SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II: A MULTIPLICATION FORMULA

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II: A MULTIPLICATION FORMULA CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstrt. Let n be mximl nilpotent sublgebr of omplex symmetri K-Moody Lie lgebr. Lusztig hs introdued bsis of U(n) lled the seminonil bsis, whose elements n be seen s ertin onstrutible funtions on vrieties of nilpotent modules over preprojetive lgebr of the sme type s n. We prove formul for the produt of two elements of the dul of this seminonil bsis, nd more generlly for the produt of two evlution forms ssoited to rbitrry modules over the preprojetive lgebr. This formul plys n importnt role in our work on the reltionship between seminonil bses, representtion theory of preprojetive lgebrs, nd Fomin nd Zelevinsky s theory of luster lgebrs. It ws inspired by reent results of Cldero nd Keller.. Introdution nd min result.. Seminonil bses. Let Λ be the preprojetive lgebr ssoited to onneted quiver Q without loops. This is the ssoitive lgebr Λ = CQ/ Q (ā ā), where Q denotes the double of Q nd Q is the set of rrows of Q (see e.g. [6, 9]). Rell tht Q is obtined from Q by inserting for eh rrow : i j in Q new rrow ā: j i. The lgebr Λ is independent of the orienttion of Q, nd it is finite-dimensionl (nd selfinjetive) if nd only if the underlying grph of Q is simply led Dynkin digrm, i.e. if {A n (n ), D n (n ), E n (n = 6, 7, 8)}. In the sequel, ll Λ-modules re ssumed to be finite-dimensionl nilpotent left modules. We denote by I the set of verties of Q, nd by Λ d the ffine vriety of nilpotent Λ-modules with dimension vetor d = (d i ) i I. The redutive group GL d = i I GL d i (C) ts on Λ d by bse hnge. Let n be mximl nilpotent sublgebr of omplex K-Moody Lie lgebr g of type. Lusztig [3, ] proved tht the enveloping lgebr U(n) is isomorphi to M = d N I M(d), where the M(d) re ertin vetor spes of GL d -invrint onstrutible funtions on the ffine vrieties Λ d (see 2. below for the definition of M(d)). This yields new bsis S of U(n) indexed by the irreduible omponents of the vrieties Λ d, lled the seminonil bsis []. Let M be the grded dul of M. A multiplition on M is defined vi the nturl omultiplition of the Hopf lgebr U(n) = M. In [9] we onsidered the bsis S of M dul to the seminonil bsis of M, nd begn to study its multiplitive properties. 2000 Mthemtis Subjet Clssifition. M99, 6G20, 7B35, 7B67, 20G05. Key words nd phrses. Universl enveloping lgebr, seminonil bsis, preprojetive lgebr, flg vriety, omposition series.

2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER.2. δ-strtifition. For Λ-module x Λ d define the evlution form δ x : M C whih mps onstrutible funtion f M(d) to f(x). Define x := {y Λ d δ y = δ x }. This is onstrutible subset of Λ d sine M(d) is finite-dimensionl spe of onstrutible funtions on Λ d. For the sme reson we n hoose finite set R(d) Λ d suh tht Λ d = x. x R(d) Eh irreduible omponent Z of Λ d hs unique strtum x Z ontining dense open subset of Z, nd the points of this strtum re lled the generi points of Z. We n then reformulte the definition of S s follows: the element ρ Z of S lbeled by Z is equl to δ x for generi point x of Z. The im of this pper is to prove generl formul for the produt of two rbitrry evlution forms δ x..3. Extensions. Let x Λ d, x Λ d, nd e := d + d. For onstrutible GL e - invrint subset S Λ e we onsider Ext Λ(x, x ) S = {[0 x y x 0] Ext Λ(x, x ) \ {0} y S}. This is onstrutible C -invrint subset of the qusi-ffine spe Ext Λ (x, x ) \ {0}, see 5.5. Thus we my onsider the onstrutible subset P Ext Λ (x, x ) S := Ext Λ (x, x ) S /C of the projetive spe P Ext Λ (x, x ). For onstrutible subset U of omplex vriety let χ(u) be its (topologil) Euler hrteristi with respet to ohomology with ompt support. For the empty set we set χ( ) = 0. Note tht we hve, by dditivity of χ, χ(p Ext Λ(x, x ) x ) = χ(p Ext Λ(x, x )) = dim Ext Λ(x, x ), see lso 7.2. x R(e).. Min result. Theorem (Multiplition Formul). With the nottion of.3, we hve χ(p Ext Λ(x, x )) δ x x = ( χ(p Ext Λ (x, x ) x ) + χ(p Ext Λ(x, x ) x ) ) δ x. x R(e) This theorem is very muh inspired by reent work by Cldero nd Keller on luster lgebrs nd luster tegories of finite type [6]. Note tht our multiplition formul is meningless if Ext Λ (x, x ) = 0 (or equivlently if Ext Λ (x, x ) = 0). We proved in [9] tht in ll ses δ x x = δ x δ x. For this reson we regrd Theorem s multiplition formul. from [9] the formul of the theorem n be rewritten s ( δ x δ x = dim Ext δ Λ (x, x mt(η) + ) C [η] P Ext Λ (x,x ) C [η] P Ext Λ (x,x ) Using the nottion δ mt(η) ) Here, the integrls re tken with respet to the mesure given by the Euler hrteristi, mt(η) stnds for the isomorphism lss of the middle term of the extension η. If [η] 0 we write C [η] for the orresponding element in the projetive spe P Ext Λ (x, x ). The following importnt speil se is esier to stte nd lso esier to prove..

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 3 Theorem 2. Let x nd x be Λ-modules suh tht dim Ext Λ (x, x ) =, nd let 0 x x x 0 nd 0 x y x 0 be non-split short ext sequenes. Then δ x δ x = δ x + δ y. A Λ-module x is lled rigid if Ext Λ (x, x) = 0. If x is rigid, it is generi nd δ x is dul seminonil bsis vetor. In the bove theorem, if x nd x re rigid modules with dim Ext Λ (x, x ) = one n show tht x nd y hve to be rigid s well, see []. Thus we obtin multiplition formul for ertin dul seminonil bsis vetors. This is nlyzed nd interpreted in more detil in []..5. Clbi-Yu property. For the proof of our result it is ruil tht in the tegory of finite-dimensionl (nilpotent) Λ-modules we hve funtoril piring Ext Λ(x, y) Ext Λ(y, x) C. In other words, for eh pir (x, y) of finite-dimensionl (nilpotent) Λ-modules there is n isomorphism φ x,y : Ext Λ(x, y) D Ext Λ(y, x) suh tht for ny λ Hom Λ (y, y ), [η] Ext Λ (x, y), ρ Hom Λ(x, x) nd [ɛ] Ext Λ (y, x ) one hs (.) φ x,y (λ [η] ρ)([ɛ]) = φ x,y([η])(ρ [ɛ] λ). This is well-known to speilists. The usul rgument is tht this is the shdow of ertin 2-Clbi-Yu properties for preprojetive lgebrs, see 7. for more detils. However, in the non-dynkin se this rgument relies on the ft tht then the preprojetive lgebr is 2-Clbi-Yu lgebr. Unfortuntely, we ould not lolize written proof for this lst ft. So we inlude in Setion 8 diret proof for the required funtoril isomorphism whih does not depend on the type of Q nd neither requires nilpoteny for the finite-dimensionl modules. 2. Flgs nd omposition series indued by short ext sequenes 2.. Definition of M(d). We repet some nottion from [9]. Rell tht I is the set of verties of Q. We identify the elements of I with the nturl bsis of Z I. Thus, for i = (i,..., i m ) sequene of elements of I, we my define i := m k= i k Z I. Let V be n I-grded vetor spe with grded dimension V Z I. We denote by Λ V the ffine vriety of Λ-module strutures on V. Clerly, if V is nother grded spe with V = V the vrieties Λ V nd Λ V re isomorphi. Therefore, it is sometimes onvenient to write Λ V insted of Λ V nd to think of it s the vriety of Λ-modules with dimension vetor d = V. If = (,..., m ) {0, } m nd V is n I-grded vetor spe suh tht k ki k = V, we define flg in V of type (i, ) s sequene f = ( V = V 0 V V m = 0 ) of I-grded subspes of V suh tht V k /V k = k i k for k =,..., m. We denote by Φ i, the vriety of flgs of type (i, ). When (,..., m ) = (,..., ), we simply write Φ i. Let x Λ V. A flg f is x-stble if x(v k ) V k for ll k. We denote by Φ i,,x (resp. Φ i,x ) the vriety of x-stble flgs of type (i, ) (resp. of type i). Note tht n x-stble flg is the sme s omposition series of x regrded s Λ-module. Let d i, : Λ V C be the funtion defined by d i, (x) = χ(φ i,,x ).

CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER It follows from [9, 5.8] tht d i, is onstrutible funtion, see lso [3, 2.]. If k = for ll k, we simply write d i insted of d i,. In generl, d i, = d j where j is the subword of i onsisting of the letters i k for whih k =. We then define M(V) s the vetor spe over C spnned by the funtions d j, where j runs over ll words in I with j = V. Agin this spe only depends on d = V, up to isomorphism, nd we lso denote it by M(d). This is equivlent to Lusztig s definition in [3, ]. 2.2. Reformultion. In order to prove Theorem nd Theorem 2 it is suffiient to show tht both sides of the respetive equlities oinide fter evlution t n rbitrry d i, see 2.. Sine δ x (d i ) = χ(φ i,x ) to prove Theorem we hve to show tht (2.) χ(p Ext Λ(x, x )) χ(φ i,x x ) = ( χ(p Ext Λ (x, x ) x ) + χ(p Ext Λ(x, x ) x ) ) χ(φ i,x ) x R(e) for ll sequenes i with i = e = dim(x ) + dim(x ). Similrly, for Theorem 2 we hve to show tht χ(φ i,x x ) = χ(φ i,x) + χ(φ i,y ) for ll sequenes i with i = dim(x ) + dim(x ). 2.3. Let V, V, V be I-grded vetor spes suh tht V + V = V, nd let i = (i,..., i m ) I m with k i k = V. Let x Λ V, x Λ V nd x Λ V. For short ext sequene ɛ : 0 x ι x x 0 define mp α i,ɛ : Φ i,x (, ) Φ i,,x Φ i,,x whih mps flg f x = (V l ) 0 l m Φ i,x of submodules of x to (f x, f x ) where f x = (V l /(V l ι(x ))) 0 l m nd f x = (V l ι(x )) 0 l m. Here we regrd f x s flg in V by identifying V with ι(x ), nd f x s flg in V by identifying V with V/ι(x ). Clerly, we hve (f x, f x ) Φ i,,x Φ i,,x for some sequenes, in {0, } m stisfying m m (2.2) k i k = V, k i k = V, k + k = ( k m). Let W V,V k= Thus we obtin mps k= denote the set of pirs (, ) stisfying (2.2). Define L i,,,ɛ = (Φ i,,x Φ i,,x ) Im(α i,ɛ), L 2 i,,,ɛ = (Φ i,,x Φ i,,x ) \ L i,,,ɛ, Φ i,x (,, ɛ) = α i,ɛ (Φ i,,x Φ i,,x ). α i,ɛ (, ): Φ i,x (,, ɛ) Φ i,,x Φ i,,x. Set W V,V,ɛ = {(, ) W V,V Φ i,x(,, ɛ) }. We get finite prtition Φ i,x = Φ i,x (,, ɛ). (, ) W V,V,ɛ

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 5 2.. For the rest of this setion let f x = (x = x 0 x x m = 0) nd f x = (x = x 0 x x m = 0) be flgs with (f x, f x ) Φ i,,x Φ i,,x for some (, ) W V,V where x Λ V nd x Λ V. For j m let ι x,j nd ι x,j be the inlusion mps x j x j nd x j x j, respetively. Note tht for ll j either ι x,j or ι x,j is n identity mp. In prtiulr, either x m = 0 or x m = 0. All results in this setion nd lso the definition of the mps β nd β (see below) re inspired by [6]. The following lemm follows diretly from the definitions nd the onsidertions in Setion 7.2. Lemm 2... Let ɛ : 0 x x x 0 be short ext sequene of Λ-modules. Then the following re equivlent: (i) (f x, f x ) L i,,,ɛ ; (ii) There exists ommuttive digrm with ext rows of the form ɛ : 0 x x x 0 ɛ 0 : 0 x 0 ι x, x 0 x 0 ι x, 0 ɛ : 0 x ι x,2 x x ι x,2 0 ι x,m 2... ι x,m 2 ɛ m 2 : 0 x m 2 ι x,m x m 2 x m 2 ι x,m 0 ɛ m : 0 x m x m x m 0 (iii) There exist elements [ɛ i ] Ext Λ (x i, x i ) for i = 0,,..., m suh tht [ɛ 0] = [ɛ] nd [ɛ j ] ι x,j = ι x,j [ɛ j ] for j =, 2,..., m. We define mp m 2 (2.3) β i,,,f x,f : x j=0 m 2 Ext Λ(x j, x j ) =: W V := j=0 Ext Λ(x j+, x j ) m 3 ([ɛ 0 ],..., [ɛ m 2 ]) ([ɛ j ] ι x,j+ ι x,j+ [ɛ j+ ]) + [ɛ m 2 ] ι x,m j=0 Observe tht [ɛ j ] ι x,j+ ι x,j+ [ɛ j+ ] is n element of Ext Λ (x j+, x j ). By p 0 : m 2 j=0 Ext Λ(x j, x j ) Ext Λ(x 0, x 0)

6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER we denote the nonil projetion mp. Lemm 2..2. For short ext sequene ɛ: 0 x x x 0, the following re equivlent: (i) (f x, f x ) L i,,,ɛ ; (ii) [ɛ] p 0 (Ker(β i,,,f x,f x )). Proof. We hve [ɛ] p 0 (Ker(β i,,,f x,f )) if nd only if x β i,,,f x,f ([ɛ x 0],..., [ɛ m 2 ]) = 0 for some ([ɛ 0 ],..., [ɛ m 2 ]) with [ɛ 0 ] = [ɛ]. This is the se if nd only if ι x,j+ [ɛ j+ ] = [ɛ j ] ι x,j+ for ll 0 j m 3, [ɛ m 2 ] ι x,m = 0 nd [ɛ 0 ] = [ɛ]. Now the result follows from Lemm 2... Next, we define mp (2.) β i,,,f x,f x : m 2 j=0 m 2 Ext Λ(x j, x j+) =: V W := j=0 Ext Λ(x j, x j) m 2 ([η 0 ],..., [η m 2 ]) ι x, [η 0 ] + (ι x,j+ [η j ] [η j ] ι x,j) Note tht ι x,j+ [η j ] [η j ] ι x,j is n element of Ext Λ (x j, x j ). Lemm 2..3. For short ext sequene η : 0 x y x 0 the following re equivlent: (i) (f x, f x ) L i,,,η ; (ii) [η] Ext Λ (x, x ) Im(β i,,,f x,f x ). Proof. By definition, [η] Ext Λ (x, x ) Im(β i,,,f x,f x ) if nd only if j= ([η], 0,..., 0) = β i,,,f x,f x ([η 0 ],..., [η m 2 ]), in other words if nd only if there exists ([η 0 ],..., [η m 2 ]) suh tht ι x, [η 0 ] = [η] nd [η j ] ι x,j = ι x,j+ [η j ] for ll j m 2. On the other hnd, by Lemm 2.. (f x, f x ) L i,,,η if nd only if for 0 i m there exist [η i ] Ext Λ (x i, x i ) suh tht [η 0 ] = [η] nd [η i ] ι x,i = ι x,i [η i ] for i m. Tking into ount tht for eh i either ι x,i or ι x,i is n identity, it is now esy to see tht the two onditions re equivlent. 2.5. As before, let x Λ V, x Λ V, nd let f x = (x = x 0 x x m = 0) nd f x = (x = x 0 x x m = 0) be flgs with (f x, f x ) Φ i,,x Φ i,,x for some (, ) W V,V. Due to the Clbi-Yu property of Λ, see.5, we hve non-degenerte nturl pirings φ V : V V C, φ W : W W C, (([η 0 ],..., [η m 2 ]), ([δ 0 ],..., [δ m 2 ])) (([ψ 0 ],..., [ψ m 2 ]), ([ɛ 0 ],..., [ɛ m 2 ])) for the spes (V, V ) nd (W, W ) defined in (2.3) nd (2.). m 2 j=0 m 2 j=0 φ x j,x j+ ([η j])([δ j ]) φ x j,x j ([ψ j])([ɛ j ])

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 7 Lemm 2.5.. For ll ([η 0 ],..., [η m 2 ]) V nd ([ɛ 0 ],..., [ɛ m 2 ]) W we hve φ V (([η 0 ],..., [η m 2 ]), β (([ɛ 0 ],..., [ɛ m 2 ]))) = φ W (β(([η 0 ],..., [η m 2 ])), ([ɛ 0 ],..., [ɛ m 2 ])). Proof. By definition, the left hnd side of the eqution is m 3 j=0 φ x j,x j+ ([η j])([ɛ j ] ι x,j+ ι x,j+ [ɛ j+ ]) + φ x m 2,x m ([η m ])([ɛ m 2 ] ι x m ), using (.), this is equl to m 2 φ x (ι 0,x x 0, [η 0 ])([ɛ 0 ]) + φ x (ι j,x x j,j+ [η j ] [η j ] ι x,j)([ɛ j ]), j= whih is by definition the right hnd side of our lim. Proposition 2.5.2. For [ɛ] Ext Λ (x, x ) the following re equivlent: (i) [ɛ] p 0 (Ker(β i,,,f x,f x )); (ii) [ɛ] is orthogonl to Ext Λ (x, x ) Im(β i,,,f x,f x ). Proof. By Lemm 7.3. nd Lemm 2.5. we hve Ker(β i,,,f x,f x ) = ( Im(β i,,,f x,f x ) ). Putting W 0 = Ext Λ (x 0, x 0 ) = Ext Λ (x, x ) nd W 0 = Ext Λ (x 0, x 0 ) = Ext Λ (x, x ), it follows tht p 0 (Ker(β i,,,f x,f )) is the orthogonl in W x 0 of W 0 Im(β i,,,f x,f x ). Corollry 2.5.3. Assume tht dim Ext Λ (x, x ) =, nd let ɛ : 0 x x x 0 nd η : 0 x y x 0 be non-split short ext sequenes. Then the following re equivlent: (i) (f x, f x ) L i,,,ɛ ; (ii) (f x, f x ) L 2 i,,,η. Proof. By Lemm 2..2 we know tht (i) holds if nd only if [ɛ] p 0 (Ker(β i,,,f x,f )). By x Proposition 2.5.2 this is equivlent to [ɛ] being orthogonl to Ext Λ (x, x ) Im(β i,,,f x,f x ). Sine dim Ext Λ (x, x ) = nd sine ɛ is non-split, this is the se if nd only if Ext Λ(x, x ) Im(β i,,,f x,f x ) = 0. By Lemm 2..3 nd the ft tht η is non-split this hppens if nd only if (f x, f x ) L 2 i,,,η. Let us stress tht Proposition 2.5.2 is very similr to [6, Proposition ]. We just hd to dpt Cldero nd Keller s result to our setting of preprojetive lgebrs. Our proof then follows the min line of their proof. In ft our sitution turns out to be esier to hndle, beuse we work with modules nd not with objets in luster tegories. 3. Euler hrteristis of flg vrieties This setion is inspired by the proof of [3, Lemm.].

8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 3.. Let V, W nd T be I-grded vetor spes with V = T W. We shll identify V/W with T. We write m = dim V. Let Φ(V) denote the vriety of omplete I-grded flgs in V. Let π : Φ(V) Φ m (T) Φ m (W) be the morphism whih mps flg (V l ) 0 l m to the pir of flgs ((T l ) 0 l m, (W l ) 0 l m ), defined by W l = W V l nd T l = p (V l ). Here p : V = T W T is just the projetion onto T, nd Φ m (T) nd Φ m (W) denote the vrieties of m-step I- grded flgs in W nd T, respetively. Note tht T j /T j+ nd W j /W j+ hve dimension 0 or for ll j. We wnt to study the fibers of the morphism π. Let (f = (T l ), f = (W l )) Im(π). For n I-grded liner mp z : T W we define flg f z = f z,f,f = (Vl z) 0 l m Φ(V) by V l z = (0, W l ) + {(t, z(t)) t T l } T l W. Clerly, we hve f z π (f, f ). We sy tht two I-grded liner mps f, g : T W re equivlent if Vf l = Vl g for ll l. In this se we write f f,f g or just f g. In other words, f nd g re equivlent if nd only if (f g)(t l ) W l for ll l. The next lemm shows tht every flg in π (f, f ) is of the form f z for some z. Lemm 3... If π(f) = (f, f ) then there exists n I-grded liner mp z : T W suh tht f = f z,f,f. Proof. Choose deomposition for the two grded vetor spes m m T = T(i) nd W = W(i) suh tht for ll l. Put T l = i=0 m T(i) nd W l = i=l l W l := W(i), i=0 i=0 m W(i) nd tke flg f = (V l ) 0 l m suh tht π(f) = (f, f ). Sine we hve short ext sequene i=l 0 W l V l T l 0, there exists unique I-grded liner mp w l : T l W l suh tht The onditions V l V l+ nd W l W l+ V l = (0, W l ) + {(t, w l (t)) t T l }. imply (3.) w l+ (t) w l (t) W l+ for ll t T l+. Hene, for t T l+ we hve W l = W(l) (t, w l+ (t)) (t, w l (t)) = (0, w l+ (t) w l (t)) W l V l. Now, define z : T W on the summnds of T by z(t) = w i (t) for t T(i).

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 9 Thus the fiber π (f, f ) n be identified with the vetor spe Hom(T, W)/ f,f of I-grded liner mps T W up to equivlene. 3.2. Let Γ = CQ /J be n lgebr where Q is finite quiver with set of verties I nd J is n idel in the pth lgebr CQ. For finite-dimensionl I-grded vetor spe U let Γ U be the ffine vriety of Γ-module strutures on U. Fix short ext sequene ɛ : 0 x x x 0 of Γ-modules with x Γ V, x Γ T nd x Γ W. To n I-grded liner mp z : T W we ssoite liner mps z l : T l W/W l defined by z l (t) = z(t) + W l. For vertex i I let z i : T i W i be the degree i prt of z. The next lemm follows from the definitions. Lemm 3.2.. If x = x x nd if f nd f re flgs of submodules of x nd x, respetively, then f z,f,f is flg of submodules of x x if nd only if the mp z l is module homomorphism for ll l. We now del with the se when the short ext sequene ɛ does not split. A module m Γ V n be interpreted s tuple m = (m()) where for eh rrow Q we hve liner mp m(): V s() V e() where s() nd e() denote the strt nd end vertex of the rrow. Thus given our short ext sequene ɛ : 0 x x x 0 we n ssume without loss of generlity tht the liner mps x() re of the form ( x x() = ) () 0 y() x () where y(): T s() W e() re ertin liner mps. The proof of the following sttement is gin strightforwrd, ompre lso the proof of [3, Lemm.]. Lemm 3.2.2. If f nd f re flgs of submodules of x nd x, respetively, then f z,f,f is flg of submodules of x if nd only if for ll 0 l m nd Q. (x ()z s() z e() x () y())(t l s() ) Wl e() 3.3. Now we pply the bove results to the se of the preprojetive lgebr Λ. Lemm 3.3.. For short ext sequene ɛ : W V,V,ɛ the fibers of the morphism 0 x x x 0 nd (, ) α i,ɛ (, ): Φ i,x (,, ɛ) Φ i,,x Φ i,,x re isomorphi to ffine spes, moreover Im(α i,ɛ (, )) = L i,,,ɛ. Proof. Both onditions in Lemm 3.2. nd Lemm 3.2.2 re liner. The lst eqution follows from the definitions, see 2.3. Corollry 3.3.2. For short ext sequene ɛ : 0 x x x 0 nd (, ) W V,V we hve χ(φ i,x(,, ɛ)) = χ(l i,,,ɛ ). Proof. This follows from Lemm 3.3. nd Proposition 7...

0 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Corollry 3.3.3. If x = x x, then for ll short ext sequenes ɛ : 0 x x x 0 nd (, ) W V,V we hve χ(φ i,x (,, ɛ)) = χ(φ i,,x Φ i,,x ) = χ(φ i,,x ) χ(φ i,,x ). Proof. If x = x x nd (, ) W V,V, then L i,,,ɛ = Φ i,,x Φ i,,x for ll ɛ, see [9]. In [9] we limed tht for (, ) W V,V,ɛ the mp Φ i,x x (,, ɛ) Φ i,,x Φ i,,x is vetor bundle, referring to [3, Lemm.]. Wht we hd in mind ws n rgument s bove. However this only proves tht the fibers of this mp re ffine spes. Nevertheless, for the Euler hrteristi lultion needed in [9] this is enough beuse of Proposition 7.... Proof of Theorem 2 Assume dim Ext Λ (x, x ) =, nd let ɛ : 0 x x x 0 nd η : 0 x y x 0 be non-split short ext sequenes. We obtin the following digrm of mps: Φ i,x (,, ɛ) α i,ɛ (, ) L i,,,ɛ Φ i,y (,, η) L i,,,η α i,η (, ) ι i,,,ɛ ι i,,,η L i,,,ɛ L2 i,,,ɛ L i,,,η L2 i,,,η Φ i,,x Φ i,,x i Φ i,,x Φ i,,x Here, the mp i is the isomorphism whih mps (f x, f x ) to (f x, f x ), nd ι i,,,ɛ nd ι i,,,η re the nturl inlusion mps. By Corollry 3.3.2, χ(φ i,x (,, ɛ)) = χ(l i,,,ɛ ) nd χ(φ i,y(,, η)) = χ(l i,,,η ). We know from Corollry 3.3.3 tht χ(φ i,x x (,, θ)) = χ(φ i,,x Φ i,,x ) = χ(l i,,,ɛ ) + χ(l2 i,,,ɛ ), where θ is ny (split) short ext sequene of the form 0 x x x x 0. By Corollry 2.5.3 the isomorphism i indues isomorphisms i,2 : L i,,,ɛ L2 i,,,η nd i 2, : L 2 i,,,ɛ L i,,,η

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II whih implies χ(l 2 i,,,ɛ ) = χ(l i,,,η). Combining these fts we get χ(φ i,x x ) = χ(φ i,x x (,, θ)) = = = (, ) W V,V (, ) W V,V (, ) W V,V (, ) W V,V = χ(φ i,x ) + χ(φ i,y ). χ(l i,,,ɛ ) + χ(l i,,,ɛ ) + χ(φ i,x (,, ɛ)) + (, ) W V,V (, ) W V,V (, ) W V,V χ(l 2 i,,,ɛ ) χ(l i,,,η ) By the onsidertions in Setion 2.2 this finishes the proof of Theorem 2. 5. The generl se χ(φ i,y (,, η)) 5.. Derivtions. For Λ-modules x nd x let D Λ (x, x ) be the vetor spe of ll tuples d = (d(b)) b Q of liner mps d(b) Hom C (x s(b), x e(b) ) suh tht (5.) (d(ā)x () + x (ā)d()) (d()x (ā) + x ()d(ā)) = 0 Q :s()=p Q :e()=p for ll p Q 0. We ll the elements in D Λ (x, x ) derivtions. Let d = (d()) Q with d() Hom C (x s(), x e() ), nd for eh Q let ( x E d () = ) () 0 d() x. () Then E d = (E d ()) Q defines Λ-module if nd only if the mps d() stisfy Eqution (5.) for ll p. In this se we obtin n obvious short ext sequene ɛ d : 0 x E d x 0. 5.2. Inner derivtions. For Λ-modules x nd x let I Λ (x, x ) be the vetor spe of ll tuples i = (i(b)) b Q of liner mps i(b) Hom C (x s(b), x e(b) ) suh tht for some (φ q) q Q0 with φ q Hom C (x q, x q) we hve i(b) = φ e(b) x (b) x (b)φ s(b) for ll b Q. The elements in I Λ (x, x ) re lled inner derivtions nd we hve obviously I Λ (x, x ) D Λ (x, x ). Let π : D Λ (x, x ) Ext Λ(x, x ) be defined by π(d) = [ɛ d ]. It is known tht the kernel of π is just the set I Λ (x, x ) of inner derivtions, hene we obtin n ext sequene 0 Hom Λ (x, x ) Hom C (x q, x q) D Λ (x, x ) π Ext Λ(x, x ) 0. q Q 0 We hoose fixed vetor spe deomposition D Λ (x, x ) = I Λ (x, x ) E Λ (x, x ). We n therefore identify Ext Λ (x, x ) with E Λ (x, x ). We lso n identify P Ext Λ (x, x ) with PE Λ (x, x ). Set E Λ(x, x ) = E Λ (x, x ) \ {0}.

2 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER 5.3. Prinipl C -bundles. Rell tht the multiplitive group C is speil in the sense tht eh prinipl C -bundle is lolly trivil in the Zriski topology, see [7, ]. Thus, if π : P Q is suh bundle, π dmits lol setions. Sine moreover the tion of C on P is free by definition, we onlude tht (π, Q) is geometri quotient for the tion of C on P, see for exmple [2, Lemm 5.6]. Note moreover tht π is flt (nd in prtiulr open) sine lolly it is just projetion. As rule, we will write C x for the C -orbit of x if x belongs to prinipl C -bundle. 5.. The vrieties EF i (x, x ). Let x Λ V nd x Λ V nd i = (i, i 2,..., i m ) be word in I m suh tht i = dim(x x ). The tion of C on V V defined by λ (v, v ) := (v, λ v ) indues n tion on the flg vriety Φ i (V V ): if x = (x 0 x x m ), then the i-th omponent of (λ x ) is {λ z z x i }. On the other hnd we hve the prinipl C -bundle E Λ (x, x ) PE Λ (x, x ). Thus we obtin by [7, 3.2] new prinipl C -bundle We onsider q : E Λ(x, x ) Φ i (V V ) E Λ(x, x ) C Φ i (V V ). ẼF i (x, x ) := {(d, x ) E Λ(x, x ) Φ i (V V ) x Φ i,ed }. This is lerly losed subset of EΛ (x, x ) Φ i (V V ), nd it is moreover C -stble sine ( ) ( ) ( ) V 0 x 0 x 0 (5.2) : E 0 λ d = V d x E λd = λd x is n isomorphism of Λ-modules. We onlude tht EF i (x, x ) := q(ẽf i(x, x )) is losed in E Λ (x, x ) C Φ i (V V ) sine q is open, see 5.3. Thus, the restrition of q to q : ẼF i(x, x ) EF i (x, x ) is gin prinipl C -bundle [7, 3.], nd in prtiulr (q, EF i (x, x )) is geometri quotient for the tion of C on ẼF i(x, x ). The projetion p : ẼF i(x, x ) PE Λ (x, x ), (d, x ) C d is onstnt on C -orbits. So we obtin morphism p : EF i (x, x ) PE Λ (x, x ) whih mps C (d, x ) to C d. In other words, we hve p (C d) = Φ i,ed. 5.5. δ-strtifition of EF i (x, x ). Let e := dim(x ) + dim(x ) nd onsider the δ-strtifition Λ e = x. We lim tht x R(e) EF i (x, x ) x := {C (d, x ) EF i (x, x ) E d x }

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 3 is onstrutible set. Indeed, we hve morphism ι: E Λ (x, x ) Λ e whih mps d to E d. Sine x is onstrutible nd GL e -invrint, E Λ (x, x ) x = {d E Λ (x, x ) E d x } is onstrutible nd C -invrint (see lso (5.2)). Now, EF i (x, x ) x = p (PE Λ(x, x ) x ) is lso onstrutible. The fibers p ([d]) re identified with the vrieties Φ i,e d. Sine they ll ome from the sme δ-strtum, they ll hve the sme Euler hrteristi. Thus from Proposition 7.. we obtin χ(ef i (x, x ) x ) = χ(pe Λ (x, x ) x ) χ(φ i,ed ) = χ(p Ext Λ(x, x ) x ) χ(φ i,ed ) for ny d E Λ (x, x ) x. It follows tht, (5.3) χ(ef i (x, x )) = x R(e) χ(p Ext Λ(x, x ) x ) χ(φ i,x ). 5.6. Proof of Theorem. We define morphism of vrieties π : EF i (x, x ) (PE Λ (x, x ) Φ i,,x Φ i,,x ) (, ) W V,V whih mps C (d, x ) to (C d, f x, f x ) where (f x, f x ) is the pir of flgs in x nd x indued by x vi the short ext sequene ɛ d : 0 x E d x 0. (Observe tht (d, x ) nd (λd, λ x ) indue the sme pir of flgs vi the sequenes ɛ d nd ɛ λd, respetively.) We denote by L the imge of π, nd by L 2 the omplement of L in (, ) W (PE V,V Λ(x, x ) Φ i,,x Φ i,,x ). The following digrm illustrtes the sitution: ẼF i (x, x ) E Λ (x, x ) Φ i (V V ) p q q : (d, x ) C (d, x ) PE Λ (x, x ) EF i (x, x ) EΛ (x, x ) C Φ i (V V ) p π : C (d, x ) (C d, f x, f x ) L L 2 = (, ) W (PE V,V Λ(x, x ) Φ i,,x Φ i,,x ) Proposition 5.6.. The following hold: () χ(l ( ) = χ(ef i (x, x )); ) (b) χ (, (PE ) WV,V Λ(x, x ) Φ i,,x Φ i,,x ) = χ(pe Λ (x, x )) χ(φ i,x x ); () χ(l 2 ) = χ(ef i (x, x )). Proof. () By the sme rgument s in Lemm 3.3., the fibers of π re isomorphi to ffine spes. This implies () by Proposition 7... (b) We hve to show tht χ(φ i,,x ) χ(φ i,,x ) = χ(φ i,x x ). (, ) W V,V

CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER This is explined in the proof of [9, Lemm 6.], see lso the remrk fter Corollry 3.3.3. () Let (P[d], f x, f x ) L 2. By Lemm 2..2, this mens tht for ll (, ) W V,V, d p 0 (Ker(β i,,,f x,f x )). By Proposition 2.5.2, this mens tht for ll (, ) W V,V, d (E Λ (x, x ) Im(β i,,,f x,f x )). Therefore there exists d E Λ (x, x ) Im(β i,,,f x,f x ) whih is not orthogonl to d, nd flg y of submodules of the module (x 0 d x ) suh tht y indues f x nd f x. We re thus led to onsider the onstrutible set C of ll pirs ((C d, f x, f x ), C (d, y )) L 2 EF i (x, x ) suh tht (d, y ) indues (f x, f x ) nd d nd d re not orthogonl for the piring between E Λ (x, x ) nd E Λ (x, x ). Let us onsider the two nturl projetions: C pr pr 2 L 2 EF i (x, x ) We re going to show tht ll fibers of both projetions hve Euler hrteristi equl to. More preisely we hve: (i) The mp pr is surjetive with fibers being extensions of ffine spes; (ii) The mp pr 2 is surjetive with fibers being ffine spes (of onstnt dimension dim E Λ (x, x ) ). Let us prove (i). Let (C d, f x, f x ) L 2. Let E Λ (x, x ) (fx,f x ) be the set of ll d E Λ (x, x ) suh tht there exists filtrtion y of the module ( x 0 d x ) whih indues f x nd f x. This is vetor spe, nd by the bove disussion we know tht it is not ontined in the hyperplne d. Thus is hyperplne in E Λ (x, x ) (fx,f x ), nd is n ffine spe. We get mp d E Λ (x, x ) (fx,f x ) Z := P(E Λ (x, x ) (fx,f x ) \ d ) pr (C d, f x, f x ) Z whih mps ((C d, f x, f x ), C (d, y )) to C d. By onstrution this mp is surjetive nd its fibers re ffine spes whih implies (i). Let us prove (ii). Let C (d, y ) EF i (x, x ). Let f x nd f x be the flgs indued by y on x nd x, respetively. Then by Lemm 2..3 nd Proposition 2.5.2, d p 0 (Ker(β i,,,f x,f x )) for some (, ) W V,V. So if d d, then (C d, f x, f x ) L 2, by Lemm 2..2. Therefore pr 2 ([d, y ]) n be identified with the projetiviztion of the set of ll d

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 5 E Λ (x, x ) suh tht d d. But the projetiviztion of the omplement of hyperplne in vetor spe of dimension m is n ffine spe of dimension m. Now, the proposition together with Eqution (5.3) imply obviously Eqution (2.), whih is equivlent to Theorem. 6. An exmple In this setion, Λ will be the preprojetive lgebr of type D, with underlying quiver We study extensions between 2 b ā T := b 2 3 b nd the simple module S. We hve dim Ext Λ (T, S ) = 2. The middle terms of the nonsplit short ext sequenes 0 T E S 0 re of the form M(λ) := (,λ) b 2 3 (ā, λ) M( ) := b b 2 3 (λ C \ {0, }), M(0) := (, ) b, M( ) := The middle terms of the non-split short ext sequenes re of the form B := R := 2 b 2 b 0 S E T 0 3 3, A :=, C := 3 (ā, ) b 2 3, b (ā, ) 2 3. b 3 2 b 3, 2 b.

6 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Note tht R is rigid, nd A, B, C belong to the orbit losure of R. Our multiplition formul yields for ny λ C \ {0, } 2δ T δ S = ( ( )δ M(λ) + δ M(0) + δ M( ) + δ M( ) ) + (( )δr + δ A + δ B + δ C ). To see this, note tht P Ext Λ (S, T ) is projetive line, with points identified to the middle terms of short ext sequenes 0 T M(λ) S 0 with λ C { }. This beomes strtified ording to the Euler hrteristis of flgs of submodules into (C \ {0, }) {0} {} { } nd we obtin the first term on the right hnd side. Also Ext Λ (T, S ) is projetive line. In this sitution there re only isomorphism lsses R, A, B, C of middle terms for non-split short ext sequenes 0 S X T 0. However, in the first se we hve (C \ {0, })-fmily of possible embeddings of S into R suh tht the quotient is T while in the other three ses there is unique embedding of this type. This gives the seond term on the right hnd side of our eqution. Using the rigid modules 2 3 F := we obtin b 2 3 b (, ), G := b ā 3 (, ), H := δ M(0) = δ M(λ) + δ H, δ A =δ R + δ F, δ M( ) = δ M(λ) + δ F, δ B =δ R + δ G, δ M( ) = δ M(λ) + δ G, δ C =δ R + δ H. ā b 2 (b, ), These equlities follow from simple lultions of Euler hrteristis of vrieties of omposition series. For exmple, to see the first equlity one should observe tht for M(0) there re two types of omposition series: Those with top S my be identified with the omposition series of M(λ) for λ generi. The remining omposition series hve top S 3 nd my be identified with the omposition series of H. Our lim follows, see 2.. So eventully δ T δ S = δ M(λ) + δ R + δ F + δ G + δ H s n expnsion in the dul seminonil bsis. 7. Reolletions 7.. 2-Clbi-Yu lgebrs nd tegories. Let k be field. A tringulted k-tegory T with suspension funtor Σ is lled d-clbi-yu tegory if ll its homomorphism spes re finite-dimensionl over k, nd there exists funtoril isomorphism T (x, y) Hom k (T (y, Σ d x), k)

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 7 for ll x, y T. It is well-known, tht for preprojetive lgebr Λ of Dynkin type the stble module tegory Λ-mod with the suspension funtor Ω is 2-Clbi-Yu tegory, see for exmple [0, 7.5]. In this se eqution (.) follows vi the nturl isomorphism [2, 6] Ext Λ (x, y) Hom Λ(x, Ω y) from the 2-Clbi-Yu property of Λ-mod. Reently it ws nnouned [5] (see lso 8.3 ()) tht (onneted) preprojetive lgebr Λ whih is not of Dynkin type is 2-Clbi-Yu lgebr, so by definition D f (Λ), the full subtegory of the bounded derived tegory of Λ of omplexes with finite-dimensionl nilpotent ohomology, is 2-Clbi-Yu tegory. In this se eqution (.) follows from the 2-Clbi-Yu property of D f (Λ) vi the nturl isomorphism Ext Λ (x, y) D f (x, y[]) for nilpotent Λ-modules x, y. 7.2. Extensions. If ɛ: 0 y y y 0 is short ext sequene of Λ-modules, we write [ɛ] for its lss in Ext Λ (y, y ). Let us rell the funtoril behvior of Ext Λ in terms of short ext sequenes. If ρ Hom Λ (x, y ) nd λ Hom Λ (y, z ) then [ɛ] ρ Ext Λ (x, y ) is represented preisely by short ext sequene 0 y y x 0 whih is obtined from ɛ s the pullbk long ρ. Similrly, λ [ɛ] Ext Λ (y, z ) is represented preisely by short ext sequene 0 z z y 0 whih is obtined from ɛ s the pushout of ɛ long λ. ɛ λ : 0 z z y 0 λ ɛ: 0 y ρ y y 0 ɛ λ ρ : 0 z e λ ɛ ρ : 0 y x ρ x x So we hve in the bove digrm [ɛ λ ] = λ [ɛ] nd [ɛ ρ ] = [ɛ] ρ nd the ssoitivity (λ [ɛ]) ρ = [ɛ λ ρ] = λ ([ɛ] ρ). 0 0 7.3. Biliner forms nd orthogonlity. Let φ: U U C be biliner form. For subspes L U nd L U define L = {u U φ(u, u ) = 0 for ll u L}, L = {u U φ(u, u ) = 0 for ll u L }. We ll u U nd u U orthogonl (with respet to φ) if φ(u, u ) = 0. If φ is nondegenerte, then (L ) = L nd ( L ) = L, nd dim L + dim L = dim L + dim L = dim U = dim U. Let φ V : V V C nd φ W : W W C be non-degenerte biliner forms, nd let f Hom C (V, W ). Then we n identify f with the dul f : V W of mp f Hom C (W, V ) if nd only if (7.) φ V (v, f (w )) = φ W (f(v), w ) for ll v V nd w W. Here we use the isomorphisms φ V : V V nd φ W : W W defined by v φ V (v, ) nd w φ W (w, ), respetively. Assume tht Eqution (7.)

8 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER holds. Thus we get ommuttive digrm V f W v f f(v) eφ V V f W eφ W eφ V φ V (v, ) f φ V (v, f ( )) φ W (f(v), ). eφ W The proof of the following Lemm is n esy exerise. Lemm 7.3.. Ker(f ) = Im(f). 7.. Euler hrteristis. For omplex lgebri vriety V, let C(V ) denote the belin group of onstrutible funtions on V with respet to the Zriski topology. If π : V W is morphism of omplex vrieties nd f C(V ), we define funtion π f on W by (π f)(w) = f = χ(f () π (w)), (w W ). π (w) C Then it is known tht π f is onstrutible. θ : W U of omplex vrieties we hve (θ π) = θ π. Moreover, for morphisms π : V W nd (The se of ompt omplex lgebri vrieties is disussed in [5, Proposition ], the generl se n be found in [8, Proposition..3].) As prtiulr se, we note the following result: Proposition 7... Let π : V W be morphism of omplex vrieties suh tht there exists some Z with χ(π (w)) = for ll w Im(π). Then χ(v ) = χ(im(π)). In prtiulr, if π : V W is morphism of omplex vrieties suh tht for ll w Im π the fiber π (w) is isomorphi to n ffine spe, then χ(v ) = χ(im(π)). 8. Ext-Symmetry for preprojetive lgebrs The min gol of this setion is to provide diret proof of the following result whih is ruil for this pper. Otherwise it is independent of the min body of the pper. Theorem 3. For quiver Q without loops let Λ be the ssoited preprojetive lgebr over field k. Then for ll finite-dimensionl Λ-modules M nd N there is funtoril isomorphism φ M,N : Ext Λ(M, N) D Ext Λ(N, M). 8.. Preliminries. Let k be field, nd let Q = (Q 0, Q, s, e) be finite quiver without loops. Here Q 0 denotes the set of verties, Q is the set of rrows of Q, nd s, e: Q Q 0 re mps. An rrow α Q strts in vertex sα = s(α) nd ends in vertex eα = e(α). By Q we denote the double quiver of Q, so Q 0 = Q 0, Q = Q {α α Q }, nd we extend the mps s, e to mps s, e: Q Q 0 by sα := eα, eα := sα for ll α Q. It will be onvenient to onsider? s n involution on Q with { β if β Q, β := α if β = α for some α Q.

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 9 Moreover, for ll β Q we define β := { 0 if β Q, else. We onsider the preprojetive lgebr Λ = kq/ ρ i i Q0 where ρ i = ( ) β ββ. β Q sβ=i Note tht Λ is qudrti, possibly infinite-dimensionl quiver lgebr, in ny se Λ is ugmented over S := k Q 0 (i.e. we hve k-lgebr homomorphisms S Λ S whose omposition is the identity on S). Note tht S is ommuttive semisimple k-lgebr whih hs nturl bsis onsisting of primitive orthogonl idempotents {e i i Q 0 }. In wht follows, ll undeorted tensor produts re ment over S. The proof relies on the following well-known result whih holds muttis mutndis more generlly for ny qudrti quiver lgebr. It follows for exmple from the proof of [3, Theorem 3.5]. Lemm 8... Let Λ be the preprojetive lgebr s defined bove. Then P : Λ R Λ d Λ A Λ d0 Λ S Λ is the beginning of projetive bimodule resolution of Λ, where S = i Q0 ke i, A = β Q kβ nd R = i Q0 kρ i re S-S-bimodules in the obvious wy, nd d 0 = β Q (β e sβ e sβ e eβ e eβ β) S-S β, d = β Q ( ) β (β β e sβ + e sβ β β) S-S ρ sβ. In the sttement of the Lemm we denote for exmple by (β ) β Q the dul bsis for the S-S-bimodule DA = Hom k (A, k). In this se we hve e i β e j = δ i,sβ δ j,eβ β. 8.2. Proof of Theorem 3. For finite-dimensionl Λ-modules M nd N it is esy to determine the omplex Hom Λ (P Λ M, N): (8.) i Q 0 Hom k (M(i), N(i)) d 0 M,N β Q Hom k (M(sβ), N(eβ)) where d 0 M,N = Hom Λ(d 0 Λ M, N) is given by d M,N i Q 0 Hom k (M(i), N(i)) (f i ) i Q0 (N(β)f sβ f eβ M(β)) β Q nd d M,N = Hom Λ(d Λ M, N) is given by (g β ) β Q ( ) β (N(β)g β + g β M(β)) β Q : sβ=i Thus, we hve funtorilly Ext Λ (M, N) = Ker(d M,N )/ Im(d0 M,N ). i Q0.

20 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Similrly, we find D Hom Λ (P Λ N, M), whih is identified vi the tre pirings ( i Q 0 Hom k (M(i), N(i))) ( i Q 0 Hom k (N(i), M(i))) k, nd ( Hom k (M(sβ), N(eβ))) ( Hom k (N(sβ), M(eβ))) k, β Q β Q to (8.2) i Q 0 Hom k (M(i), N(i)) d, N,M with d, N,M = D Hom Λ(d N, M) given by β Q Hom k (M(sβ), N(eβ)) ((ϕ i ), (f i )) ((ɛ β ), (g β )) i Q 0 Tr(ϕ i f i ) β Q Tr(ɛ β g β ) d 0, N,M i Q 0 Hom k (M(i), N(i)) (ϕ i ) i Q0 (( ) β (N(β)ϕ sβ ϕ eβ M(β))) β Q nd d 0, N,M = D Hom Λ(d 0 N, M) given by (ɛ β ) β Q ɛ β M(β) β Q : eβ=i β Q : sβ=i N(β)ɛ β i Q0. In ft, we hve by definition d, N,M ((ϕ i) i Q0 )((g β ) β Q ) = Tr(ϕ i ( ( ) β (M(β)g β + g β N(β)))) i Q 0 = β Q sβ=i β Q ( ) β (Tr(ϕ eβ M(β)g β ) + Tr(g β N(β)ϕ sβ )) = β Q ( ) β Tr((N(β)ϕ sβ ϕ eβ M(β)) g β ) nd d 0, N,M ((ɛ β) β Q )((f i ) i Q0 ) = Tr(ɛ β (M(β)f sβ f eβ N(β))) β Q = Tr(( ɛ β M(β) N(β)ɛ β ) ϕ i ). i Q 0 β Q eβ=i β Q sβ=i

SEMICANONICAL BASES AND PREPROJECTIVE ALGEBRAS II 2 Thus, we hve funtorilly D Ext Λ (N, M) = Ker(d0, nd (8.2) re isomorphi: i Q0 Hom k (M(i), N(i)) d0 M,N N,M d M,N β Q Hom k (M(sβ), N(eβ)) )/ Im(d, N,M ). The omplexes (8.) i Q0 Hom k (M(i), N(i)) Θ M,N i Q0 Hom k (M(i), N(i)) d, N,M β Q Hom k (M(sβ), N(eβ)) d0, N,M i Q0 Hom k (M(i), N(i)) with Θ M,N ((g β ) β Q ) = (( ) β g β ) β Q. This finishes the proof of Theorem 3. 8.3. Remrks. () Assume Q is onneted. Then it is known tht in the sitution of Lemm 8.. we hve n isomorphism of bimodules: Ker d = { DΛ if Q is Dynkin quiver, 0 else. This follows for exmple from [3, Thm..8. & Thm..9] in the first se nd from [3, Prop..2] together with [, Thm. 9.2] in the seond se. Thus, P is projetive bimodule resolution of Λ, if Q is not Dynkin quiver. A similr lultion s in 8.2 shows tht P is self-dul in the sense of Boklndt [,.], thus the bounded derived tegory D b (Λ-mod) is 2-Clbi-Yu [, Thm..2]. (2) We leve it s n (esy) exerise to derive from the lultions in Setion 8.2 Crwley-Boevey s formul [7, Lemm ], nd in the non-dynkin se the eqution dim Hom Λ (M, N) dim Ext Λ(M, N) + dim Ext 2 Λ(M, N) = (dim M, dim N) Q, where (, ) Q is the symmetri qudrti form ssoited to Q. Aknowledgements. We like to thnk Bernhrd Keller nd Willim Crwley-Boevey for interesting nd helpful disussions. Referenes [] R. Boklndt, Grded Clbi Yu Algebrs of dimension 3. rxiv:mth.ra/0603558. [2] K. Bongrtz, Some geometri spets of representtion theory. Algebrs nd modules, I (Trondheim, 996), 27, CMS Conf. Pro., 23, Amer. Mth. So., Providene, RI, 998. [3] S. Brenner, M.C.R. Butler, A.D. King, Periodi lgebrs whih re lmost Koszul. Algebr. Represent. Theory 5 (2002), no., 33 367. [] M. C. R. Butler A. D. King, Miniml resolutions of lgebrs. J. Algebr 22 (999), no., 323 362. [5] J. Chung, R. Rouquier, Pper in preprtion. [6] P. Cldero, B. Keller, From tringulted tegories to luster lgebrs. rxiv:mth.rt/050608. [7] W. Crwley-Boevey, On the exeptionl fibres of Kleinin singulrities. Amer. J. Mth. 22 (2000), no. 5, 027 037. [8] A. Dim, Sheves in topology. Universitext. Springer-Verlg, Berlin, 200. xvi+236 pp. [9] Ch. Geiß, B. Leler, J. Shröer, Seminonil bses nd preprojetive lgebrs. Ann. Si. Éole Norm. Sup. () 38 (2005), no. 2, 93 253. [0] Ch. Geiß, B. Leler, J. Shröer, Auslnder lgebrs nd initil seeds for luster lgebrs. rxiv:mth.rt/050605, to pper in J. London Mth. So. [] Ch. Geiß, B. Leler, J. Shröer, Rigid modules over preprojetive lgebrs. Invent. Mth. 65 (2006), no. 3, 589 632. [2] A. Heller, The loop-spe funtor in homologil lgebr. Trns. Amer. Mth. So. 96 (960), 382 39. [3] G. Lusztig, Quivers, perverse sheves, nd quntized enveloping lgebrs. J. Amer. Mth. So. (99), no. 2, 365 2.

22 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER [] G. Lusztig, Seminonil bses rising from enveloping lgebrs. Adv. Mth. 5 (2000), no. 2, 29 39. [5] R.D. MPherson, Chern lsses for singulr lgebri vrieties. Ann. of Mth.(2) 00 (97), 23 32. [6] C. M. Ringel, The preprojetive lgebr of quiver. In: Algebrs nd modules II (Geirnger, 966), 67 80, CMS Conf. Pro. 2, AMS 998. [7] J.P. Serre, Espes fibrés lgébriques. Seminire C. Chevlley 958, 36. Christof Geiß Instituto de Mtemátis Universidd Nionl Autónom de Méxio Ciudd Universitri 050 Méxio D.F. Méxio E-mil ddress: hristof@mth.unm.mx URL: www.mtem.unm.mx/~hristof/ Bernrd Leler Lbortoire LMNO Université de Cen F-032 Cen Cedex Frne E-mil ddress: leler@mth.unien.fr URL: www.mth.unien.fr/~leler/ Jn Shröer Mthemtishes Institut Universität Bonn Beringstr. 535 Bonn Germny E-mil ddress: shroer@mth.uni-bonn.de URL: http://www.mth.uni-bonn.de/people/shroer/