CURVE SINGULARITIES ARISING FROM THE REPRESENTATION THEORY OF TAME HEREDITARY

ALGEBRAS

Helmut Lenzing

I . Introduct ion

Tameness of a f i n i t e dimensional k-algebra A, where k is a commutative f i e l d , is defined with reference to one-parameter fami l ies (Mx)x E X of A-modules, supposed to contain nearly a l l indecomposable A-modules. I t seems therefore natural to invest igate the geometry of these parametrizing curves X, and ask to which extent they determine the representation theory of A.

In th is paper we w i l l r e s t r i c t to the case, where A is heredi tary. Assuming A add i t i ona l l y connected, only one parameter curve X = X(A) w i l l be needed in order to parametrize the 'continuous part ' of the representation theory of A, the category reg(A) of socalled regular A-modules. We refer to the work of V. Dlab and C. M. Ringel [8, 11]~see also [9, 31]~for a detai led invest igat ion of some of these curves X(A) and the i r i n te r re la t i on with the structure of regular A-modules.

Here, we are presenting a formal de f i n i t i on for X = X(A) by specify ing i t s category coh(X) of coherent sheaves. Moreover, we w i l l give a general descr ipt ion of the geometry of X including structure sheaf OX, dual iz ing sheaf OX(- I ) , global sections F(X,F), sheaf cohomology HI(x,F) and f i e l d K(X) of rat ional funct ions on X. We also include a b r i e f account on the d i v i so r theory of X.

As a resu l t , the parameter curve X of A turns out to be a non-commutative pro ject ive curve with s i ngu la r i t i e s . The s i tua t ion is pa r t i cu la r l y simple, i f add i t i ona l l y the base f i e l d k is assumed to be a lgebra ica l ly closed. Here, the s ingu la r i t y type of X (which counts the m u l t i p l i c i t y of the s ingular points) i s given by a Dynkin diagram Ap,q, D n, E6, E 7, E8, which determines X, i . e . the

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category coh(X), uniquely up to isomorphism (Theorem 6.2). Moreover, i f A denotes the s ingu la r i t y type of X(A), A has extended Dynkin type ~. Accordingly, the parameter curve X(A) does not depend on the 'o r ien ta t ion 'o f A(Proposition 6.1).

For each coherent sheaf F E coh(X), both global sections ?(X,F) and cohomology HI(x,F) are f i n i t e dimensional l e f t A-modules. Moreover, F is uniquely determined (up to isomorphism) by th is pair (F(X,F), HI(x,F)) of A-modules. As a resu l t , the categories cob(X) and mod(A) determine each other completely (Theorem 4.10). Here, mod(A) denotes the category of a l l f i n i t e dimensional l e f t A-modules. I t is convenient to express th is resul t in the language of t i l t i n g theory [7, 19, 6, 34, 18]: the structure sheaf 0 X serves as a ' t i l t i n g module', and the category mod(A) emerges from coh(X) by t i l t i n g with Ox(cf. of the functor 'global sections' the category sheaves is equivalent to the category reg(A) remaining indecomposable coherent sheaves F

4.11). To be more speci f ic , by means COho(X) of a l l f i n i t e length coherent of a l l regular A-modules. Al l the

are loca l l y project ive (Proposition 5.7) , determined e i ther by the A-module of global sections r(X,F), necessarily pre- project ive, or else the cohomology module HI(x,F), which is a prein ject ive A-module. From the c lass i f i ca t ion of these modules (see [9]) i t is possible to construct indecomposable loca l l y project ive sheaves on X of rank 2,3,4,6 i f the base f i e l d is a lgebra ica l ly closed, and X has s ingu lar i t y type D n, E6, E 7, E8, respect ively.

The parameter curve of the Kronecker algebra Z, the path algebra of the quiver • ~ . , is the project ive l ine P1(k) (Proposition 6.3). Hence the category coh(P1(k)) is completely control led by the representation theory of Z, i . e . by the c lass i f i ca t ion of pairs of rectangular matrices up to simultaneous conjugation, effected by L. Kronecker in 1890. In th is way, the c lass i f i ca t ion of vector bundles on the project ive l ine becomes a corol lary of Kronecker's c lass i f i ca t ion (cf . Corollary

6.4). For the history of th is theorem, see [38].

I t is shown in [27] that for the base f i e l d ~ of complex numbers the parameter curves of type Ap,p, D n (n > 4), E 6, E 7, E 8 are just F.Kleins simple curve s ingu la r i t ies [23] PI(C)/G ar is ing from the action of the polyhedral groups on the project ive l ine PI(C). Another poss ib i l i t y - which holds true for each a lgebra ica l ly closed base f i e l d k - is to consider the parameter curves for tame hereditary algebras as the project ive l ine P1(k) together with a configuration of singular points control led by a Dynkin diagram. For a detai led account on th is point of view we refer to [35].

Section 2 introduces the category coh(X) as the quotient category ~(A)/~Fo(A) of a l l f i n i t e l y presented abelian group valued functors on preprojective r ight A-modules modulo f i n i t e length functors. We note that the de f in i t ion of X and the in terpretat ion of F/~ o as a category of sheaves is postponed unt i l Section 5.

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Theorem 2.1 establishes a fact fami l ia r from sheaf theory [ I ] , namely cob(X) is a Krull-Schmidt category moreover has morphism spaces which a l l are f i n i t e dimensional over k. Section 3 establishes that F/IF ° = coh(X) has Auslander-Reiten sequences (Theorem 3.1). Section 4 contains the i n te r re la t i on between coh(X) and mod(A) discussed before (Theorem 4.10). We want to point out that also A. H. Schofield deserves f u l l c red i t for most of the results presented in th is Section.

Section 5 gives the sheaf theoret ic explanation of ~ o based mainly on the results of j o i n t work with D. Baer and W. Geigle [5]. For instance the f i e l d of rat ional functions K(X) of X is determined by mod(K(X)) = coh(X)/COho(X), where - as before - COho(X) denotes the category of a l l f i n i t e length (coherent) sheaves on X. A l te rna t i ve ly , K(X) arises as the endomorphism ring of the unique inde- composable tors ion- f ree d i v i s i b l e A-module [31] or as a sui table ring of f ract ions ( in the sense of [14]) of morphisms between preproject ive A-modules [5].

Section 6 deals with the case of an a lgebra ica l ly closed base f i e l d k. Based on

the c lass i f i ca t ion of indecomposable A-modules by Nazarova [36J and Donovan, Fre is l ich [37], see also Dlab, Ringel [9] we characterize the parameter curves

X(A) by means of the i r s ingu lar i ty type A(X) , which is always a Dynkin diagram

(Theorem 6.2). Section 7 contains an account on the d iv isor theory of X" for

the case of an a lgebra ica l ly closed base f i e l d . As a resul t , a complete system

of preproject ive A-modules of defect - I together with a complete system of

pre in ject ive A-modules of defect I forms an abelian group, the Picard group

Pic(X) , which describes the extensions of preproject ive by regular modules and

turns out to be isomorphic to the d iv isor class group of X (Proposit ion 7.4).

Pic(X) is abelian of rank one and general ly has torsion elements.

For a general account on noncommutative algebraic geometry we refer to [28]. We have t r ied however to fo l low the classical ' (commutative) treatment - as exposed for instance in [20] - quite c losely. In our judgement the reader w i l l f ind no d i f f i c u l t i e s to apply the classical notions sheaves, cohomology, d iv isor theory etc. to the present context. Only minor modif ications w i l l be needed. On the other hand, the notion of a quotient category, developed by J.-P. Serre, A. Grothendieck, P. Gabriel [12, 29] is an indispensable tool for the present treatment. In fact , a main motivation for th is research was to 'expla in ' the resul t due to W. Geigle [15] that , for A tame heredi tary, the category of a l l f i n i t e l y presented abelian group valued functors on mod(A) has Krull dimension two, where Krull dimensic, n, in a sense near to [12], is defined with reference to quotient categories.

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2. Indecomposable coherent sheaves

Throughout th is section A is a hereditary Ar t in algebra, assumed to be connected and not of f i n i t e representation type. Let

F (A) = f .p . (prep(A°P),Ab) ,

resp. ~o(A) = f . l . (prep(A°P),Ab) , denote the category of a l l addi t ive functors, which are f i n i t e l y presented, respec- t i v e l y of f i n i t e length, from the category of preproject ive r i gh t A-modules to the category of abelian groups. We note that prep(A °p) has internal cokernels, thus is

coherent. Hence ~(A) is an abelian category. Due to the existence of Auslander-

Reiten sequences in prep(A°P), Fo(A) is contained in F(A). Moreover, Fo(A) is

closed in F(A) under the formation of subobjects, quotients and extensions, i .e .

Fo(A) is a Serre subcategory of F(A). We are now going to invest igate the quot ient category (cf . [12])

m ~ o = F(A)~Fo(A)

which - in the tame case - has the nice in terpre ta t ion as the category coh(X) of coherent sheaves on the parameter curve X = X(A) of A (see section 5).

We recal l that F/F o is an abelian k-category, again. Moreover, there is a canonical exact functor

~: F - - , F / F o , F ~ F

which is a b i jec t ion on objects. Morphisms in fo rmu I a

HO~F/Fo(F,G) = I im F/F ' , G' E F °

This formula also explains how ~ acts on morphisms of IF. For fu r ther information on quot ient categories we refer to [12,29].

F/F o are defined by means of the

Hon~(F',G/G').

In the previous de f in i t i ons i t is often convenient to replace prep(A °p) by the f u l l subcategory P(A °p) of a complete system of pairwise nonisomorphic indecompo- sable preproject ive r i gh t A-modules. For instance, for P preproject ive, the functor Ex t (P , - ] , the res t r i c t i on of Ext , (P, - ) to prep(A°P), has f i n i t e support in P(A°P), since DExtI(p'x)A = HomA(X,DTrP) 7 3 ] . Hence E~'t(P,-] -- 0 in F ~ o . S im i la r l y , for A in mod(A°P), Hom(A,-] w i l l denote the res t r i c t i on of HomA(A,-) to preproject ive modules. Clear ly , Hom(A,-] = 0 i f A has no preproject ive d i rec t summand [ 2].

As is well known [ 2] , we may assume that A is a f in i te-d imensional algebra over some commutative f i e l d k. The main resu l t of th is section reads as fo l lows:

2.1. Theorem. For each F,G i n IF /F ° the k-vectors~ace Hon~/Fo(F,G ) fis f i n i t e

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dimensional, hence ~/F o is a Krul l-Schmidt category. Moreover the 'sheaves'

~m(P,- ] wi th P indecomposable prepro ject ive, E~t(Q,-] with Q indecomposable p re in jec t i ve , E~t(R,-] wi th R indecomposable regular

const i tu te a complete l i s t of indecomposable objects in FflF o.

For the proof of (2.1) i t is essential to know that F/~: o is not only abel ian, but carr ies the extra st ructure of a t rans la t ion category:

The Auslander-Reiten functors 'dual of the transpose' DTr and 'transpose of the dual ' TrD are inducing autoequivalences (Use Lemma 2.2 a),b) for the proof that

• , - I preserve f i n i t e l y presented functors. )

: ~ o ~ O ' ~ ~ (F o DTr) ~ - I T : F/IF o ~F / ] : O, ~ H (F o TrD) ~

of ~/@~o' which are ' inverse' to each other, see [ 4, prop. 2.1]. Guided by Riedtmann's concept of a t rans la t ion quiver [29], we refer to the pair (~/]Fo,%) as a t rans la t ion category and to T as the t rans la t ion functor of F/F o. (As we w i l l see la te r (Theorem 3.1) ~ o has, in fac t , Auslander-Reiten sequences, and serves as Auslander-Reiten t rans la t ion for F ~ o . )

By i t e ra t i on , we may define n : ~/~o ' twisted sheaves' by the formula

F(n) = T-n(F)

for each F in F ~ o.

FftF o for each integer n, and hence

2.2. Lemma. Suppose A j s indecomposable i n mod(A°P). - I

a) % H~m(A,-] ~ H~m(DTrA,-] i f A is not pro ject ive.

b) #I H~m(A,-] ~ E~t(DAt, - ] i f A is pro ject ive.

c) ~I E~t(A,-] ~ ~ t ( D T r A , - ] .

Here, A t denotes the A-dual HomA(A,A) of A. We also note that , for A inde- composable pro ject ive, DA t is the in jec t i ve hul l of the simple top A/rad A of A. Hence A ~ DA t is the Nakayama permutation (see [13]) .

Proof. Concerning a) and c) note that Hom(A,TrDP) = Hom(DTrA,P) and ExtI(A,TrP) = ExtI(DTrA,P) hold i f A is not pro ject ive (P E prep(A°P)). For A pro ject ive, assert ion b) fol lows from Hom(A,TrDP) = TrDP @ A t = Ext(DP,A t ) = Ext(DAt,p). D

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We reca l l that Kan extension e: (prep(A°P),Ab) ~ (mod(A°P),Ab) along prep(A ° p ) ~ mod(A °p) is def ined as the l e f t ad jo in t of the r e s t r i c t i o n (mod(A°P),Ab) ~ (prep(A°P),Ab). e is r i gh t exact and car r ies Hom(P,-] to Hom(P,-) fo r each prepro jec t i ve module P. Consequently eExt (A, - ] = Ext (A, - ) fo r each module A.

2.3. Lemma. Kan extension preserves exactness o f sequences 0 ~ F ~ G ~ H ~ 0 in ~ , provided H ~ H~m(P,-] i n ~/]F ° f o r some prepro jec t i ve A-module P.

Proof. Let S CF be a simple functor concentrated at Po C P(A°P). The Auslander- Reiten sequence 0 ~ Po ~ PI ~ P2 ~ 0 leads to a p ro jec t i ve reso lu t ion

0 ~ Hom(P2,- ~ Hom(P1,-] ~ Hom(Po,-] ~ S ~ 0

of S. This sequence remains exact under Kan extension e, hence the l e f t der ived functors Lie o f e are zero on S fo r i > I . Hence Lie(E) = 0 i f E c o and i > I .

Turning to the general s i t u a t i o n , by the d e f i n i t i o n of morphisms in ~/F o an isomorphism H ~ H~m(P,-] comes from a morphism ~: H' ~ Hom(P,-] in F , where H/H', Ker ~, Coker m are a l l in F . (Observe that Hom(P,-] has no simple sub- o func tors) . This impl ies (Lie)(H) = 0 fo r i > I . o

The main technical step in the proof of (2.1) is the next p ropos i t ion .

2.4. Propos i t ion. For every A,B E mod(A °p) we have

= 0 . Ex ~ o (H~m(A,-], EFt(B,- ] )

Proof. Assume that the sequence

q: 0 ~ E~t(B,- ] ~ ~ ~ ~m(A , - ] ~ 0

is exact . By the d e f i n i t i o n of F/~F o, ~ is induced by a morphism f : U ~ F/F' in ~ , where Ex t (B , - ] /U , F' and Ker f are in ~o" We may assume that B has no prepro jec t i ve d i r ec t fac to r . Therefore Ext (B, - ] is exact on Auslander-Reiten se- quences in prep(A°P). Thus Ex t (B , - ] , hence to an exact sequence

~: 0 ~ U ! F / F ' ~ G ~ 0

wi th ~ isomorphic to n.

Since G ~ H~m(A,-], and app l ies . Therefore leads to a morphism

U, has no simple subfunctors. This leads

A is p repro jec t i ve wi thout loss of gene ra l i t y , Lemma 2.3 eU e ( f l e (F /F ' ) is a monomorphism. Kan extension of U ~ Ext (B, - ]

eU e ( j~ Ex t (B , - ) , which extends to h: e (F /F ' ) ~ Ext (B, - ) by

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i n j e c t i v i t y of Ext(B,-) in (mod(A°P),Ab). (Note that DExt(B,-) = Hom(-,DTrB) is pro jec t ive) . With k: F/F' ~ Ext(B,- ] denoting the res t r i c t i on of h to prep(A°P), we have k o f = j , hence ~ ° ~ is an isomorphism in ~ o ' and n s p l i t s .

We use the notat ion mo~>(A) for the f u l l subcategory of mod(A), consist ing of a l l f i n i t e dimensional A-modules without any preproject ive indecomposable d i rec t factor . (These modules have been termed ' tors ion modules' by Ringel [31]. Because of the presence of too many tors ion theories in our discussion, we are forced to avoid th is terminology, here.)

2.5. Proposit ion. (cf . [ 5]) The functor

~: mod#(A°P) °p ~ ~/~F o, A ~ E~t(A,-]

is f u l l and f a i t h f u l . Moreover ~ preserves exactness of sequences 0 ~ A' ~ A ~ A" ~ 0 with A,A',A" in mod#(A°P).

o _ _

Proof. Since Hom(A,-] = 0 for each A in moom(A°P), the exactness property in question fol lows t r i v i a l l y . For non-zero A in modm(A °p) we in fe r from DExt(A,P) = Hom(P,DTrA) that Ext(A,- ] has i n f i n i t e support in P(A°P), hence ~(A) ~ 0. In combination with the exactness of # th is proves that # is f a i t h f u l . (Observe that modm(A °p) is closed under the formation of quotients in mod(A°P)).

I t remains to prove that ~ is f u l l . Since Ext(A,- ] has no simple subfunctors, each ~: EFt(B,-] ~ ~ t ( A , - ] , with A,B in modm(A°P), is induced by some f : U ~ Ex t (A , - ] , where U is a subfunctor of Ext(B,- ] with Ext(B,- ] /U of f i n i t e length. We i n fe r from Lemma 2.3 that Kan extension to mod(A °p) preserves the inc lus ion U~ Ext (B, - ] . By i n j e c t i v i t y of Ext(A,-) in (mod(A°P),Ab), f therefore extends to a morphism f : Ext(B,-) ~ Ext(A,- ) . Now the classical formula [22]

Hom(Ext(B,-), Ext (A, - ) ) = Hom(A,B)

proves that f , hence a, is induced by some u: A ~ B. (As usual the notat ion Hom refers to the morphisms in the pro jec t ive ly stable category).

2.6. Corol lary. Hom~F/~Fo(E~t(A,-], ~m(B,- ] )= 0 for each A,B in mod(A°P).

Proof by combination of (2.5) and (2.2).

Proof of theorem 2.1. Proposit ion 2.5 in combination with Lemma 2.2 proves that the 'sheaves' H~m(P,-], with P indecomposable preproject ive, and E~t(A,- ] , with A inde- composable in modm(A°P), a l l have local endomorphism rings and f i n i t e dimensional

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morphism se ts .

I t t h e r e f o r e s u f f i c e s to prove t h a t

= H~m(mer f , - ] @ E ~ t ( C o k f , - ] ,

p rov i ded Hom(Po,_ ] -o~ Hom(P1,_ ] ~ F ~ 0 is a f i n i t e p r e s e n t a t i o n o f F (w i t h Po'

PI p r e p r o j e c t i v e ) . In f a c t , decompos i t i on o f the exac t sequence

O ~ P-* P I f Po ~ A ~ 0

i n t o s h o r t exac t sequences 0 ~ P ~ PI u p -~ O, 0 ~ P v Po

and P p r e p r o j e c t i v e , leads to a d iagram

~ A ~ O, w i t h f : v o u

0 0 + +

0 ~ H~m(A,-] ~ a~m(Po, - ] - ° ~ H~m(P,-] ~ E F t ( A , - ] ~ 0 I1 II +-ou +

0 ~ H~m(A,-] ~ a~m(Po, - ] - ° ~ a~m(P1, - ] ~ ~ ~ 0 + +

H~m(P,- ] ~ G

0 0

w i t h exac t rows and columns in ~/]F o. The Ker-Coker-Lemma proves the e x i s t e n c e o f an

exac t sequence 0 ~ E ~ t ( A , - ] ~ F ~ H~m(P,- ] ~ 0 which s p l i t s by P r o p o s i t i o n 2 .4 . []

3. A u s l a n d e r - R e i t e n sequences in F / F °

By Theorem 2 .1 , F/~= ° i s a K r u l l - S c h m i d t c a t e g o r y . I t is t h e r e f o r e n a t u r a l t o ask

whe ther ~/]F ° has a l so A u s l a n d e r - R e i t e n sequences:

3 .1 . Theorem. The c a t e g o r y ~/~=o o f ' cohe ren t sheaves ' has A u s l a n d e r - R e i t e n

sequences. More p r e c i s e l y , f o r each indecomposable

Re i ten sequences

0 ~ F ~ X ~ F(1)

0 ~ F ( - I ) ~ X ( - I ) ~ F

Hence the t r a n s l a t i o n f u n c t o r % o f F/~c o f o r IF/ ~ .

0

' shea f ' F t h e r e a re Aus lande r -

~ 0

-* O.

serves as A u s l a n d e r - R e i t e n t r a n s l a t i o n

Consequen t l y , a non -ze ro ' Isheaf ' cannot be p r o j e c t i v e nor i n j e c t i v e in ~/~c o.

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Proof. Up to t r a n s l a t i o n we may assume tha t F = ~ t ( A , - ] w i th A indecomposable

in moc~>(A°P). (Lemma 2 .2 ) . The Auslander-Rei ten sequence 0 ~ DTrA ~ B ~ A ~ 0 in

mod(A °p) leads to an exact sequence

0 ~ E~t (A , - ] ~ E~t (B , - ] ~ E~t(DTrA~-] ~ 0

in ~/IF o which has indecomposable end-terms (Thm.2.1) and is n o n - s p l i t (Prop. 2 .5 ) .

Using the c l a s s i f i c a t i o n of indecomposable 'sheaves' @ by Theorem 2.1, every non-

isomorphism E~t (A, - ] ~ G extends to E~t (B, - ] as a consequence of Co ro l l a r y 2.6 and Propos i t ion 2.5. n

The above proof moreover shows:

3.2. P ropos i t i on . The f u l l embedding

#: modm(A°P) °p ~ ~/IF o, A ~ E~t (A, - ]

preserves Auslander-Rei ten sequences.

3.3. Lemma. For E E ~o and P p r e p r o j e c t i v e , we have Ext , (E, Hom(P,-])

f o r i = 0,1.

= 0

Proof. We may assume tha t E is s imple , hence E has a p r o j e c t i v e r eso lu t i on

0 ~ Hom(P2,-] ~ Hom(P1,-] ~ Hom(Po,-] ~ E ~ O,

induced by an Auslander-Rei ten sequence 0 ~ Po ~ PI ~ P2 ~ 0 of p rep ro j ec t i ve

modules. Now apply Hon~(-, Hom(P,-]) and ca l cu la te homology.

The f o l l ow ing r e s u l t complements (3.2) and (2 .5 ) .

3.4. P ropos i t i on . The func to r

~: prep(A°P) °p ~ ~/IF o, P ~ HUm(P,-]

is a f u l l embedding and preserves exactness of sequences 0 ~ P' ~ P ~ P" ~ 0 p r e p r o j e c t i v e modules. Moreover, ~ preserves Auslander-Rei ten sequences.

o f

Proof. ~ is f u l l : A morphism ~: HUm(P,-] ~ ~ m ( P , - J , w i th P,P p r e p r o j e c t i v e , is induced by some morphism f : U ~ Hom(P,-] , where U c Hom(P,-] and Hom(P,-]/U has f i n i t e length . By Lemma 3.3, f extends to f : Hom(P,-] ~ Hom(P,-] , necessar i l y

induced by some u: P ~ P.

is f a i t h f u l : u: P ~ P has a f a c t o r i z a t i o n u = P ~ P' ~ P w i th v a mono- morphism and w an epimorphism. Assuming '{(u) = O, the diagram

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~(v) ~ m ( P , - ] . . . . . . . . H~m(P',-] , mx ' t (P/P ' , - ]

/

~(w) /

~dm(P,-]

, 0

has a commutative complet ion by some m: E~xt(P/P', - ] ~ H~m(P,-]. Since ~ = 0 by

Coro l l a r y 2 .6 , H~m(P',- ] = 0 hence P' = O, and u = 0 f o l l o w s .

Since Ex' t (P,- ] = O, f o r P p r o j e c t i v e , the requ i red exactness of ~ fo l l ows t r i v i a l l y . In p a r t i c u l a r , f o r each Auslander-Rei ten sequence 0 ~ P' ~ P ~ P" ~ 0 of

p r e p r o j e c t i v e modules, the induced sequence

0 ~ HUm(P",-] ~ H~m(P,-] ~ H~m(P',-] ~ 0

is exac t . I t i s , in f a c t , an Auslander-Rei ten sequence in ~/~=o: By the c l a s s i f i c a t i o n of indecomposable 'sheaves' G by Theorem 2.1, every non-isomorphism G ~ ~ m ( P ' , - ]

l i f t s to H~m(P,-] using Coro l l a ry 2.6 and the f ac t t ha t T is a f u l l embedding, n

3.5. Remark. Passing to a s l i g h t l y d i f f e r e n t con tex t , namely functors def ined g l o b a l l y

on mod(A), the study of f i n i t e l y presented functors means the study of shor t exact sequences in mod(A), w i th s i l l p le functors ( roughly) corresponding to Auslander-

Reiten sequences. Hence, in t h i s con tex t , ~o may be considered as the par t o f

determined by Auslander-Rei ten theory , and the passage from ~ to F ~ 0 means to f o rge t about Auslander-Rei ten sequences. In f a c t , i t was the main mot i va t i on f o r the

present research to de tec t , what happens behind tha t par t of representa t ion theory determined by Auslander-Rei ten theory . We re fe r to [15, 16] f o r a re la ted approach, mot ivated by P. Gab r i e l ' s i n t e r p r e t a t i o n of Kru l l -d imens ion [12 ] .

I t came to us as a surpr ise tha t ( in the context covered by Theorem 3.1) ~ /F ° has again Auslander-Rei ten sequences. Despite the f ac t t ha t the general s i t u a t i o n s t i l l needs i n v e s t i g a t i o n , t h i s phenomenon, which holds t rue regardless i f A is tame or w i l d , under l ines once more the u t i l i t y of the concept in t roduced by M. Auslander and I . Reiten in [ 3] a lso in the study o f r e p r e s e n t a t i o n - i n f i n i t e a lgebras.

4. Global sec t ions , cohomology and t i l t i n g from sheaves to modules

The aim of t h i s sect ion is to show tha t the category F /~ ° is complete ly c o n t r o l l e d by mod(A). With t h i s purpose in mind, we in t roduce the s t ruc tu re sheaf 0 = ~ t ( D A , - ] , and def ine g lobal sect ions and cohomology fo r F in ~/]F ° by means o f

209

F(X,F) = Hon~/]:o(O,F) , HI(x,F) = Ext~/JFo(O,F).

These are l e f t A-modules, since EndjF/~o(O) = A °p by Proposi t ion 2.5, and are in fac t

f i n i t e -d imens iona l . Concerning global sect ions, th is statement is covered by Theorem 2.1. For H I (x ,F ) , the asser t ion is contained in Proposi t ion 4.1. Moreover, £(X,O) = A A and HI (x ,o) = 0 (Lenrna 4 .2) .

I We note tha t - as before - ExfC/~:o(F,@) is def ined in the sense of Yoneda. We a lso

remark that our choice o f s t ruc ture sheaf is somewhat a r b i t r a r y . Since F ~ o is a t r ans l a t i on category, each O(n), f o r instance 0 ( - I ) = ~ m ( A , - ] , w i l l answer the same purpose. HUm(A,-] w i l l besides appear to be the more natural choice, judging from graded module theory , c f . [ 5].

The i n t e r r e l a t i o n between ~/JF o and mod(A) is best understood by means o f a t i l t i n g procedure. By Theorem 4.10, ~/JF o emerges from mod(A) by t i l t i n g a l l p re i n j ec t i ve A-modules to the ' l e f t hand side' of a l l remaining indecomposable A-modules, and inser t ing some add i t i ona l morphisms. However, there is no t i l t i n g module in mod(A) e f f ec t i ng th is ' t i l t i n g from modules to sheaves'. The s i t ua t i on is be t te r fo r the inverse process, since the s t ructure sheaf serves as a ' t i l t i n g module' (Remark 4.11). The reader w i l l observe the in f luence of the t i l t i n g theor ies developed by Brenner-But ler [ 7 ] , Happel-Ringel [19] , Bongartz [ 6 ] , Tachikawa [34] , though the present s i t ua t i on is not covered by these i nves t i ga t i on in a formal sense. The re l a t i onsh ip to Happel's new i n t e r p r e t a t i o n of t i l t i n g s , presented at the conference [18] , seems to be more c lose, but s t i l l needs fu r the r ana lys is ,

We would l i k e to po in t out , that a lso A. H. Schof ie ld deserves f u l l c red i t f o r a l l the resu l ts contained in th is sect ion. In fac t , the p ic ture on sheaves, global sections and cohomology presented here, is the resu l t of discussions wi th him during the conference, in a j o i n t e f f o r t to answer a question of P. Gabriel concerning the complete s t ruc ture o f cob(X). (At th is t ime, only the s t ruc ture of COho(X), the category o f a l l f i n i t e length sheaves, and of coh(X)/COho(X) were at the author 's d isposa l , c f . [ 5 ] ) . We a lso note that A. H. Schof ie ld has a d i f f e r e n t (but apparent ly equ iva len t ) procedure, a t tach ing sheaves to (tame) hered i ta ry a lgebras, based on the process of a t taching universal inverses to cer ta in maps of r ings and modules. Also, the dec is ive Proposi t ion 4.1 is his observat ion.

I f there is an exact sequence O n ~ F ~ O, fo r some n, F is ca l led generated by global sect ions. An equ iva lent asser t ion is the existence of an exact sequence 0 ~ O' ~ 0" ~ F ~ O, where 0 ' , 0" are in add(O), i . e . are d i r e c t factors o f some O n . Among indecomposable 'sheaves' exac t l y the EFt(A,- ] with A indecomposable p re in j ec t i ve or regular are generated by global sect ions. Moreover, the category

210

( ~ o ) m of 'sheaves' generated by global sections is closed under extensions. A l l these assertions fo l low from Theorem 2.1 and Corol lary 2.6.

By dua l i t y , the category modm(A°P) Op is equivalent to the category mode(A) of f in i te-d imensional l e f t A-modules without pre in jec t ive indecomposable d i rec t factor .

4.1. Proposit ion (Scho f i e l d ) . The functor 'global sect ions' induces an equivalence F: (F~o)m ~ mode(A), F~F(X,F) from the category of 'sheaves' generated by global sections to the category of l e f t A-modules without pre in jec t ive indecomposable summands. Moreover, F(X,0) = A.

Proof. The functor ~' : mode(A) ~ (E/~=o)m, A ~ E~t(DA,-] is an equivalence of categories by Proposition 2.5. And F is an inverse to # ' , as fol lows from

(2.5) Hom(~t(DA,-] , E~t(DA,-]) Hom(DA,DA) = Hom(A,A) = A fo r A in mode(A).

I t fol lows from the next Lemma that F is moreover exact on exact sequences 0 ~ F' ~ F ~ P' ~ 0 with F' generated by global sections.

4.2. Lemma. HI(x,F) = 0 i f F is generated by global sections.

Proof. We may assume I f A is not i n j ec t i ve , by t rans la t ion (2.2) and Proposit ion 2.4 we ar r ive at

Ext~=/Fo(O, E%t(A,-]) = EXtlF/~=o(H~m(A,-], E%t(TrDA,-]) = 0.

For A i n j ec t i ve , P = (DA) t is pro ject ive, hence by (2.2)

Ext0=/iFo(O, EFt(A,- ] ) = Ext~=/jFo(~m(A,-], ~m(P, - ] ) .

This group is also zero by an easy var iant of Proposit ion 2.4. (Observe that for a project ive P, the functor Hom(P,-) is i n jec t i ve in (mod(A°P), Ab)).

F = E~t(A,- ] , with A indecomposable regular or p re in jec t ive .

To deal with sheaf cohomology properly, we w i l l need an addit ional property of the functor ~ of Proposition 2.5.

4.3. Proposit ion. For each A,B E mod~(A°P), # induces an isomorphism

Ext~(A,B) ~ Ex t~ /~o(~ t (B , - ] , E~t (A, - ] ) , n ~ # ( ~ )

ev ident ly functor ia l in A and B.

Proof. We recal l that (F/~o)m is closed under extensions. By Proposition 2.5 i t therefore remains to show that for each exact sequence

211

¢(g) # ( f ) N: 0 ~ E~t(A,- ] , E~t(C,-] , E~t(B,- ] ~ 0

the corresponding sequence q: 0 ~ B ~ C ~ A ~ 0 is exact. Applying global sect ions to N, we obtain from (4.2) the exact sequence 0 ~ TrA T -~T rC T r f TrB ~ O, which proves the asser t ion . []

4 .4. Proposi t ion (Serre d u a l i t y ) . For each isomorphism

t~- (F,G) ~ HOn]F/~o(G, TF) DEx /.Fo

which is f unc to r i a l in F and G.

F,G i n ]F/]F o there is a canonical

Proof. Up to t r ans l a t i on we may assume F = E~t (B , - ] , G = E~t(A,- ] where A and B are w i thout p repro jec t i ve indecomposable d i r ec t fac to r . By (4.3) DExt(F,G) D(Ex t I (A ,B) ) , DExtI(A,B) ~ Hom(B,DTrA) canon ica l l y [13],and f i n a l l y Hom(B,DTrA)

Hom(~IG,F) fo l lows from Proposi t ion 2.5. []

4.5. Coro l la ry . The 'sheaf ' 0 ( - I ) = FFom(A,-] have

D HI(x,F) ~ HomlF/~o(F, 0 ( - I ) )

f u n c t o r i a l l y in F E~/~=o.

t~ (F , - ) and Ext~/~o(-,G) 4.6. Coro l la ry . Ex /iF °

4.7. Coro l la ry . For an indecomposable 'sheaf ' equ iva len t .

a) F is generated by global sect ions.

b) F(X,F) , O.

c) HI(x ,F) = O.

serves as a dua l i z i ng sheaf, i . e . we

are r i g h t exact func tors .

F, the fo l l ow ing asser t ions are

Proof. (a) ~ (b) From (2.1) and (2.6) we deduce F = E~t(A,- ] f o r some A, and F is generated by global sect ions. (a) ~ (c) is Lemma 4.2. (c) ~ (a): I f F is not generated by global sect ions, F = H~m(P,-] fo r some prepro jec t ive module P as a consequence of (2 .1 ) . By (3.4) Hom(H~m(P,-], H~m(A,-]) = Hom(A,P) # 0 and the asser t ion fo l lows from (4 .5) . []

4.8. Propos i t ion. The cohomology func to r

H I ( x , - ) : F /~ o -* mod(A), F ~ HI(x ,F)

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induces an equivalenc e between the category (F/~o)~ of 'coherent sheaves' F with- out non-zero global sections and the category prej(A) of a l l preinject ive l e f t A- modules. Moreover, HI(X, -) induces an isomorphism

E~/IFo(H~m(DQ,-], H~m(DQ,-]) ~ Ext~(Q,Q)

for each pair Q,Q of preinject ive l e f t A-modules.

Proof. By means of the dual i ty functor D, the category prej(A) is dual to prep(A°P). According to (3.4. ) , (2.1) and (2.6) the functor

~' : prej(A) ~ (~/]Fo)~, Q~ H~m(DQ,-]

is an equivalence of categories. Moreover, for Q pre in ject ive, we in fer from (4.5), (3.4) that HI(x, H~m(DQ,-]) = D Hom(~m(DQ,-], H~m(A,-]) = Q. This proves the f i r s t assertion. Since ( /~c O) is closed under extensions, the las t assertion follows from the previous argument and Proposition 3.4. m

4.9. Remark. We may rephrase (4.8) par t ly as fol lows: Each 'coherent sheaf' F/T ° without non-zero global sections is determined by i t s cohomology, since

F in

F m H~m(DHI(x,F),-]. S imi lar ly we in fer from (4.7) and (4.1) that every 'coherent sheaf' O with t r i v i a l cohomology is determined by i t s global sections, since G ~ E~t(DZ'(X,G),-]. As a resu l t , each indecomposable F in ~/IF o is e i ther deter- mined by global sections or by cohomology (Corollary 4.7).

We also note that we may get r id of excessive dual izat ions, replacing Hom and Ext by tensorproduct and Tor. For instance, in the s i tuat ion above, we have

T E~t(D?(X,G),-] = (- @A £(X'G))N and

T ~m(DHI(X,F), - ] = Tor~(-,HI(x,F)) ~.

In order to summarize the previous results in a suggestive form, define A and B as the fu l l subcategory of mod(A) consistin£ of a l l p re in ject ive (a l l preprojec- t i ve or regular) indecomposable modules, respect ively.

4.10. Theorem. For each connected hereditary Art in algebra of i n f i n i t e representation type, the category ind(F/iF o) of indecomposable 'coherent sheaves' i s equivalent to the category $ constructed from the category ind(A) of indecomposable l e f t A-modules as follows

' the objects of $ are the objects of ind(A)

Hom$(X,Y) = HomA(X,Y) i_f X,Y are both in A or both in B and Hom$(X,Y) Ext~(X,Y) otherwise.

213

the composition in $ is given by the Yoneda-composition of Ext~(i = 0,1).

By transport of structure, the t ranslat ion % of F ~ o acts on obje_cts X o f $ by ~(X) = DTrX, i f X is non-projective, and %(X) = DX t i f X is project ive. Moreover, we have Ext$(X,Y) = ExtA(X,Y) provided X,Y are both in A or both in 8, and Ext$(X,Y) = HomA(X,Y) otherwise. Dual i ty DExt$(X,Y) = Hom$(Y,TX) holds func to r ia l l y in X,Y. The addit ive completion add(S) of $ is abelian. Assuming A add i t iona l ly basic, the d i rect sum of a complete system of indecomposable project ive ( in jec t ive) A-modules in add(S) serves as the structure sheaf O, the dualizing sheaf 0( - I ) , respect ively.

Proof. Define #: $ ~ i n d ( ~ o) on objects by #(X) = HomA(DX,-] for A in A and @(X) = E~{t(DX, - ] for A in 8. The action of # on morphisms u: X ~ Y is given by Proposition 3.4 i f X,Y are both in A, by Proposition 2.5 i f X,Y are both in B. For A E A, B E B we have Hom$(B,A) = Ext~(B,A). = D HomA(A,DTrB) = O. A variant of the proof of Proposition 2.5 proves that Yoneda's lemma defines an isomorphism Hom$(A,B) = Ext~(A,B) ~ Hon~/IFo(H~m(DB,-], E~t(DA,-]), functor ia l in A E A, B E 8.

This defines # as a functor, and shows in combination with (2.6) that # is an equivalence of categories. []

The relat ionship between mod(A) and ~ o may hence be depicted as fol lows. Here, P, R, J stand for a complete system of preproject ive, regular or prein ject ive indecomposable l e f t A-modules, respect ively.

ind(A): ~ P

J

ind (~=/iFo) : J C Figure I

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I t is understood tha t in both cases there are only (non-zero) morphisms from the l e f t hand to the r i g h t hand side.

4.11. Remark. The comparison of (4.10) wi th the t i l t i n g theor ies fo r A r t i n algebras [ 6, 7,18,19,34] shows tha t the ' s t ruc tu re sheaf ' 0 serves as a t i l t i n g module f o r

~/~=o' and mod(A) emerges from ~/IF o by t i l t i n g wi th O. Let us reca l l from [ 6,19] t h a t , f o r an A r t i n algebra A, a module T is ca l led a t i l t i n g module i f i t s a t i s f i e s the fo l l ow ing three proper t ies : (a) Ex t I (T ,T ) = O, (b) E x t ' , - ) = O, (c) There is an exact sequence 0 ~ A ~ T' ~ T " ~ 0 wi th T ' ,T " in add(T). Also reca l l tha t (c) may be replaced by the more convenient cond i t ion ( c ' ) s ta t i ng tha t the indecomposable d i r e c t fac tors o f T cons t i t u te a basis f o r the Grothendieck group Ko(mOd(A)).

Obviously, cond i t ion (c) does not make sense fo r F/IF o. This prevents tha t the present s i t u a t i o n is covered by [ 6,19] in a s t r i c t sense. Note however, tha t 0 s a t i s f i e s the analogues of condi t ions (a) , ( b ) , ( c ' ) f o r ~/IF o, as fo l lows from (4 .2 ) , (4.6) and (5 .2) . This suggests to develop t i l t i n g theory in a framework not r es t r i c t ed to modules over A r t i n algebras, based e n t i r e l y on condi t ions (a) , (b) , (c ' ) .

We close th i s sect ion w i th another property, f a m i l i a r in a lgebra ic geometry o f p ro jec t i ve spaces:

4.12. Propos i t ion . For each 'coherent sheaf' F ( i n F/F o) there is an in teger N, depending on F, such that F(n) is generated by 31obal sect ions fo r each n > N.

5. Euler c h a r a c t e r i s t i c and rank

Let us denote by

dim: mod(A) ~ Ko(mOd(A)), A ~ dim A

[ ] : F/F o ~ Ko(~/Fo),F ~ IF]

the natural maps of mod(A) and ~/JF ° in to t h e i r respect ive Grothendieck groups. For each F in ~ o i t s Euler c h a r a c t e r i s t i c in Ko(mod(A)) is def ined as

x(F) = dim F(X,F) - dim HI (x ,F ) .

~ecal l f o r t h i s purpose tha t F(X,F) and HI (x ,F) are f i n i t e l y generated l e f t A-modules. We i n f e r from (4.6) tha t × is add i t i ve on shor t exact sequences, hence induces a morphism h: Ko(F/~: o) ~ Ko(mOd(A)) o f abel ian groups s a t i s f y i n g h ( [F ] ) = ×(F). We also note tha t both Ko(mod (A)) and Ko(~/~: o) are equipped wi th

215

b i l i n e a r forms, s a t i s f y i n g

< dim A, dim B > = dimkHom(A,B) - dimkExt(A,B)

< IF ] , [G] > = dimkHom(F,G) - dimkExt(F,G)

f o r a l l A,B in mod(A) and a l l F,@ in ~ o ' respect ive ly . (Use Coro l la ry 4 .6) . By the d e f i n i t i o n of the b i l i n e a r form of KoOF/JFo), the t r a n s l a t i o n func to r ~ of ~/~o induces an isometry of Ko(~/JFo), also denoted ~, hence s a t i s f y i n g T ( [F ] ) = [TF]. The f o l l ow ing propos i t ion describes a phenomenon f a m i l i a r from t i l t i n g theory [ 6,19]:

5.1. Proposi t ion. The Euler c h a r a c t e r i s t i c induces a group isomorphism

h: Ko(F/IF o) ~ Ko(mod(A))

(a) h ( [F ] ) = x(F)

(b) < x , y > = < h (x ) , h(y) >

(c ) h ( [ ~ F ] ) = c h ( [ F ] )

where c: Ko(mod(A)) ~ Ko(mOd(A)) t r a n s l a t i o n func to r of ~ /F o.

s a t i s f y i n g

fo r each F i n ~/]F o,

fo r each x ,y inn Ko(~/Fo),

fo r each F i n ~/]Fo,

denotes the Coxeter t ransformat ion and the

Proof. As fo l lows from g l .d im A < I there is a group homomorphism k: Ko(mod(A)) ~ Ko(F/~= o) s a t i s f y i n g k(di_m A) = [E~t (DA,- ] ] - [~m(DA, - ] ] fo r each A in mod(A). I t fo l lows from (4.9) tha t k o h ( [F ] ) = IF] f o r each indecomposable 'sheaf ' F, and h o k(dim A) = dim A fo r each indecomposable A-module A. The v e r i f i c a t i o n of (b) and (c) is also s t ra igh t fo rward from (4.10). n

By means o f (5.1) the Euler c h a r a c t e r i s t i c ×: F/IF ° ~ Ko(mOd(A)) def ines Ko(mOd(A)) as ' the ' Grothendieck group of ~/~o" Ko(m°d(A)) is f ree abel ian w i th a 'natura l bas is ' cons is t ing of d imS I . . . . . dim S d f o r a complete system S I . . . . . S d of simple A-modules. With respect to the above basis we may and w i l l i d e n t i f y X(F) w i th an element of ~d , whenever i t turns out to be convenient.

n I n r I f 0 = 01 @ . . . @ O r is a decomposition of the ' s t ruc tu re sheaf ' 0 in to

indecomposable d i rec t f ac to rs , i t fo l lows from (4.1) tha t F(X,O I ) . . . . . F(X,O r ) is a complete system of indecomposable p ro jec t i ve A-modules. Note tha t x(O i ) = F(X,Oi ) . Hence by hered i ty of A we have:

5.2. Coro l la ry . Let 01 . . . . . 0 d be a complet_ee system of (pai rwise nonisomorphic) d i r ec t fac tors of the ' s t ruc tu re sheaf' 0. Then d is the number of isomorphism classes of simple A-module§. Moreover the system X(O I ) . . . . . X(0 d) is a £ - b a s i s f o r

Ko(mOd(A)) = Ko~/Fo) .

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5.3 We assume from now on that the algebra A is in addi t ion tame and as before hereditary and connected. By de f i n i t i on A is tame [ 9,31] i f the b i l i nea r form on Ko(mOd(A)), considered before, leads to a posi t ive semi-def in i te quadratic form q on the rat ional Grothendieck group ~ @~ Ko(mod(A)). We refer to the work of V. Dlab and C. M. Ringel [ 9,10,31] for a detai led account on the representation theory of tame heredi tary algebras. Here, we only note the fo l lowing fac ts , which are charac ter is t i c for the tame case (cf . [31 ] ) :

( I ) There exists a non-zero ~ - l i n e a r map 6: Ko(mOd(A)) ~ ~ , cal led the defect, which is invar ian t under the Coxeter transformation c. We may choose ~ in such a way that there is a project ive A-module P, necessari ly indecomposable, with ~(P) = - I . (Notation: ~(A) = 6(dim A) for A in mod(A)).

(2) By means of ~, preproject ive, pre in jec t ive and regular A-modules A are characterized by 6(A) < O, 6(A) > 0 and 6(A) = O, respect ively.

(3) The category reg(A) of regular A-modules is an exact subcategory of mod(A), hence an abelian length category, and each regular module R has f i n i t e length (= regular length of R) in reg(A).

(4) reg(A) decomposes uniquely into a coproduct reg(A) = I I regx(A) x C X

of un iser ia l subcategories regx(A). X may be choosen to be the set of a l l regular Auslander-Reiten components. For x in X, regx(A) is the addi t ive closure of the modules contained in the component x.

(5) Each 'component' regx(A) has only a f i n i t e number n x of (isomorphism types of) simple objects (= simple regular modules), a l l conjugate under the Auslander- Reiten t rans la t ion DTr, which acts on reg(A) hence on each 'component' regx(A) as a category equivalence. Moreover, with the exception of at most three 'components' Xl,X2,X 3, cal led non-homogeneous, we have n x = I . The remaining components, and the modules contained there in , are cal led homogeneous.

(6) There is a simple regular module S, necessari ly homogeneous, such that

~(A) = dim Hom(S,A) - dim Ext(S,A) End(S) End(S)

for each A-module A. Moreover, i f h = dim S, for each x £ X, the sum of the dimension vectors of a complete system of simple objects in regx(A) is an integer mul t ip le of h.

(7) X, the set of a l l regular Auslander-Reiten components, has the a l te rnat ive descr ipt ion as the set of DTr-orbits of (isomorphism classes) of simple regular

217

A-modules.

5.4. We are now going to sketch, how to def ine a sheaf 0 x dimensional k-£1gebras on X and how to i n te rp re te ~/]F o as the ca.tegory coh(X) o f coherent OX-mOdules. For a de ta i l ed account we re fe r to the expos i t ion in [ 5]. The po in t o f view, adapted here, d i f f e r s however from [ 5] since a l l ' l o c a l i z a t i o n s ' are e f fec ted by the format ion of quot ien t categor ies in the sense of Grothendieck [ t 2 ] , whereas in [ 5] the 'ca lcu lus o f f r ac t i ons ' in the sense of [14] dominates.

The approach is based on the fo l l ow ing resu l t from [ 5] , c f . a lso [15] fo r a re la ted phenomenon in (mod(A), Ab).

5.4. Propos i t ion. Assume A is a connected, tame, hered i ta ry Ar t in algebra. The functor

~: reg(A) ~ / ] T o, R ~ E~t(DR,-]

induces an equivalence between the category o f a l l regular l e f t A-modules and the

o f not necessar i ly f i n i t e -

category U = (F~o)o o f a l l f i n i t e length objects of C = ~/]F o. Moreover, an indecomposable object of F/]F o e i t h e r has f i n i t e length , or else has no subobject of f i n i t e length.

5.5. By (5.3) U decomposes uniquely in to a coproduct U = ] ] U of un ise r ia l x E X x

subcategories. A subset U of X is ca l led open i f U = ~ or U is the complement of a f i n i t e set. This def ines the Zar isk i topology on X. For U c X, observe that < U > = ] ] U is a Serre subcategory of C =~:/~o. Let C(U) denote the

x E X ' - U x

quot ien t category C/< U > . For U c V, we have < U > = < V >~ hence a 'canonica l ' functor ruv: C(V) -~ C(U), C ~ C. Note that C(X) = C.

Recall the d e f i n i t i o n of 0 = E~t(DA,-] E ~ / ~ o . For U * @ open in X, def ine Ox(U) = HOmC(u)(O,O)°P and, of course, Ox(~) = O. For U c V, the canonical functor ruv: C(V) ~ C(U) def ines ' r e s t r i c t i o n maps ~ Ox(V) ~ Ox(U), obv iously subject to t r a n s i t i v i t y . This def ines 0 X as a presheaf of k-algebras. 0 X i s , in f ac t , a sheaf as fo l lows by s t ra igh t fo rward v e r i f i c a t i o n from the decomposition U = ] [ U .

x £ X x

Since the category prep(A °p) is l e f t noether ian in the r ingoid-sense [ 5 ,25 ] , 0 X is a sheaf o f noether ian k-algebras. More p rec ise ly , Ox(X) = A, and OX(U) is a k-algebra of i n f i n i t e dimension, which is hered i ta ry of Krul l dimension one and noether ian on both sides,, i f U is an a f f i n e open subset of X. This means that U is a non-empty open subset of X, but d i f f e r e n t from X.

218

S i m i l a r l y , f o r F in ~/~F o and U open in X, def ine Fx(U) = Homc(u)(O,F) and r e s t r i c t i o n maps Fx(V) ~ Fx(U), f o r U ~ V, by means of the func to r rVU. Again, F X is a sheaf on X. Moreover, f o r each open subset U of X, Fx(U) is a l e f t Ox(U)-module, which is noether ian. I t eas i l y fo l lows that each F X is a coherent l e f t Ox-module. Also, the assignment FM~ F X extends to an equivalence of categories F/~= o ~ coh(X), F ~ F X.

5.6. For each x C X denote by C x the quot ien t category C~ < X ~ {x} >. The s ta lk 0 x o f the s t ruc tu re sheaf 0 X a t x turns out to be isomorphic to Hom Cx(O,O) °p. S i m i l a r l y , f o r each F in F/n= o the s ta lk Fx of F X at x may

be ca lcu la ted as HOmcx(O,F). C x is equ iva lent to the category mod(O x) of a l l

f i n i t e l y generated l e f t Ox-modules. In p a r t i c u l a r , U x may be i d e n t i f i e d w i th mod(Ox) o, the category of a l l l e f t Ox-modules of f i n i t e length. 0 x is semi-per fect , hered i ta ry noetherian of Kru l l dimension one wi th n x equal to the number of (isomorphism types of ) simple Ox-modules. Hence the s ta lk 0 x is Mor i ta -equ iva len t to a local r ing exac t l y f o r the homogeneous points in X.

5.7. Since X is i r r e d u c i b l e in the Zar isk i topology we may def ine the s ta lk F~ of F (or F X) in the generic po in t ~ by means of

F~ = lim~ Fx(U), U

where U runs through the set of a l l non-empty open subsets of X, ev iden t l y d i rected by inc lus ion V m U. (Note tha t according to our d e f i n i t i o n s ~ does not belong to X, but sometimes i t may be convenient to at tach ~ to X). A l t e r n a t i v e l y , F~ may be ca lcu la ted as Hom C~(O,F) °p, where C~ = C / < X > is a semi-simple category [5] equ iva len t to m~d(O~).

Hence the r ing O~ is semi-simple, i . e . a f u l l matr ix r ing over some skew f i e l d K = K(X). Sometimes we w i l l term O~ the ra t iona l func t ion r ing and K the ra t iona l func t ion f i e l d of X. I t is shown in [ 5] tha t K is isomorphic to the endomorphism skew f i e l d End(AQ) of the unique indecomposable to rs ion f ree d i v i s i b l e l e f t A-module Q, introduced by C. M. Ringel in his study of i n f i n i t e -d imens iona l A-modules. Hence i f the base f i e l d k is a l g e b r a i c a l l y c losed, we have K = k(T) the f i e l d of ra t iona l func t ions in one indeterminate T over k [30] . We may rephrase the preceeding as fo l l ows : The quot ien t category

coh(X) / COho(X)

of coherent sheaves modulo f i n i t e length sheaves is equ iva len t to mod(K(X)), the category of f i n i t e dimensional modules over the ra t iona l func t ion f i e l d of X.

219

For a semi-perfect r ing A each f i n i t e l y generated project ive A-module P r

decomposes into a d i r ec t sum @ Pi of r indecomposable pro jec t ive A-modules i=1

Call r the rank of P. Notation: rk A P. We define the rank of a coherent sheaf

F as rk F = rko~ F~ = length 0~ F~.

5.7. Proposit ion. I f ×: F ~ ° ~ Ko(mOd(A)) denotes the Euler charac ter is t ic and 6: Ko(mOd(A)) ~ the defect, we have

Pi"

rk F = ~(x(F)

for each coherent sheaf F. F has rank zero i f and only i f F has f i n i t e length i n E/~= o. I f F is an indecomposable coherent sheaf of non-zero rank r , i t s r es t r i c t i on to each a f f ineopen subset is p ro jec t i ve , moreover F x i s Ox-project ive of rank r for each x (~ X.

Proof. The fo l lowing facts are proved in [ 5]: I f R E mod(A °p) is regular , E~{t(R, - ] is zero in C~ = coh(X)/COho(X). Hom(P,-] has length -~(P) in C~ i f P is pro ject ive. ~ t ( Q , - ] has length B(Q) in C~ i f Q is pre in jec t ive . By means of (4.10) rk F = ~(×(F)) fol lows from the de f i n i t i on of the Euler charac ter is t i c . The second assertion is a consequence of (5.4). I f F is indecomposable of rank r > 0, we may assume by t rans la t ion that F = F(6m(P,-] with P preproject ive. The last assertion now fol lows from [ 5]. []

5.8. We in fe r from (5.7) that an indecomposable coherent sheaf F is e i ther of f i n i t e length (and concentrated at some x C X) or else is l oca l l y pro ject ive. Since an indecomposable preproject ive (pre in jec t ive) module is uniquely determined by i t s dimension vector dim A, each indecom~osable l oca l l y pro ject ive coherent sheaf F is uniquely determined (up to isomorphism) by i t s Euler charac te r is t i c . Since rk F = ~(×(F)) for each F, the rank is invar ian t under t rans la t ion .

Figure 2 depicts the s i tua t ion i f A is the path algebra of a quiver of extended D7" Since the rank is %-invar iant , and given on T-orb i ts by the scheme Dynkin type

1 2 2 2 2 I we have 4 T-orb i ts of rank one and two respect ively. From well-known propert ies of the defect [ 9] we fu r ther conclude that fo r A of extended Dynkin type E 6, E 7, E 8 the maximal rank of an indecomposable l oca l l y project ive sheaf is 3,4 and 6, respect ively.

6. The quiver case

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221

closed. Hence A is the path algebra k[F] of a connected quiver F of extended Dynkin type.

6.1. I f p is a source (or sink) of F, we may inver t the di rect ion of a l l arrows of F s tar t ing (respect ively ending) at p. I terat ion of th is procedure leads to the notion of an admissible change of or ientat ion of F and hence of A = k[F]. I f A I emerges from A by an admissible change of or ientat ion, the in te r re la t ion between mod(A~ p) and mod(A Op) is most conveniently described by means of a sui tably chosen preprojective t i l t i n g module T £ mod(A°P), forming a complete s l ice (see [19]).

As a resu l t , we may iden t i f y P(A~ p) with a f u l l subcategory A of P(A°P), which is co f in i te in P(A Op) and closed under successors in P(A°P). I .e . only f i n i t e l y many objects of P(A Op) are not contained in A. Further Hom (A,P) ~ 0 with A in A and P in P(A °p) implies that P is already in A. From these facts one eas i ly deduces that for each f i n i t e l y presented functor F: P(A °p) ~ Ab also the res t r i c t ion to A is f i n i t e l y presented.

This allows to define the res t r i c t ion ~(A) ~ ( A 1 ) , F ~ F]A which c lear ly induces an isomorphism

~(A)/]Fo(A) -~ F(A I )AYo(AI),

since A = P(A~ p) is co f in i te in P(A). To summarize

6.1. Proposition. I f A I emerges from A by an admissible change of or ientat ion, A and A I have isomorphic categories ~(A)AFo(A) and ~(AI)/~o(A I) of coherent sheaves, and - a f o r t i o r i - isomorphic parameter curves X(A) and X(AI). In par t icu lar X(A) = X(A°P).

Note that we use the term curve for the pair (X, coh(X)), not for the pair (X, 0x). This amounts to iden t i f y curves X,Y with equivalent categories coh(X), coh(Y) of coherent sheaves, i .e . with Morita-equivalent structure sheaves 0 X, 0y. (Call 0 X and 0y Morita-equivalent provided mod(0 x) and mod(0y), the i r respective categories of coherent modules, are equivalent.) This choice of de f in i t ion is motivated by the fact that T(A)/]Fo(A) determines the parameter curve (X, cob(X)) uniquely (cf . Section 5).

6.2. I f A is a quiver algebra of extended Dynkin type ~, where ~ means one of the Dynkin diagrams Ap,q(1 < p < q), Dn(n > 4), E 6, E 7, E 8, the curve X(A) only depends on A, according to (6.1). Hence each Dynkin type ~ determines a curve X(~) uniquely determined by ~ up to isomorphism.

222

Conversely, the curve X determines i ts Dynkin type as fo l lows: Suppose X = X(A) with A of extended Dynkin type Ap,q, D n, E 6, E 7, E 8. We recal l from Section 5 that for each x in X, the stalk 0 is semi-perfect with a f i n i t e x number n x of (isomorphism classes) of simple modules. Thus the m u l t i p l i c i t y n x of x is the number of (isomorphism classes) of simple objects in the 'component' regx(A). From the c lass i f i ca t i on of a l l non-homogeneous regular simple A-modules by Dlab and Ringel (see the tables in [ 9]) we obtain the fo l lowing information:

We may always choose three points respect ive ly , such that each x E X Moreover, exact ly the cases

Xl,X2,X 3 of m u l t i p l i c i t y n I < n 2 < n 3 d i f fe ren t from Xl,X2,X 3 has m u l t i p l i c i t y I .

Ap,q: (1,p,q) I < p < q

Dn+2: (2,2,n) 2 < n

E6: (2,3,3)

E7: (2,3,4)

E8: (2,3,5)

w i l l occur for (n I , n 2, n3). Hence the s i ngu la r i t y type &(X) = (n I , n 2, n 3) X = X(A) is always a Dynkin diagram (with base-point) Ap,q, D n, E 6, E 7, E 8. Moreover, A has extended Dynkin type ~. To summarize:

of

6.2. Theorem. The map X ~ A(X), where A(X) is the s i ngu la r i t y type of X, establishes a b i jec t ion between the set of isomorphism classes of parameter curves X = X(A) with A a connected tame hereditary k-algebra, and Dynkin diagrams Ap,q(p < q), Dn(n > 4), E6, E7, E8, respect ively.

For A tame heredi tary connected, the Dynkin diagram ~(X), where X is the parameter curve of A, w i l l also be termed the s i ngu la r i t y type of A.

6.3. We recal l that the Kronecker algebra Z = Z(k) is the path algebra k[- ~ -] of the quiver - ~ -. Z is tame hereditary of s i ngu la r i t y type At, I . Hence the curve X(Z) has no s ingular (= non-homogeneous) points, i . e . points x of m u l t i p l i c i t y n x > I .

6.3. Proposit ion. For each base f i e l d k, not necessari ly a lgebra ica l l y closed, the parameter curve of the Kronecker algebra z(k) is the pro ject ive l i ne P1(k).

Proof. Consider the usual ~ -g rad ing

k[X,Y] = e H n n E

223

of the polynomial algebra k[X,Y] by to ta l degree of polynomials. As is eas i ly ve r i f i ed , the S-modules

X Pn = Hn-1 # Hn (n = 0,1,2 . . . . )

are a complete system of preproject ive indecomposable Kronecker modules. (Here, X and Y also refer to the mu l t i p l i ca t i on by X and Y, respect ive ly) . Moreover, each map

~n,m: Hm-n ~ H°m~(Pn'Pm)' f ~ mul t i p l i ca t i on by f

is an isomorphism of k-vectorspaces for each pair n,m > 0 [26].

Hence the category P = P(~) has the fo l lowing abstract in te rp re ta t ion :

- the objects of P are the pos i t ive integers 0,1,2, . . . .

- the morphisms of P are given by P(n,m) = Hm_ n.

- the composition of P is given by the mu l t i p l i ca t i on of k[X,Y].

As a resu l t , the category ~(s) = f .p . (P(s°P) , Ab) is equivalent to the category

mod~ (k[X,Y]) +

of f i n i t e l y presented, ~+-graded kIX,Y]-modules with degree zero maps. (Note that ~ soP). Hence F(Z)/IFo(S) coincides with the quot ient category of mod~ (k[X,Y])

modulo the Serre subcategory of a l l ~+-graded k[X,Y]-modules having f i n i t e length. By Serre's c lassical resu l t [32] th is is the category coh(P1(k)) of coherent sheaves on the project ive l i ne P1(k). Hence cob(X) ~ coh(P1(k)). Since in both cases the points of the respective curves are in l- l-correspondence with the simple coherent sheaves, we also have a natural homeomorphism X ~ P1(k) of the underlying spaces, inducing the given isomorphism coh(X) ~ coh(P1(k)). []

According to our de f i n i t i on of the parameter curve X = X(z), given in Section 5, which excludes the generic point ~, P1(k) has to be interpeted as m-Proj(k[X,Y]), the maximal project ive spectrum of the #+-graded algebra k[X,Y]. Hence the points of P1(k) are the maximal graded prime ideals of k[X,Y]. Of course, i f k is a lgebra ica l l y closed, P1(k) takes the more fami l i a r form {k x k} ~ {O)/k*.

6.4. Since preproject ive (pre in jec t ive) indecomposable Kronecker modules have dimension type (n,n + I ) , resp. (n + I , n) , n = 0,1,2 . . . . . and defect - I , resp. I , the indecomposable l oca l l y pro ject ive coherent sheaves F on the project ive l i ne a l l have rank one. Moreover, they are uniquely determined by t he i r Euler character charac ter is t ic x(F) = (n, n + I ) , hence by t he i r degree d(F) = n E ~ . In th is way Grothendieck's c l ass i f i ca t i on of holomorphic vector bundles on P1(~), [17] hence

224

modulo Serre's GAGA [33] the c lass i f i ca t i on of algebraic vector bundles (= l oca l l y project ive coherent sheaves on PI(~)) , becomes a consequence of L. Kronecker's c l ass i f i ca t i on [24] of f i n i t e dimensional Z-modules, i . e . of pairs of matrices up to simultaneous conjugation. Moreover, the c l ass i f i ca t i on of indecomposable vector bundles on PI(¢) depends via (4.10) mainly on the easy part of Kronecker's c l a s s i f i c a t i o n , i .e . on the determination of a l l indecomposable preproject ive or pre in jec t ive Z-modules. This 'expla ins ' the p o s s i b i l i t y for "a short elementary proof of Grothendieck's theorem on algebraic vector bundles over the project ive l ine" [21].

6.4. Corol lary. (Kronecker (1890), Grothendieck (1957)). Each l oca l l y project ive coherent sheaf F o n_n P1(k) decomposes into a f i n i t e d i rec t sum of rank one sheaves. Moreover, the degree map F:~ d(F) establishes a l- l-correspondence between l oca l l y project ive coherent sheaves F of rank one and integers n £ ~ respect ively.

6.5. By (6.3) the parameter curve of the Kronecker algebra and the pro ject ive l i ne P1(k) have the same categories of coherent sheaves. However, the commutative and the noncommutative point of view lead to d i f f e ren t notions of structure sheaf, global sect ions, cohomology and tw is t ing operation. The usual tw is t ing cf . [20] ~: coh(P1(k)) -* coh(P1(k)),FF* F I t ] is induced by the s h i f t operat ion,

- I g M n ~ 0 Mn+ I , on graded k[X,Y]-modules. As is eas i ly ve r i f i ed o 2 = % , n > O n > O where T denotes the t rans la t ion functor of section 2. Hence the tw is t ing operations are related by

F(n) = F[2n]

for each coherent sheaf F and integer n.

Further, ~ I = H°m(Po'-] is the c lassical choice for the structure sheaf. Our choice instead i s , up to t rans la t ion (cf . Section 4) ,

0 = ~I e ~ l [ I ] = H~m(Po,-] @ H~m(PI,-]

As a resu l t , global sections £ and Fcl and cohomology H I and H Icl compare according to

F(X,F) = rc l (X,F [ - I ] ) e rc l (X,F)

1 (X,F [ -1 ] ) 0 1 (X,F). HI(x,F) = Hcl Hc l

Accordingly, the non-commutative version of global sections and cohomology contains more information on g(cf . Remark 4.9) .

6.6. Each f i n i t e subgroup G of the project ive general l i near group PGL(2.~), i . e . each polyhedral group G acts on the project ive l i ne PI(~) by M~bius transformations. From the point of view of geometry and invar ian t theory the resu l t ing quotients

225

PI(C)/G have been c lass i f i ed by F. Klein in his famous book on the icosahedron [23].

6.6. Theorem (F. Klein (1884). The quot ient X = PI(£)/G is a compact Riemann surface with at most three s ingular points x , y , z . The s i ngu la r i t y type A(X) o f X, given b_y_y the m u l t i p l i c i t i e s (nx,ny,nz) , where n x < ny < n z, is a Dynkin diagram Ap,p(p > I ) , Dn(n > 4) , E 6, E 7, E 8, which determines X up to isomorphism.

Here, the m u l t i p l i c i t y n x of x C X, x = Gp, p E P~({), is defined as the order of the s t ab i l i ze r group G(p) of p. We also note that A(X), X = PI({) /G, is jus t

the Dynkin type of the polyhedral group G. Hence, for A = (p ,q , r ) , G is generated by a,b,c with relat ions a p = b q = c r =abc = I.

By means of the theory of skew group algebras i t is proved in [27] that the parameter curves of the tame hereditary connected C-algebras - with the exception Of the s ingu ]a r i t y types Ap,q with p , q - are jus t Kle in 's simple curve s i ngu la r i t i es l i s ted in (6.6).

For k a lgebra ica l l y closed of a rb i t ra ry charac te r i s t i c , there is an a l te rna t i ve way to describe the parameter curves of Proposit ion 6.2 as the pro ject ive l i ne to- gether with a conf igurat ion of s ingular points contro l led by a Dynkin diagram ( in the sense of 6.2). There is a corresponding process of des ingu lar iza t ion , which leaves the funct ion f i e l d K(X) invar ian t , and leads - in a f i n i t e number of steps - to P l (k) . For a detai led account we refer to [35]. Thus each parameter curve X of (6.2) is b i r a t i o n a l l y equivalent to P1(k), i . e . has the f i e l d k(T) of rat ional funct ions in the indeterminate T as i t s f i e l d of rat ional funct ions. Since also HI(x, 0 X) = 0 (see (4 .2) ) , i t is in accordance with c lassical terminology to attach genus zero to each parameter curve (6.2) .

7. Theory of d iv iso rs : Picard group

7.1. Let X = X(A) be the parameter curve of A. For each x in X, l e t S(I) ~(n) be a complete system of simple regular A-modules of type x, i . e .

X 2 . • . ~ b x

belonging to the component regx(A), and denote by S x the d i rec t sum S (1)x " " ' s ( n ) x Since we suppose the base f i e l d k of A to be a lgebra ica l ly closed, the dimension vector dim S x does not depend on x [ 9]. Accordingly, we may calculate the defect of a A-module A by means of

6(A) = dimkHomA(Sx,A) - dim Ext~(Sx,A).

We define the group of d iv isors Div(X) on X as the free abelian group on X. A d i v i so r D = Z nxX is pos i t ive i f n x ~- 0 fo r each x E X. Each d i v i so r D

x E X

226

has a decomposit ion D = D+ - D_, where D+, D_ are both p o s i t i v e .

By means of Theorem 4.10 we can and w i l l i d e n t i f y regu la r A-modules and f i n i t e length sheaves on X, which are necessar i l y coherent . The Grothendieck group

Ko(reg(A)) = Ko(COho(X)) is f ree abe l ian w i th a basis cons i s t i ng of a complete

system of a l l s imple regu la r A-modules (= s imple sheaves on X). The map, assoc ia t ing to each simple regu la r A-module S i t s type x E X, induces a natura l epimorphism

t : Ko(reg(A)) ~ Div(X) o f abe l ian groups. In t h i s way, each regu la r module R ( f i n i t e length sheaf E) def ines a d i v i s o r JR] (resp. [E l ) on X.

7.2. We def ine Pic(X) as the set of a l l isomorphism classes o f l o c a l l y p r o j e c t i v e coherent rank one sheaves on X. I f no confusion a r i s e s , we w i l l use the no ta t ion

L C Pic(X) a lso to i nd i ca te tha t L is a l o c a l l y p r o j e c t i v e coherent rank one sheaf on X.

7.2. Lemma. Given L E Pic(X) and x C X, there is a simple sheaf E ( resp. E') type x, un iquely determined up to isomorphism, such tha t Ex t I (E ,L ) m 0 (resp. Hom(L,E') ~ 0) . Moreover, E' = %E.

of

Proof. Since D Ext(E,L) = Hom(L,% E) i t su f f i ces to prove existence and uniqueness f o r E. By t r a n s l a t i o n we may assume in the context of Theorem 4.10 tha t L = P is

n S( i ) a p r e p r o j e c t i v e A-module of defect 6(P) = - I . I f S = @ i=I

a complete system of s imple regu la r modules o f type x, we have n E x t ( S ( i ) , p ) Hence ~ dim = I , and the asser t ion f o l l o w s .

i= I

We thus ob ta in n o n - s p l i t exact sequences

O ~ L ~ L(x) ~ E ~ 0

o ~ L ( - x ) ~ L ~ E' ~ O.

Observe tha t L(x) and L ( -x ) are in Pic(X) the Euler c h a r a c t e r i s t i c shows tha t L(x) and p a i r (L ,× ) . As a r e s u l t , f o r each x C X b i j e c t i o n w i th inverse L ~ L ( - x ) .

is the d i r e c t sum of

6(P) = - d imkExt(S,P). D

again. Moreover, the c a l c u l a t i o n o f L ( -x ) are uniquely determined by the

the map Pic(X) ~ P ic (X) , L i~ L(x) is a

7.3. Lemma. For each L in Pic(X) and x ,y Moreover, ~ (L (x ) ) = (~L ) ( x ) .

in X we have L (x ) ( y ) = L ( y ) ( x ) .

Proof. Consider the push-out diagram

227

0 i, J, L (x ) ,~ E ,,, ~ 0 x

t u 0 ~ ~ L D E x =-0

1 ~ E

Y

1 0

0

L L

i L(y)

1 E

Y

I 0

where Ex, Ey are s imple sheaves o f type x , y r e s p e c t i v e l y . We may assume x • y . The midd le row does not s p l i t , s ince o the rw ise E x ~ Ey. Hence [ E P ic(X) and [ = ( L ( x ) ) ( y ) = ( L ( y ) ( x ) f o l l o w s . The l a s t a s s e r t i o n f o l l o w s from the ~ - i n v a r i a n c e

o f regx (A) , u

7 .4 . By means o f (7 .3 ) the a c t i o n XxPic(X) ~ P i c ( X ) , ( x , L ) ~ L ( x ) , extends un ique ly

to a group a c t i o n

Div(X) x Pic(X) ~ P i c ( X ) , (D,L) ~ L(D) ,

s ince Div(X) i s f r e e abe l i an on X. This a c t i o n may be c h a r a c t e r i z e d as f o l l o w s :

I f D = D+ - D_, w i t h D+, D both p o s i t i v e , t he re are f i n i t e leng th sheaves E+, E_

w i t h [E+] = D+, [E ] = D t o g e t h e r w i t h shor t exac t sequences

O ~ L ~ [ - ~ E+ ~ 0

0 ~ L(D) ~ L ~ E -. 0 .

Moreover , Div(X) acts t r a n s i t i v e l y on P i c (X ) : Given L o, L I E P i c ( X ) , in the c o n t e x t

o f (4 .10) we may assume t h a t Lo = Po~ LI = PI a re p r e p r o j e c t i v e A-modules o f de fec t - , 0 - I . Accord ing to [ 5] t he re are sho r t exac t sequences 0 -~ Po -~ P ~ Ro '

0 ~ PI ~ P ~ RI ~ 0 wi i :h R o, R I r e g u l a r and P p r e p r o j e c t i v e o f de fec t - I . With D = [R o] - [R I ] we o b t a i n L I = Lo(D) f rom the above d e s c r i p t i o n o f the a c t i o n o f Div(X) on P i c (X ) .

Since Div(X)

L o E P ic(X)

is a b e l i a n , t r a n s i t i v i t y i m p l i e s t h a t the s t a b i l i z e r group o f

Divo(X) = {D E Div(X) I Lo(D) = L o}

228

does not depend on the choice of L o. We ca l l Divo(X) the group of p r i nc ipa l

d i v i s o r s . Div(X) /Divo(X) is the d i v i s o r class group of X. To summarize:

7.4. P ropos i t i on . The group ac t ion

Div(X) x Pic(X) ~ P ic (X) , (D,L) ~ L(D)

is t r a n s i t i v e . For each L o E Pic(X) the s t a b i l i z e r group of L o is the group of p r i n c i p a l d i v i s o r s on X. By means of the b i j e c t i o n

Div(X) /Divo(X) ~ P ic (X) , [D] ~ Lo(D)

Pic(X) becomes an abe l ian group, isomorphic to the d i v i s o r c lass group w i th L o serv ing as the zero element. Moreover T(L(D)) = (TL)(D) holds f o r each D E Div(X) and L E P ic (X) .

Let K be a d i v i s o r w i th TL o = Lo(K). From ~(Lo(D)) = (%(Lo))(D) = Lo(K + D) = (Lo(D))(K) we ob ta in tha t %L = L(K) f o r each K E P ic (X) . Each d i v i s o r K w i th t h i s p roper ty w i l l be ca l l ed a canonical d i v i s o r f o r X. Note tha t the d i v i s o r c lass

[K] E Div(X) /D ivo(X) of K is uniquely determined.

7.5. As in the c lass i ca l contex t a d i v i s o r D is p r i n c i p a l , i . e . L(D) = L f o r each L E P ic (X) , i f and only i f D ar ises as the d i v i s o r of a non-zero r a t i ona l func t ion f E K(X)*:

We begin w i th the observat ion t ha t each non-zero morphism s: L ~ [ , w i th L, [

in P ic (X) , is necessar i l y a monomorphism wi th a cokernel E of f i n i t e length . (Consider the rank of k e r ( s ) , coke r ( s ) ) . We may hence de f ine the d i v i s o r of s as

d i v ( s ) = [E] . C lea r l y , d iV(SlS) = d iV(S l ) + d i v ( s ) i f s,s I are both non-zero, and t h e i r composi t ion is de f ined.

We r e f e r to [ 5] f o r a proof t ha t - f o r each f i xed L E Pic(X) - the f i e l d K(X) cons is ts of a l l l e f t f r ac t i ons f = s - l t , where s: L ~ [ , t : L ~ [ both are

monomorphism s t a r t i n g a t L, having a cokernel of f i n i t e length . We note tha t necessar i l y [ belongs to P ic (X) , again. Since s - l t = s ; I t I in K(×) i f and on ly i f there e x i s t s, s I both non-zero w i th a f i n i t e length cokernel and s a t i s -

f y ing s ls = s s I , s l t = s t I , the d i v i s o r of f = s - l t is def ined uniquely by d i v ( f ) = d i v ( t ) - d i v ( s ) . Moreover, the formula d i v ( f g ) = d i v ( f ) + d i v (g ) holds f o r a l l f , g c K(× )* . According to (7.4) we have L(D) = L i f and only i f there are shor t exact sequences 0 ~ L ~ [ ~ E+ ~ O, 0 ~ L ~ [ ~ E_ ~ 0 w i th E+, E_ both o f f i n i t e length and D = [E+] - [E ] , hence d i v ( s - l t ) = D, which proves the asse r t i on .

229

7.5. Proposi t !on. I f ( p , q , r ) denotes the s i n g u l a r i t y type of X, the Picard group Pic(X) is the abel ian group on three generators x , y , z w i th re la t i ons px = qy = rz .

Proof. I f S is a simple sheaf of type x wi th m u l t i p l i c i t y n x = I , we have x (L ( x ) ) = x(L) + x(S) . Hence [x ] E Div(X) /Divo(X) has i n f i n i t e order. Moreover, we have [x ] = [y ] in Div(X) /Divo(X) fo r a l l non-s ingu lar points x , y of X. Assuming x s ingu la r of m u l t i p l i c i t y p > I , l e t E i ( i E~p) be a complete system of simple sheaves of type x , s a t i s f y i n g ~IE i = Ei+ I . Given L E P ic (X) , we may assume Ext1(E i ,L) # O. Hence thus obta in x (L (px ) ) = x(L) +

x (L (px ) ) = x(L) + x(E) i f E holds in D iv (X) /D ivo(X) .

0 • Hom(L(x), E i ) = D Ext(Ei+ I , L (x ) ) f o l l ows . We x ( E i ) . Hence by (7.1) we get x (L (px ) ) =

i E P

is simple of non-s ingu lar type y. Hence p [x ] = [y ]

7.6. As a r e s u l t , the Picard group Pic(X) is an abel ian group of rank one which genera l l y has to rs ion . The fo l l ow ing table summarizes the in format ion on Pic(X):

s i n g u l a r i t y type of

Ap,q = (1 ,p ,q)

Dn_ 2 = (2 ,2 ,n) n even n odd

E 6 = (2,3,3)

E 7 = (2,3,4)

E 8 = (2 ,3 ,5)

Pic(X)

8 Z: d ' where

@ ZZ2 @ 77 2

2Z ¢ ;~ 2 Z: eT~ 3 ~ eZZ 2 7/

d = g d(p,q}

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Helmut Lenzing Fachbereich Mathematik der Universit~t-GH D-4790 Paderborn W.-Germany

Received 20.11.1984