ¨ FR rŽ XGi . as an arbitrary isomorphism F Ž X i . ª F Ž X i . for all i g I. Second, for each g g G define ¨ FR rŽg GX i . so that the following diagram commutes: v

v

v

¨ FRvrŽ G X i.

6

F˜Ž X i .

can. ;

F

v

Žg Xi .

¨ FRvrŽgGX i .

6

v

6

F Ž Xi .

F Žg Xi . .

Ž3. We set ind1 H b Žpro RrG. to be the full subcategory of ind H b Žpro RrG. consisting of FX for some X g ind H b Žpro R .. Remark 2.8. Ž1. The definitions of ¨ FR rŽg GX i . above are well defined because F Žg X i . s F Žh X j . for some g, h g G, and i, j g I implies g X i sh X j by Remark 2.4Ž1., from which i s j and g s h follow by the choice of  X i N i g I 4 . Ž2. The diagram obtained from that in the definition above by replacing X i by X turns out to be commutative for all X g ind H b Žpro R ., which guarantees F g Žy. s F for all g g G by Lemma 2.3Ž1.. v

v

v

v

395

DERIVED EQUIVALENCE

THEOREM 2.9.

Assume that G acts freely on ind H b Žpro R .. Then F : ind H b Ž pro R . ª ind1 H b Ž pro RrG.

is a Galois co¨ ering with group G. Proof. Recall that F is a Galois covering with group G iff Ž1. Ž2. Ž3.

F is a covering functor; F ? g Žy. s F for all g g G; F: ind H b Žpro R . ª ind1 H b Žpro RrG. is surjective on objects;

Ž4.

G acts transitively on Fy1 Ž X . for all X g ind1 H b Žpro RrG..

and Statement Ž2. follows by construction ŽRemark 2.8Ž2... Statement Ž3. is trivial. Statement Ž4. follows from Lemma 2.5Ž2.. Hence it only remains to show statement Ž1.. Let X g ind H b Žpro R . and Y g ind1 H b Žpro RrG.. Then it follows from Lemma 2.3Ž3. that F induces an isomorphism

[

ind H b Ž pro R . Ž X , Z . ª ind1 H b Ž pro RrG . Ž FX , Y . .

FZsY

Next we show that F induces an isomorphism

[

ind H b Ž pro R . Ž Z, X . ª ind1 H b Ž pro RrG . Ž Y , FX . .

FZsY

Note that we have the right adjoint Fr of the pull-up functor F : Mod R

F

Fr

Mod RrG,

6

6

v

v

H Ž Mod R .

Fv Fr

H Ž Mod RrG . ,

6

6

defined as Fr Ž M .Ž a. [ Ł x g a M Ž x .. These also induce S-functors

where again Fr is the right adjoint of F . Note that, for X g H b Žpro R ., we have F X s Fr X. By definition of ind1 H b Žpro RrG., there is some Z g ind H b Žpro R . such that FZ s Y. Then v

v

H b Ž pro RrG . Ž Y , F X . ( H Ž Mod RrG . Ž F Z, Fr X . v

v

( H Ž Mod R . Ž F F Z, X . v

v

( H Ž Mod R .

ž[ ggG

(

Ł

g

Z, X

/

H b Ž pro R . Ž g Z, X .

ggG

(

[H ggG

b

Ž pro R . Ž g Z, X . .

396

HIDETO ASASHIBA

The last isomorphism follows from the fact that the action of G on H b Žpro R . is locally bounded. This gives the desired isomorphism. Remark 2.10. Since the action of G on ind H b Žpro R . is locally bounded by Lemma 2.2, we can form the quotient ind H b Žpro R .rG and F induces an isomorphism ind H b Ž pro R . rG ª ind1H b Ž pro RrG . . The following is useful in computation of examples. LEMMA 2.11. If G acts freely on R and G is torsion-free, then the action of G on ind H b Žpro R . is free. Proof. Ž1. G acts freely on isoclasses of indecomposables in mod R. Let g g G and X be an indecomposable object in mod R. Put supp X [  x g R N X Ž x . / 04 . Assume that g X ( X. Then supp g X s supp X. Thus g induces a permutation on the finite set supp X. Therefore there is some n g N such that g n fixes some object of supp X. Since G acts freely on R, g n s 1. Thus g has a finite order. Hence g s 1 because G is torsion-free. Ž2. G acts freely on ind H b Žpro R .. Let g g G and let X be an indecomposable object in H b Žpro R .. Since X / 0 in H b Žpro R ., H n Ž X . / 0 for some n g N. Assume that g X ( X. Then we have a quasiisomorphism X ª g X, which induces an isomors phism H n Ž X . ª Yi for some ˜ H n Žg X . (g H n Ž X .. Let H n Ž X . ( [is1 g s s indecomposables Yi in mod R. Then since [is1 Yi ( [is1 Yi , there is a permutation p on the set  1, . . . , s4 such that, for all i, we have g Yi ( Yp Ž i. . a Hence for some a g N and for some i, g Yi ( Yi . Then by Ž1. above, we have g a s 1. Hence g s 1.

3. PUSH-DOWN OF A TILTING SPECTROID In this section we investigate how to get derived equivalences from those between coverings. Throughout the section R is a locally bounded spectroid and E is a tilting spectroid for R. ŽWe do not need to assume that E is locally bounded.. Further, G is a group of automorphisms of R and F: R ª RrG is the canonical Galois covering. We keep the notation introduced in the previous section. We always assume that G acts freely on ind H b Žpro R .. Note that every object of E is indecomposable. Denote by E the full subcategory of ind H b Žpro R . consisting of ¨ R Ž X . for some X g E. Then Ž ¨ R N E .Žy.: E ª E is an isomorphism. DEFINITION 3.1. E is called G-stable Žup to isomorphisms. if, for each X g E and each g g G, g X is isomorphic to some object in E.

397

DERIVED EQUIVALENCE

Remark 3.2. E is G-stable up to isomorphisms iff E is G-stable in ind H b Žpro R . in the sense of Sect. 2. DEFINITION 3.3. Assume that E is G-stable. Then E has the induced G-action. We define a G-action on E by identifying E and E using the isomorphism given by ¨ R N E. Namely, for each g g G, the new action, denoted by g )Žy., is defined by ¨ R Ž g ) X . s g ¨ R Ž X . and ¨ R Ž g ) f . s g R ¨ Ž f . for all f : X ª Y in E. Remark 3.4. Ž1. For all X g E, we clearly have g X ( g ) X. More precisely, w XR [ Ž ¨ gR) X .y1 ?g ¨ XR : g X ª g ) X is an isomorphism and we have a commutative diagram g

X

f

6

g

g

wXR

w YR

6

6

g) f

6

g) X

Y

g )Y

for all f : X ª Y in E. Ž2. The G-action on E defined above is locally bounded by Lemma 2.2 because EŽ g ) X, Y . ( EŽg ¨ R Ž X ., ¨ R Ž Y .. s H b Žpro R .Žg ¨ R Ž X ., ¨ R Ž Y .. for all X, Y g E and all g g G. THEOREM 3.5. Let R be a locally bounded spectroid, E a tilting spectroid for R and G a group of automorphisms of R acting freely on ind H b Žpro R .. If E is G-stable, then a locally bounded G-action is defined on E as abo¨ e, and RrG is deri¨ ed equi¨ alent to ErG. Proof. First note that E is also a G-stable tilting spectroid for R and ErG ( ErG. Hence we may assume that E s E. Let E9 be the full subcategory of ind1 H b Žpro RrG. consisting of FX with X g E. We show that Ž RrG, E9, ErG. is a tilting triple. Ža. For all T, U g E9 and for all n / 0, H b Žpro RrG.ŽT, U w n x. s 0. In fact, we have T s FX, U s FY for some X, Y g E. By Lemma 2.3Ž3. the left-hand side is isomorphic to H b Ž pro RrG . Ž F X , Ž F Y . w n x . ( H b Ž pro RrG . Ž F X , F Ž Y w n x . . v

v

v

(

[H

b

Ž pro R . Ž X , g Ž Y w n x . .

[H

b

Ž pro R . Ž X , Ž g Y . w n x .

ggG

(

v

ggG

s 0. The last equality follows from the assumption on E.

398

HIDETO ASASHIBA

Žb. triŽ E9. = XR r G . Here, for each full subcategory A of H b Žpro RrG., we denote by ² A: the smallest full subcategory of H b Žpro RrG. containing A and closed under isomorphisms. Then since F is an S-functor, we have the following inclusions of object classes of categories: v

XR r G :² F Ž XR .: :² F Ž tri Ž E . .: : tri Ž F Ž E . . s tri Ž E9 . . v

v

v

Žc. E9 ( ErG. It is enough to show that F N E: E ª E9 is a Galois covering with group G. But this immediately follows from the G-stability of E by Theorem 2.9. By Ža., Žb., and Žc. above, Ž RrG, E9, ErG. is a tilting triple. DEFINITION 3.6. Let r : A ª A9 be an isomorphism of spectroids and H, H9 groups of automorphisms of A, A9 whose actions are locally bounded on A, A9, respectively. Assume that there is an isomorphism a : H ª H9 of groups. Then Ž1. r is Ž H, H9.-compatible Žwith respect to a . if, for each h g H, we have r h s a Ž h. r ; Ž2. r is Ž H, H9.-compatible on objects Žwith respect to a . if, for each h g H and each x g A, we have r hŽ x . s a Ž h. r Ž x .. LEMMA 3.7. are equi¨ alent:

In the same setting as in the definition abo¨ e, the following

Ž1. r is Ž H, H9.-compatible. Ž2. Ža. r is Ž H, H9.-compatible on objects, and Žb. r induces an isomorphism ArH ª A9rH9. Proof. It is enough to show that the equivalence of Ž1. and Ž2.Žb. under the condition Ž2.Ža.. Let L9: A9 ª A9rH9 be the canonical Galois covering. Then Ž2.Žb. is equivalent to saying that L9 r : A ª A9rH9 is a Galois covering with group H. Under the condition Ž2.Ža., this holds iff Ž L9 r h. f s Ž L9 r . f for all h g H and all f : x ª y in A. By the definition of quotient spectroids, this is equivalent to the fact that h9Ž r f . s h9a Ž h.y1r hf for all h g H, all f : x ª y in A and all h9 g H9, which clearly is equivalent to Ž1.. COROLLARY spectroids with automorphisms ind H b Žpro R ., RrG and SrH

3.8. Let Ž R, E, S . be a tilting triple of locally bounded an isomorphism c : E ª S, and let G, H be groups of of R, S, respecti¨ ely. Assume that G acts freely on that E is G-stable, and that c is Ž G, H .-compatible. Then are deri¨ ed equi¨ alent.

Proof. This follows from Theorem 3.5 and Lemma 3.7.

DERIVED EQUIVALENCE

399

In Sect. 5 and 6, we compute Ž H, H9.-compatibility using quivers. For this sake we prepare the following lemma. First we need some definitions. DEFINITION 3.9. Let Q, Q9 be quivers and J, J9 ideals of kQ, kQ9, respectively. Ž1. For a morphism f : Q ª Q9, kf: kQ ª kQ9 denotes the functor induced from f. If kf Ž J . F J9, then kf: kQrJ ª kQ9rJ9 denotes the functor induced from kf. Ž2. Let H be a group of automorphisms of Q. Then H induces a group  kh N h g H 4 of automorphisms of kQ. If khŽ J . F J for all h g H, then H induces a group  khN h g H 4 of automorphisms of kQrJ. DEFINITION 3.10. Let f : Q ª Q9 be an isomorphism of quivers and H, H9 groups of automorphisms of Q, Q9, respectively, with an isomorphism a : H ª H9 of groups. Then f is called Ž H, H9.-compatible Žwith respect to a . if fh s a Ž h. f for all h g H. LEMMA 3.11. In the same setting as in the definition abo¨ e, assume that kf: kQrJ ª kQ9rJ9 is an isomorphism of spectroids for some admissible ideals J, J9 of kQ, kQ9, respecti¨ ely, that groups K, K 9 of automorphisms of kQrJ, kQ9rJ9 are induced from H, H9, respecti¨ ely, and that an isomorphism b : K ª K 9 is induced from a Ž i.e., b Žkh. s k a Ž h . for all h g H .. If f is Ž H, H9.-compatible Ž with respect to a ., then the isomorphism kf is Ž K, K 9.compatible Ž with respect to b .. Proof. The proof is straightforward.

4. APPLICATION: REPETITIONS In this section we give the first application of our technique, which gives a generalization of w19, Theorem 3.1x. THEOREM 4.1. Let Ž A, E, B . be a tilting triple of locally bounded spectroids with an isomorphism c : E ª B and n any integer. Then we ha¨ e Ž1. The tilting spectroid Eˆ for AZ ŽTheorem 1.5. is ² nAn :-stable and c Z Ž² is nAn :, ² n Bn :.-compatible. Ž2. AZ r² nAn : and B Z r² n Bn : are deri¨ ed equi¨ alent. In particular, when n s 1, tri¨ ial extensions T Ž A. and T Ž B . are deri¨ ed equi¨ alent. Proof. Ž1. It is enough to show the assertion for n s 1. Ža.

Eˆ is ² nA :-stable.

400

HIDETO ASASHIBA

In fact, let T s ŽT i, d i . g E and m g Z. Then nA

Ž T i mA

AZ Ž y, 1w mx?. . s T i mA AZ Ž nAy1 Ž y . , 1w m x? . ( T i mA AZ Ž y, nA1w mx? . s T i mA AZ Ž y, 1w mq1x?. .

Thus

nA

Žb.

ŽT w m x . ( T w mq1x. Hence Eˆ is ² nA :-stable.

c Z is Ž² nA :, ² n B :.-compatible.

It is enough to show the commutativity of the following diagram: cZ

BZ

6

Eˆ

nB

nA ) Žy .

6

6

cZ

Ž)).

B .

6

Eˆ

Z

By the computation in Ža. above, we have nA )T w m x s T w mq1x for all T g E and for all m g Z. This proves the commutativity of Ž)). on objects: c Z Ž nAw m x . s Ž c T .w mq1x s n B c Z ŽT w m x .. To make it possible to compute Z nA )Žy. on morphisms, we choose ind H b Žpro AZ . and ¨ s ¨ A N Eˆ as follows. First we may assume that each component T i of each object T s ŽT i, d i . of E is a direct sum of representables, i.e., for all i g Z there i are some a i Ž x . g N j  04 such that T i s [x g A AŽy, X .Ž a Ž x .. . Next we show the following. Claim. If the ² nA :-orbits of the isoclasses w T w0x x and wU w0x x intersect for some T, U g E, then T s U. In fact, assume that n A T w0x (n A U w0x for some m, n g Z. Then T w m x ( U w nx. Hence it follows from m

n

H b Ž pro AZ . Ž T w m x , U w nx . / 0 and

H b Ž pro AZ . Ž U w nx , T w mx . / 0

that n g  m, m q 14 and m g  n, n q 14 , i.e., m s n. In particular, T mA AZ Ž 1w mx Ž y . , 1w m x? . ( U mA AZ Ž 1w m x Ž y . , 1w mx? . , which shows T ( U in E and T s U. The claim shows that there is a complete set w X i x N i g I 4 of representatives of ² nA :-orbits in the set of isoclasses of Ind H b Žpro AZ . containing m T w0x N T g E4 . Now choose objŽind H b Žpro AZ .. [  n A X i N i g I, m g Z4 ,

401

DERIVED EQUIVALENCE

and for each T w m x g Eˆ choose ¨ T w m x : T w mx ª ˜ n A T w0x as the canonical one. i i Namely, for T s ŽT , d ., ¨ T w m x is given by m

T i mA nAym : T i mA AZ Ž y, nAm 1w0x? . ª T i mA AZ Ž nAym Ž y . , 1w0x? . . Under these choices of ind H b Žpro AZ . and ¨ , the action nA )Žy. is described as follows. First note that

¡H Ž pro A. Ž T , U . , ~ . ( DHH Ž pro A . Ž U, T . , ¢0, b

H

b

Ž pro

Z

A

.ŽT

w mx

,U

w nx

b

if n s m, if n s m q 1, otherwise.

Hence we may write H b Ž pro AZ . Ž T w m x , U w nx .

¡ f ~ s w ¢0,

w mx

N f g H b Ž pro A . Ž T , U . 4 ,

if n s m,

w mx

N w g D H Ž pro A . Ž U, T . 4 ,

if n s m q 1, otherwise.

b

Claim. Let T, U g E and H b Ž pro A . Ž U, T . . f g H b Ž pro A . Ž T , U . , w g DH Then Ži.

nA )Žy. sends f w mx: T w mx ª U w mx to f w mq1x: T w mq1x ª U w mq1x;

Žii.

nA )Žy. sends w w m x: T w mx ª U w mq1x to w w mq1x: T w mq1x ª U w mq2x.

and Z

Z

In fact, let T s ŽT i, dTi ., U s ŽU i, dUi . be in E. Put ¨ s ¨ A , w s w A . Then w T w m x s Ž ¨ T mq 1 .

y1

? nA ¨ T wmx

s Ž T m nAymy1 . s T m nA .

y1

? n A Ž T m nAym .

402

HIDETO ASASHIBA

Hence, by Remark 3.4, it is enough to show the commutativity of the diagram T w mx

nA

Ž a w m x.

6

nA

nA

Tm nA

Um nA

6

6

a w mq1x

6

T w mq1x

U w nx

U w nq1x

for all m, n g Z with n s m or m q 1 and for all

ag

½

if n s m,

H b Ž pro A . Ž T , U . ,

if n s m q 1.

b

H Ž pro A . Ž U, T . , DH

Using the assumption on the objects of E, the verification of this is reduced to the case where T, U g XA , which is not hard to check and is left to the reader. Now we can show the commutativity of Ž)). as follows. Ži. For each f g H b Žpro A.ŽT, U ., m g Z, we have c Z Ž nA ) f w m x . s c Z Ž f w mq1x . s Ž c f .w mq1x s n B ŽŽ c f .w mx . s n B c Z Ž f w mx .. Žii. For each w g DH H b Žpro A.ŽU, T ., m g Z, we have

c Z Ž nA ) w w m x . s c Z Ž w w mq1x . s D Ž c TU .

ž ž

s nB

ž ž DžŽc

U y1 T

.

/w/

w mx

y1

/w/

/ sn c B

w mq1 x

Z

Ž w w mx . .

As a consequence, c Z is Ž² nA :, ² n B :.-compatible. Ž2. This follows from Ž1. above by Corollary 3.8 and Lemma 2.11.

5. APPLICATION: SELF-INJECTIVE ALGEBRAS OF TYPE A n As the next application of our technique, we classify representation-finite self-injective algebras of type A n up to derived equivalence. In this section we simply write g X for g ) X because this creates no confusion. Let L be a representation-finite self-injective algebra of type A n . This type of algebra was completely classified by Riedtmann w22x, to which we refer for the terminologies and notation used here. We exclude the trivial

DERIVED EQUIVALENCE

403

case that L s k. Let C be a configuration of Z A n and G an admissible group of automorphisms of Z A n stabilizing C such that ŽZ A n . CrG is isomorphic to the Auslander]Reiten quiver of L. Define an automorphism t of Z A n by t Ž p, q . [ Ž p y 1, q .. When n is odd, define an automorphism f of Z A n by f Ž p, q . [ Ž p q q y Ž n q 1.r2, n q 1 y q ., the reflection at the central line of Z A n . We set m to be the number of isoclasses of simple L-modules and e the period of C . Then e divides both n and m: n s er, m s es. Set g s tye . Then g stabilizes C and ² g s : s ²Žty1 . e s : s ²t m :. By w22, 3.3x the group G may be written as G s ² g s : or G s ² g sf :. By definition, L is called a wreathlike algebra in the former case, and is called a Mobius algebra in the latter case. In the latter case n ¨ must be odd and e s n, thus m s ns, whence both Ž n y 1.r2 and mrn ˜C which has the are natural numbers. As in w22, 6.2x, C defines a quiver Q integers as vertices and whose arrows are divided into two groups: a-arrows and b-arrows. Further, any automorphism h of Z A n stabilizing C induces ˜C , which we denote also by h. Then G can be an automorphism of Q ˜C , and the quiver Q˜Cr²t n : is regarded as a group of automorphisms of Q identified with the Brauer qui¨ er Q C corresponding to C . Then ²t n :-orbit of a-arrows Žresp. b-arrows. are called a-arrows Žresp. b-arrows. of Q C . ˜C is The automorphism of Q C corresponding to the automorphism t of Q ˜ denoted by t . Let I be the ideal of kQC defined by all possible zero relations ab s 0 s ba and commutativity relations a aŽ x . s b bŽ x . for all ˜C , where a aŽ x . Žresp. b bŽ x . . is the unique path consisting of vertices x in Q ˜C . Set R [ kQ˜CrI. Then Ž a-arrows resp. b-arrows. from x to x q n in Q again G can be regarded as a group of automorphisms of R. The ordinary ˜XC of R is obtained from Q˜C by deleting arrows x ª x q n, and R quiver Q ˜XCrI9, where I9 is obtained from I by deleting such is presented as R ( kQ arrows using the relations a aŽ x . s b bŽ x . with aŽ x . s 1 or bŽ x . s 1. By w22, Theorem 6.2x, we have a Galois covering functor R ª L with group G. First we investigate the wreathlike case. DEFINITION 5.1. Let m, n G 1. By Nm, n we denote the Žbasic. connected self-injective Nakayama algebra with m isoclasses of simple modules and of Loewy length n q 1. ª

Remark 5.2. Let A n be the quiver 1 ª 2 ª ??? ª n. Z

Then Nm, n ( B r²g defined by

ª

, where B s kA n and g is an automorphism of B Z

m:

g Ž pw ix . [

½

1w iq1x ,

if p s n, wix

Ž p q 1. ,

otherwise.

404

HIDETO ASASHIBA

The next proposition generalizes a Rickard theorem on derived equivalences of Brauer tree algebras w19, Theorem 4.2x Žcf. a theorem of Gabriel and Riedtmann w6, 1.8 Theorem 1x.. PROPOSITION 5.3. Let L be a self-injecti¨ e algebra and m, n G 1. Then the following are equi¨ alent: Ž1. L is deri¨ ed equi¨ alent to the self-injecti¨ e Nakayama algebra Nm, n ; Ž2. L is a wreathlike algebra of type A n with m isoclasses of simple modules. Proof. By Keller and Vossieck w13, 2.3 Examplex Žor Rickard w19, Corollary 2.2x, Ž1. implies Ž2.. Ž2. « Ž1.. Assume that L is a wreathlike algebra of type A n with m isoclasses of simple modules. We have to show that L is derived equivalent to Nm, n . Cyclic paths in Q C consisting of a-arrows Žresp. b-arrows. are called a-orbits Žresp. b-orbits.. Then by Riedtmann w22, Proposition 3.5x, if e / n, then Q C has exactly one exceptional orbit, i.e., either an a-orbit or a b-orbit which is stable under t e, which may be assumed to be an a-orbit. If e s n, then choose one a-orbit that is not a loop, and call it exceptional. By deleting one arrow from each cycle in Q C we obtain a quiver Q of the form in Figure 1 satisfying the following: v

Q has n vertices;

the first horizontal full a-path L from the above corresponds to the exceptional a-orbit; v

the number of vertices of the subquiver Q i is equal to e for all i s 1, . . . , r; v

v

Q i and Q j are isomorphic for all i, j s 1, . . . , r;

v

each Q i is a full subquiver of the quiver in Figure 2.

FIGURE 1

405

DERIVED EQUIVALENCE

FIGURE 2

Each horizontal or vertical full a- or b-path is called a branch of Q. We call a vertex x in Q a junction if x belongs to two branches. Let A be the spectroid defined by the quiver Q with all possible relations ab s ba s 0. Then by the definition of repetitions, we can construct a display-functor ˜XC ª AZ with Ker F s I9. Thus F gives identifications Žw7, p. 74x. F: kQ Z r R s A and g s nA . We now construct a nice tilting spectroid E for A. We write xˆ[ AŽy, x . for all x g A for short. Let x g A. There is a unique shortest path from a vertex x 0 g L to x in Q: a1 # a

???

a

"

x2

b

??? ª ??? ª x t s x,

6

x1

6

" ! a

6

b

6

???

6

a0 # b 6

x0

6

!b

where x 0 , x 1 , . . . , x ty1 are junctions if t ) 0. We then put Ž b a0 .ˆ

x 1ˆ

Ž a a1 .ˆ

x 2ˆª ??? ª x tˆª 0 ª ??? ,

6

ž

6

Tx [ ??? ª 0 ª x 0ˆ

/

where x 0ˆ is in degree zero. Let E be the full subcategory of H b Žpro A. defined by Tx N x g A4 . We show that Ž A, E, B . is a tilting triple. Ža. triŽ E . = XA . In fact, by construction, we have a triangle of the form xˆwyt x ª Tx ª Tx ty 1 ª xˆw1 y t x. Hence Ža. follows. Žb.

H b Žpro A.ŽTx , Ty w i x. s 0 for all x, y g A and all i / 0.

406

HIDETO ASASHIBA

First remark by construction that if AŽ x, y . ( Hom AŽ xˆ, yˆ. / 0 for some x, y g A, then x and y are in the same branch of Q and x is a predecessor of y in Q Ži.e., there is a path from x to y in Q .. Further the k-dimension of Hom AŽ xˆ, yˆ. is at most 1. By this remark, the assertion is clear if < i < G 2. Case i s y1. Assume that H b Žpro A.ŽTx , Ty wy1x. / 0. Let 0 / f g H b Žpro A.ŽTx , Ty wy1x., f s Ž f j ., f j : x jq1ˆª y jˆ. Put p [ min j G 0 N f j / 04 . Then by the remark above, x p s y p and this is a proper predecessor of x pq 1. Thus f p g Hom AŽ x pq1ˆ, x pˆ. s 0, a contradiction. Case i s 1. Assume that 0 / f g H b Žpro A.ŽTx , Ty w1x., f s Ž f j ., f j : x jˆª y jq1ˆ. Put p [ pŽ f . [ max j G 0 N f j / 04 , and put f 9 [ Ž f jX ., where f jX s

½

fj ,

if j - p,

0,

if j G p.

By the remark above, x p s y p and f p factors through y pˆª y pq1ˆ. This shows that f and f 9 are homotopic. Since pŽ f 9. - pŽ f ., induction shows that f s 0, a contradiction. Žc.

E ( B.

Define a linear order F on the set of vertices of Q lexicographically as follows: Let x, y g Q and let

Ž b b 0 .ˆ

y 1ˆ

Ž a a1 .ˆ

x 2ˆª ??? ª x tˆs xˆª 0 ª ???

Ž a b1 .ˆ

/ y ˆª ??? ª y ˆs yˆª 0 ª ??? / .

6

0

x 1ˆ

6

y

Ž b a0 .ˆ

6

ž T s ž ??? ª 0 ª y ˆ

6

Tx s ??? ª 0 ª x 0ˆ

2

u

Ži. If x 0 is a predecessor of y 0 , then we define x F y. Žii. Assume that x 0 s y 0 , . . . , x i s yi for i G 0. If yi s y Ži.e., if Tyiq1 s 0., then define x F y. In case x i / x and yi / y, if x iq1 is a predecessor of yiq1 , then define x F y. Then in particular for x, y g L, x F y iff there is a path from x to y in L; for other branches z 0 ª z1 ª ??? ª z p , we have z1 - z 2 - ??? - z p - z 0 . Claim.

x ) y implies H b Žpro A.ŽTx , Ty . s 0.

In fact, assume that x ) y and that 0 / f g H b Žpro A.ŽTx , Ty ., where f s Ž f i . g C b Žpro A., f i : x iˆª yiˆ. Put p [ max i N f i / 04 . If x p / y p , then there is a path from x p to y p because f p / 0, which means that x F y, a contradiction. Hence x p s y p . It then follows from x ) y that y p / y. Here note that zˆ are uniserial for all z g Q, and the dimension

407

DERIVED EQUIVALENCE

vector of yiˆ is as follows for all 1 F i F u:

Ž dim yiˆ. z s

½

1 0,

if z is both a successor of yiy1 and a predecessor of yi , otherwise,

for all z g Q. Let h be the map y pˆª y pq1ˆ in Ty . Since Ž f i . g C b Žpro A. and f pq 1 s 0, we must have hf p s 0. Considering the dimension vectors of y pˆ and y pq1ˆ, this forces f p s 0, a contradiction. Claim.

x F y implies dim H b Žpro A.ŽTx , Ty . s 1.

In fact, assume that x F y and put p s max i N x i s yi 4 . From the assumption, x p s x implies y p s y. Thus we have the following three cases: Ži. x p s x and y p s y; Žii. x p / x and y p s y; or Žiii. x p / x and y p / y. In each case we easily calculate the C b Žpro A.ŽTx , Ty . as follows: Cases Ži. and Žii.. C b Žpro A.ŽTx , Ty . s kf, where f s Ž . . . ,0, 1 x 0ˆ , . . . , 1 x pˆ , 0, . . . . with 1 x 0ˆ in degree zero. Case Žiii.. The map x pˆª x pq1ˆ in Tx has the form Žg a .ˆ, and the map x pˆª y pq1ˆ in Ty has the form Žg aqb .ˆ for g s a or b and for some a, b g N. Then C b Žpro A.ŽTx , Ty . s kf, where f s . . . , 0, 1 x 0ˆ , . . . , 1 x pˆ , Ž g b .ˆ, 0, . . .

ž

/

with 1 x 0 ˆ in degree zero. Therefore in any case we have dim H b Žpro A.ŽTx , Ty . F 1. Now the morphism f constructed above is not homotopic to zero because Ž b a 0 .ˆ is not a section. Hence the claim follows. The two claims above show that E ( B. By Ža., Žb., and Žc. above, Ž A, E, B . is a tilting triple. We define an isomorphism c : E ª B as follows. Name the vertices of Q as ¨ Ž0., . . . , ¨ Ž n y 1. so that ¨ Ž p . - ¨ Ž q . iff p - q, and name the ª vertices of the quiver A n of B as w Ž i . [ i y 1 for all i s 1, . . . , n. Then by the claims above E is presented by the quiver T¨ Ž0. ª T¨ Ž1. ª ??? ª T¨ Ž ny1. with no relations. Define c by the quiver isomorphism determined by T¨ Ž i. ¬ w Ž i . for all i s 0, . . . , n y 1. ˆ B Z . is a tilting triple, where Eˆ is the full Now by Theorem 1.5, Ž AZ , E, bŽ Z. subcategory of H pro A with the object set

 T w mx [ T mA

AZ Ž y, 1w m x?. < T g E, n g Z 4 .

408

HIDETO ASASHIBA

Next we show that Eˆ is ² g :-stable. By construction for each p g  0, . . . , n y 14 we have g

Ž AŽ y, ¨ Ž p . . mA

AZ Ž y, 1w ix . .

s g AZ Ž y, ¨ Ž p .

wix

s AZ y, g Ž ¨ Ž p .

ž

. wix

./

¡A Žy, ¨ Ž p q e . . , s~ Z

wix

¢A Žy, ¨ Ž p q e y n. Z

if p q e - n, w iq1 x

.,

if p q e G n.

Let q g Z. Then we have q s ni q p for a unique p g  0, . . . , n y 14 and for a unique i g Z. Introduce the notation ¨ˆŽ q . [ ¨ Ž p .w i x and T¨ˆŽ q. [ T¨wŽixp. . Then the above equality is written as g AZ Žy, ¨ˆŽ q .. s AZ Žy, ¨ˆŽ q q e .. for all q g Z. Therefore g acts as g

T¨ˆŽ q. s T¨ˆŽ qqe.

for all q g Z.

Hence Eˆ is ² g :-stable. Similarly, put w ˆ Ž q . [ w Ž p .w i x for all q g Z as above. Then c Z is induced from the quiver isomorphism determined by T¨ˆŽ q. ¬ w ˆ Ž q . for all Ž .. Ž q g Z. Define an automorphism h of B Z , by hŽ w q [ w q ˆ ˆ q e .. Then by Lemma 3.11 it follows from the formula of g-action above that c Z is Ž² g :, ² h:.-compatible. Put H [ ² h s :. Then c Z is Ž G, H .-compatible. Hence, by Corollary 3.8 and Lemma 2.11, L ( AZrG is derived equivalent to B Z rH ( Nm, n . Remark 5.4. Ž1. If n divides m, say m s ns Že.g., this occurs if e s n., then the implication Ž2. « Ž1. follows from Theorem 4.1 because L has the form AZ r² nAs : for some algebra A which is tilted from the algebra ª B s kA n , and B Z r² n Bs : ( Nm, n . Ž2. In the proof of Žc. above, note that A is derived equivalent to the ª algebra B [ kA n from the beginning Žfor instance by Assem and Happel w1, Theoremx.. Therefore H b Žpro A. , H b Žpro B ., and by Happel w9x H b Žpro B . is equivalent to the stable category mod B Z , which has no oriented cycle. Further, the k-dimension of Žmod B Z .Ž X, Y . is at most 1 for all indecomposables X, Y in mod B Z . Using these it is enough for Žc. to verify that H b Žpro A.ŽTx , Ty . / 0 for all x, y g Q with x F y. Next we classify Mobius algebras up to derived equivalence. Before ¨ doing so we have to give ‘‘canonical’’ forms of Mobius algebras Žcf. w22, ¨ Fig. 36x and w23, Figure 2x..

409

DERIVED EQUIVALENCE

DEFINITION 5.5. An algebra is called a canonical Mobius algebra if it is ¨ isomorphic to the algebra M p, s for some p, s g N defined by the quiver in Figure 3 with relations Ži. paths of length p q 2 are equal to 0;

Ž ii .

½ Žiii.

b 0w iq1xa pw ix s 0,

for all i s 0, . . . , s y 2, and

a 0w iq1xb pw ix s 0,

½

a 0w0xa pw sy1x s 0, b 0w0xb pw sy1x s 0;

a pw ix ??? a 0w ix s b pw ix ??? b 0w ix for all i s 0, . . . , s y 1. ª

Remark 5.6. Let A pp be the quiver

???

Ž y1 . a py1

b0

0

p.

6

2

a2

b1

6

6

Ž y2 .

6

a1

b2

6

1

6

a0

???

6

b py1

6

Ž yp .

ª

Then M p, s ( B Z r² n Bs f B :, where B [ kA pp , and f B is an automorphism of B Z defined by f B Ž q w ix . [ Žyq .w ix. PROPOSITION 5.7. Let L be a Mobius algebra of type A n with m isoclasses ¨ of simple modules. Then L is deri¨ ed equi¨ alent to the canonical Mobius ¨ algebra M p, s with n s 2 p q 1, m s sn Ž thus p s Ž n y 1.r2, s s mrn..

FIGURE 3

410

HIDETO ASASHIBA

Proof. Since L is a Mobius algebra, the automorphism f of Z A n ¨ stabilizes C . Thus f induces an automorphism Žagain denoted by f . of Q C exchanging a-arrows and b-arrows. It is easy to see that there is a unique vertex x g Q C satisfying f Ž x . s x. Put ¨ Ž0. [ x. By deleting one arrow from each cycle in Q C we obtain a subquiver Q of Q C of the form in Figure 4 satisfying the following: Q has n vertices; f Ž ¨ Ž0.. s ¨ Ž0. and f induces an isomorphism from Qy to Qq exchanging a-arrows and b-arrows; both Qy and Qq are full subquivers of the quiver in Figure 2. v v

v

Let A be the spectroid defined by the quiver Q with all possible relations ab s ba s 0, and let fA the automorphism of AZ defined by fAŽ q w ix . [ f Ž q .w ix. Let L be the full subquiver of Q defined by the a-branch and the b-branch connected to the vertex ¨ Ž0.. Then, by the definition of repeti˜XC ª AZ with Ker F s I9. tions, we can construct a display functor F: kQ Z Thus F gives identifications R s A , g s nA , and f s fA . Therefore G s ² g sf : s ² nAs fA : and L ( AZ rG. The rest is quite similar to the proof of Proposition 5.3, and the details are left to the reader. Defining Tx g H b Žpro A. for all x g Q in the same way, Ž A, E, B . is shown to be a tilting triple with an isomorphism c : E ª B defined by a quiver isomorª phism, where B s kA pp , and E is the full subcategory of H b Žpro A. defined by Tx N x g Q4 . Further the tilting spectroid Eˆ of the induced ˆ B Z . is shown to be G-stable and c Z to be Ž G, H .tilting triple Ž AZ , E, compatible, where H [ ² n Bs f B :. Hence L ( AZ rG is derived equivalent to M p, s ( B Z rH. Summarizing the two propositions above we obtain the following. THEOREM 5.8. Let L be a representation-finite self-injecti¨ e algebra of type A n with m isoclasses of simple modules. Ž1.

If L is a wreathlike algebra, then it is deri¨ ed equi¨ alent to Nm, n .

FIGURE 4

DERIVED EQUIVALENCE

411

Ž2. If L is a Mobius algebra, then it is deri¨ ed equi¨ alent to ¨ MŽ ny1.r2, m r n . Ž3. A self-injecti¨ e algebra G is stably equi¨ alent to the algebra L iff G is deri¨ ed equi¨ alent to the algebra L. 6. EXAMPLES In this section we present an example illustrating a course of the proof of Proposition 5.3 and give our solution of Broue’s ´ problem in the introduction for the principal block of kSLŽ2, 4. in characteristic 2. This is an alternative approach to the same solution as given in Rickard w20x. Our technique might be used to solve the problem for the groups SLŽ2, p n . with p prime. Also in this section we simply write g X for g ) X because this creates no confusion. EXAMPLE 6.1. Let L be a nonsymmetric wreathlike algebra defined by the quiver in Figure 5 with relations a 3 y b 2 s ab s ba s 0. Then n s 6, e s 2, r s 3, m s 4, and s s 2. The Brauer quiver Q C of L is as in Figure 6, where we represent a-arrows by solid arrows and b-arrows by dotted arrows. The algebra A defined in the proof of Theorem 4.1 looks as shown in Figure 7. The tilting spectroids E and Eˆ constructed in the proof of the theorem are as in Figure 8, where E has no relations and Eˆ has relations that all paths of length 7 are zero, and g s s ª acts as g T¨ˆŽ q. s T¨ Ž qq4. . Thus E ( kA6 and, by Theorem 5.8, L is derived equivalent to the self-injective Nakayama algebra N4, 6 in Figure 9. Let F be a unique square-free tilting A-module. Consider F as a tilting spectroid by identifying it with the full subcategory of H b Žpro A. , D b Žmod A. consisting of indecomposable direct summands of F. Then note that the tilting spectroid Fˆ is not G-stable, and hence there is no Ž G, H .-compatible isomorphism from F. EXAMPLE 6.2. Assume that the characteristic of k is 2. Then the principal block L of kSLŽ2, 4. is derived equivalent to its Brauer correspondent G.

FIGURE 5

412

HIDETO ASASHIBA

FIGURE 6

FIGURE 7

FIGURE 8

FIGURE 9

Proof. The algebra L is given by the left quiver in Figure 10 with commutativity relations b 2 b 1 a 2 a 1 s a 2 a 1 b 2 b 1 and zero relations a 1 a 2 s 0 s b 1 b 2 . The algebra G is given by the right quiver in Figure 10 with commutativity relations a 2 a 1 s g 1 g 2 , b 2 b 1 s a 1 a 2 , and g 2 g 1 s b 1 b 2 and zero relations b 1 a 1 s 0 s a 2 b 2 , g 1 b 1 s 0 s b 2 g 2 , and a 1 g 1 s 0 s g 2 a 2 Žsee Erdmann w4x, Rickard w20x, Okuyama w15x, or Koshita w14x..

413

DERIVED EQUIVALENCE

FIGURE 10

Let A be the spectroid defined by the left quiver in Figure 11 with relations ga s 0 s db , and let B the spectroid defined by the right quiver in Figure 11 with relations ga s 0 s db . Then L ( AZ rG and G ( B Z rH, where G s ² g :, H s ² h:, and both g and h are defined by

pw ix ¬

½

w x

Ž p q 3. i , Ž p y 3.

w iq1 x

for p s 1, 2, 3, ,

for p s 4, 5, 6.

Define a full subcategory E of H b Žpro A. by the following six objects: Ž2ˆ., Ž aˆbˆ.

Ž «ˆ, zˆ.

6

6

Ž3ˆ., Ž5ˆ., Ž6ˆ., Ž2ˆ[3ˆ 1ˆ., and Ž5ˆ[6ˆ 4ˆ., where the underline stands for the place of degree zero. Then it is easy to verify that Ž A, E, B . is a tilting triple, that the tilting spectroid Eˆ is G-stable, and that c Z is Ž G, H .-compatible for some isomorphism c : E ª B. Hence L ( AZrG is derived equivalent to G ( B ZrH. The obtained tilting spectroid E9 for L is the same as in Rickard w20x. Remark 6.3. Put T n Ž A. [ AZ r² nAn : for a locally bounded spectroid A. Consider the setting in the example above. Then we can call the L and G as T 1r2 Ž A. and T 1r2 Ž B ., respectively, because g 2 s nA and h 2 s n B .

FIGURE 11

414

HIDETO ASASHIBA

Instead of taking AZ , B Z , we can use T Ž A., T Ž B . as double coverings of L, G, respectively. Then all computations are executed within the limits of algebras.

ACKNOWLEDGMENT First I would like to thank Prof. T. Okuyama for showing me the quiver presentations of the principal block of kSLŽ2, 4. and its Brauer correspondent, a careless look at which gave me the motivation of this work. Next I would like to thank Prof. Bernhard Keller for the helpful conservation in the Sommerschule on derived equivalences held at Pappenheim in 1994, where I was able to make a rough sketch of the work, and to thank Volkswagen Stiftung for the financial support for the Sommerschule. Further most of the work until Sect. 4 in the paper was done and the revised version was completed, while I was visiting the University of Bielefeld in the summer of 1995, and in 1996, respectively. I would like to thank Prof. C. M. Ringel and all members of the group of representation theory of algebras in the University of Bielefeld for the hospitality and for nice conversations, and to thank Sonderforschungsbereich 343 for the support. Finally would like to express my thanks to the referee for useful comments, one of which suggested the present form of the proof of Lemma 1.3.

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DERIVED EQUIVALENCE

415

14. H. Koshita, Quiver and relations for SLŽ2, 2 n . in characteristic 2, J. Pure Appl. Algebra 97 Ž1994., 313]324. 15. T. Okuyama, Derived equivalence and perfect isometry I, in ‘‘Proc. 4th Symposium on Representation Theory of Algebras, Izu, Japan, February 1993’’ 85]94 Žin Japanese.. 16. T. Okuyama, Projective indecomposable modules of SLŽ2, p 2 ., RIMS Kokyuroku 877, June 1994, 62]72 Žin Japanese.. 17. Z. Pogorzały, On a class of self-injective locally bounded categories, preprint, 1996. 18. J. Rickard, Morita theory for derived categories, J. London Math. Soc. 39 Ž1989., 436]456. 19. J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 Ž1989., 303]317. 20. J. Rickard, Derived equivalence for the principal blocks of A 4 and A 5 , preprint, July 1990. ¨ 21. Chr. Riedtmann, Algebren, Darstellungskocher, Uberlangerungen und zuruck, ¨ ¨ Comm. Math. Hel¨ . 55 Ž1980., 199]224. 22. Chr. Riedtmann, Representation-finite selfinjective algebras of type A n , in Lecture Notes in Mathematics, Vol. 832 Ž1980., 449]520. 23. E. Scherzler and J. Waschbusch, A class of self-injective algebras of finite representation ¨ type, in Lecture Notes in Mathematics, Vol. 832 Ž1980., 545]572. 24. J. Waschbusch, Universal coverings of selfinjective algebras, in Lecture Notes in ¨ Mathematics, Vol. 903 Ž1981., 331]349.