2. Preliminaries In this section we give some basic definitions and known results, which will be useful in the sequel.

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1 M ath. es. Lett. 4 (2007), no. 4, c International Press 2007 KOSULITY AND THE HILBET SEIES OF PEPOJECTIVE ALGEBAS Pavel Etingof and Ching-Hwa Eu. Introduction The goal of this paper is to prove that if Q is a connected non-dynkin quiver then the preprojective algebra Π Q (k) of Q over any field k is Koszul, and has Hilbert series ( Ct + t 2 ), where C is the adjacency matrix of the double Q of Q. We also prove a similar result for the partial preprojective algebra Π Q,J (k) of any connected quiver Q, where J I is a subset of the set I of vertices of Q. By definition, Π Q,J (k) is the quotient of the path algebra of k Q by the preprojective algebra relations imposed only at vertices not contained in J. We show that if J then Π Q,J (k) is Koszul, and its Hilbert series is ( Ct + D J t 2 ), where D J is the diagonal matrix with (D J ) ii 0 if i J and (D J ) ii, i / J. Moreover, we show that both results are valid in a slightly more general framework of modified preprojective algebras, considered in [3]. We note that the first result is known in most cases [6, 5, 9]. In particular, it is known in general in characteristic zero ([5]), and in most positive characteristic cases [6, 9]. Our argument, however, is elementary, and different from the arguments of [5, 9], which are based on the theory of tensor categories. 2. Preliminaries In this section we give some basic definitions and known results, which will be useful in the sequel. 2.. Hilbert series of bimodules. Let k be a field (of any characteristic). Let I be a finite set, and let i I k be the algebra of k-valued functions on I. An -bimodule W may be thought of as an I I-graded vector space W i,j I W i,j. Then for two -bimodules U, W we have ( ) (2..) W U i,j s I W i,s U s,j. k The tensor algebra T (W ) is defined as m 0 W m, with the tensor products over. Now let W d 0 W [d] be a + -graded -bimodule, with finite dimensional homogeneous subspaces. eceived by the editors December 3,

2 590 PAVEL ETINGOF AND CHING-HWA EU Definition We define the Hilbert series h W (t) to be a matrix-valued series, with the entries h W (t) i,j dim W [d] i,j t d. d0 From (2..), it follows that h W U (t) h W (t)h U (t) Free products. Let A, B be k-algebras containing. Definition We define the free product A B of A and B over to be the algebra T (A B)/J, where J is the ideal generated by elements ) r A r B, where r and r A, r B are its images in A and B, respectively, and 2) a a 2 a a 2, a, a 2 A; b b 2 b b 2, b, b 2 B. Example If A T (V )/(r i ), B T (W )/(s j ) and r i, s j are the defining relations, then A B T (V W )/(r i, s j ). Example Assume that we have decompositions of bimodules A A +, B B +. Then ( ) m (2.2.4) A B A B + A + B m0 Lemma Suppose A, B are + -graded algebras, with A[0] B[0]. Let h A (t) α, h B(t) β. Then h A B(t) α β. Proof. Let A +, B + be the positive degree parts of A, B. Then we have h m 0 B + h A+ (t) α α, h B + (t) β β, ( «β α m(t) A + β α m0 ) m ( β β ) α. α By (2.2.4), we have ( h A B(t) ( α) β ) α ( β) β α (( β)( α) βα) α β Quadratic and Koszul algebras. For the material in this subsection, we refer to [8]. Let V be an -bimodule, and E V V an -subbimodule. To this data we may attach the quadratic algebra A : T (V )/(E). Definition Assume that the bimodule V is equipped with a filtration by nonnegative integers. Then we define A : T (grv )/(gre). The following lemma is obvious.

3 KOSULITY AND THE HILBET SEIES OF PEPOJECTIVE ALGEBAS 59 Lemma We have the termwise inequality h A (t) h A (t). In other words, for every i, j A and d 0 we have dim A [d] i,j dim A[d] i,j. Definition Let A be a quadratic algebra with A[0]. A is a Koszul algebra if Ext i A (, ) (or dually T ora i (, )) is concentrated in degree i. Theorem (The Golod-Shafarevich inequality) Let A be a + -graded algebra with A[0], with the bimodule of generators V in degree and relations E in degree 2. Let C and D be the matrices defined by C i,j dim V i,j, D i,j dim E i,j. Then: (i) If Ct+Dt 0 termwise, then h 2 A (t) Ct+Dt. 2 (ii) If h A (t) Ct+Dt, then A is Koszul. 2 Proof. From the exact Koszul complex, 0 K A E A V A 0 extended by the kernel K on the left, using the Euler-Poincaré principle, we obtain the equation h A (t)( Ct + Dt 2 ) h K (t) 0, so under the assumption of (i) the inequality h A (t) Ct+Dt 0 follows. To prove 2 (ii) note that in the above exact sequence, h A (t) Ct+Dt implies h 2 K (t) 0, therefore K 0. Applying the functor to this sequence (without the last term A ), we get the complex computing T ori A (, ): So and hence A is Koszul. 0 0 E 0 V i 0 T ori A V i (, ) E i 2 0 i > 2, 3. Hilbert series of preprojective algebras 3.. Partial preprojective algebras. Let Q be a finite quiver with vertex set I (we allow multiple edges and self-loops). For each a Q let h(a), t(a) I denote the head and tail of the edge a. Let V be the k-vector space spanned by the edges of Q; it is naturally an -bimodule, where i I k. ecall that the path algebra kq is the algebra T V. Let Q the double of Q, obtained from Q by keeping the same vertex set and adding a new edge a for j to i for each edge a from i to j. Let J I be a subset of white vertices (the other vertices are colored black). Definition 3... We define the partial preprojective algebra Π Q,J (k) k Q/ aa a a a Q,h(a) I\J a Q,t(a) I\J In particular, if J then we write Π Q (k) for Π Q, (k) and call it the preprojective algebra of Q.

4 592 PAVEL ETINGOF AND CHING-HWA EU emark In a similar way, for any commutative ring K one defines the algebra Π Q,J (K). When no confusion is possible, we will denote Π Q,J (K) by Π Q,J The Hilbert series for an extended Dynkin quiver. Let Q be an extended Dynkin quiver (in particular, we allow the quiver Ã0, which has one vertex and one edge). Proposition The Hilbert series h ΠQ (k)(t) equals Ct+t 2, where C is the adjacency matrix of Q. Proof. First we show this for chark 0. (In this case the result is well known, and the argument actually works if p is a good prime for Q, in particular for any p > 5). Let Γ SL 2 (k) be the finite group attached to Q via the McKay s correspondence, f i the primitive idempotents of the irreducible representations of Γ, f f i, and i A Γ k[x, y] the skew group algebra. Then we have the following result, due to G. Lusztig [4] (see also [2]). Proposition Π Q (k) i,j f i Af j faf By applying the functor kγ to the exact Koszul complex of k[x, y] (over k) we obtain the exact complex K : 0 A(2) A A[] A kγ 0 kγ (here (2) denotes the shift in grading by 2). K f K f has the form By Proposition 3.2.2, the complex 0 Π Q (k)(2) Π Q (k) Π Q (k)[] Π Q (k) 0. We conclude that this complex is exact, and from the Euler-Poincaré principle obtain the equation h (t) h ΠQ (k)(t) h ΠQ (k)(t)ct + h ΠQ (k)(t)t 2. Now consider the case chark 0. Let T be the torsion part of Π Q (). Then Π Q ()/T is a free -module, and by the characteristic zero result has the Hilbert series (indeed, it suffices to take tensor product with C). Further, T k Π Q (k) is Ct+t 2 an ideal, and the quotient algebra Π Q (k)/t k (Π Q ()/T ) k has the same Hilbert series. Since C is an extended Car- tan matrix, its largest eigenvalue is 2 and hence we get that Π Q (k)/t k has Gelfand- Kirillov dimension 2. By [], Π Q (k) is a prime Noetherian algebra of Gelfand-Kirillov dimension 2. But for any prime Noetherian algebra A of Gelfand-Kirillov dimension d and any nonzero two-sided ideal I A, one has GKdim(A/I) d, [7, Corollary ]. Therefore, we see that T k 0, and hence T 0, as desired.

5 KOSULITY AND THE HILBET SEIES OF PEPOJECTIVE ALGEBAS Star-shaped quivers. In this subsection we prove the Hilbert series formula for star-shaped quivers with node being a white vertex. Lemma Let Q be a quiver with vertex set I {,..., n + } and arrows n + ai,,...ai,r i i I, and with a set of white vertices J {i,... i m {, n + }. Then i / J h ΠQ,J (t) Ct+D J t where D 2 J is a diagonal matrix with (D J ) ii 0 i J. Proof. First, we observe that Π Q,J A... A n+ where A i is the partial preprojective algebra of the quiver Q i with vertex set I and only the arrows n + ai,,...ai,r i i (if i n +, these arrows are self-loops). By Lemma 2.2.5, it is enough to prove the result in the case when this free product has only one factor, i.e. for n, and for n 2, r 2 0. If all vertices are white, then Π Q,J is a path algebra and the result is clear from counting paths. So it remains to consider the case n 2, r 2 0, where is a black vertex. In this case, the quiver Q has edges a,..., a r going from 2 to, and we have one defining relation a a a r a r 0. Denote the algebra Π Q,J in this case by A(r). Consider first the case r. In this case the algebra A(r) has only one quadratic element a a (up to scaling) and no cubic elements, so the formula easily follows. Now consider the case r >. Let us introduce the filtration on A : A(r) by setting deg(a ) deg(a ), and deg(a i ) deg(a i ) 0 for i >. In this case A A() B, where B is the path algebra of the quiver with edges a i, a i, i 2,..., r. It then follows from Lemma and the r case that the desired Hilbert series formula holds for A. By Lemma 2.3.2, this implies that h A (t) Ct + D J t 2. Then by Theorem 2.3.4, we see that h A (t) Ct + D J t 2, as desired Main results. Theorem Let Q be a connected quiver and J a nonempty set of white vertices. Then h ΠQ,J (t) Ct+D J t 2. In particular, Π Q,J is a Koszul algebra. Proof. The second statement follows from the first statement and Theorem So we only need to prove the first statement. We will prove by induction in the number of vertices that the statement is true for any quiver Q whose every connected component contains a white vertex. If Q has just one vertex, the statement is clear. To make the induction step, assume the formula is right for n vertices, and that Q has n + vertices. Let I be the vertex set of Q. Select any white vertex w in I and consider the subquiver Q 0 with vertex set I 0 I, consisting of w and the vertices adjacent to it,

6 594 PAVEL ETINGOF AND CHING-HWA EU whose arrows are all the arrows of Q which touch w. Also, let Q be the quiver with the vertex set I \ {w}, where the vertices in I 0 are colored white and the other ones are colored in the same way as in Q, and with the set of arrows Q \ Q 0. Finally, let Q 0, Q be the quivers with vertex set I and with the same arrows as in Q 0, Q respectively, and let J 0, J be the sets of white vertices of Q 0 and Q. Introduce a filtration on Π Q,J, by setting a grading, such that the arrows inside Q 0 have degree and the other ones have degree 0. Then Π Q,J Π b Q0,J 0 Π bq,j. Therefore, by Lemma 2.3.2, we get h ΠQ,J (t) h Π Q,J (t) h Π bq0,j 0 Π bq,j (t). Now, by Lemma 3.3. for Q 0, h ΠdQ0 (t),j 0 C 0t+D 0t where C 2 0 is the adjacency matrix of Q0, and D 0 is the diagonal matrix, such that (D 0 ) ii if i is a black vertex in I 0, and 0 otherwise. Also, applying the induction assumption to Q, we find h Π (t) cq,j (C C 0)t+D t, where D is the diagonal matrix with (D ) 2 ii if i is a black vertex / I 0, and 0 otherwise. Therefore, by Lemma we have h ΠQ,J (t) Ct+D J t. From this, it follows 2 Ct+D J t 0. Now, by Theorem 2.3.4, 2 the result follows. Theorem The Hilbert series h ΠQ (t) for a connected non-dynkin quiver Q is Ct+t 2. In particular, Π Q is Koszul. Proof. Again, the second statement follows from the first one and Theorem 2.3.4, and we only prove the first statement. For this, we use the following easy (and well known) lemma. Lemma Any connected non-dynkin quiver Q contains an extended Dynkin subquiver Q E. Let I E be the vertex set of Q E (it is possible that an arrow a between i, j I E belongs to Q but not to Q E ). Let Q be the quiver with vertex set I, such that the vertices in I E are white and the other ones are black, and the set of arrows Q \ Q E. Then every connected component of Q contains at least one white vertex. Introduce a filtration on Π Q by setting a grading, such that the arrows in Q E have degree and the other ones have degree 0. Then Π Q Π QE b Π Q,I E, where Q E is the quiver Q E with adjoined vertices of I \ I E. Now, by Proposition 3.2., h Π bqe C E t+d E t where D 2 E is the diagonal matrix, such that (D E ) ii is if i I E and 0 otherwise, and C E is the adjacency matrix of of the double of Q E. Also, by Theorem 3.4., h ΠQ,IE (C C E )t+( D E )t. Hence, 2 by Lemma 2.2.5, we obtain h ΠQ h Π Q h Π bqe Π Q,IE Ct + t 2. From this, it follows that Ct+t 2 0. Hence, Theorem implies the result Modified preprojective algebras. It turns out that our results hold for a slightly more general class of preprojective algebras. Namely, let Q be a quiver with vertex set I, and Q its double. Let J I. Let γ be a k -valued function on the set

7 KOSULITY AND THE HILBET SEIES OF PEPOJECTIVE ALGEBAS 595 of edges of Q which begin or end at I \ J. Define the modified partial preprojective algebra Π γ Q,J (k) to be the quotient of k Q by the relation γ a aa γ a a a 0. a Q,h(a) I\J a Q,t(a) I\J Obviously, this is a generalization of the usual partial preprojective algebras Π Q,J (k), which are obtained if γ. If J, Π γ Q,J (k) is called a modified preprojective algebra and denoted by Πγ Q (k). Such algebras are considered in [3]. Theorem Theorem 3.4. and Theorem hold for the algebras Π γ Q,J (k) and Π γ Q (k). Proof. The proof of Theorem 3.4. carries out verbatim to the case of general γ. To prove Theorem for Π γ Q, we only need to consider the extended Dynkin case, since it is the only case when the proof needs to be changed. Also, note that if Q is a tree, then the algebras Π γ Q are pairwise isomorphic for all choices of γ (by rescaling the edges). So it is necessary to consider only the case of type Ãn. In this case the edges are a,..., a n and a,..., a n (where a i goes from i to i + ), and the defining relations are γ a i a i a i γ ai a i a i 0, where i is computed modulo n. These relations show that Π γ Q is spanned by paths in which there is no expressions a i a i. Counting such paths, we easily get that h Π γ Q ( Ct + t2 ), which by Theorem implies the result. Acknowledgments P.E. is grateful to V. Ostrik for a useful discussion. The work of P.E. was partially supported by the NSF grant DMS and by the CDF grant M-2545-MO- 03. eferences [] D. Baer, W. Geigle, and H. Lenzing, The preprojective algebra of a tame hereditary algebra, Comm. Algebra 5 (987) [2] W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Mathematical Journal 92 (998), no. 3,. [3] M. Kleiner, The graded preprojective algebra of a quiver, Bulletin of London Math 36 (2004) [4] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (99) [5] A. Malkin, V. Ostrik, and M. Vybornov, Quiver varieties and Lusztig s algebra, Adv. Math. 203 (2006) [6]. Martinez-Villa, Applications of Koszul algebras: the preprojective algebra in epresentation theory of algebras, in CMS Conf. Proceedings, Cocoyoc, Vol. 8, , AMS, Providence, I (996). [7] J. McConnell and J. obson, Noncommutative Noetherian ings, Vol. 30, Graduate Studies in Mathematics (200).

8 596 PAVEL ETINGOF AND CHING-HWA EU [8] A. Polishchuk and L. Positselski, Quadratic algebras, AMS (2005). [9] V.Ostrik, Module categories over representations of SL q(2) and graphs II. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 0239, U.S.A. address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 0239, U.S.A. address: