SKEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS

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1 Math. J. Okayama Univ. 43(21), 1 16 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS Dedicated to Helmut Lenzing on the occasion of the 6-th birthday Roberto MARTÍNEZ-VILLA Abstract. Skew group algebras appear in connection with the study of singularities [1], [2]. It was proved in [4], [6], [1] the preprojective algebra of an Euclidean diagram is Morita equivalent to a skew group algebra of a polynomial algebra. In [7] we investigated the Yoneda algebra of a selfinjective oszul algebra and proved they have properties analogous to the commutative regular algebras, we call such algebras generalized Auslander regular. The aim of the paper is to prove that given a positively graded locally finite -algebra Λ = P i Λ i and a finite grading preserving group G of automorphisms of Λ, with characteristic not dividing the order of G, then G acts naturally on the Yoneda algebra Γ = L k Ext k Λ(Λ, Λ ) and the skew group algebra Γ G is isomorphic to the Yoneda algebra Λ G = L k Ext k Λ G(Λ G, Λ G). As an application we prove Λ is generalized Auslander regular if and only if Λ G is generalized Auslander regular and Λ is oszul if and only if Λ G is so. It was proved by H. Lenzing the indecomposable modules over a quiver algebra Q with a field and Q an Euclidean diagram, are parametrized by lein curve singularities P 1 (C)/G arising from the action of polyhedral groups on projective line. The polyhedral groups are the finite subgroups of SL(2, C), [8] they act naturally on C[x, y] sending homogeneous elements to homogeneous elements, for example the cyclic group Z n = { (ξ ) } ξ 1 ξ is a complex primitive n-th root of unity acts on C[x, y] by x ξx, y ξ 1 y. The coordinate ring of P 1 (C)/G is C[x, y] G. In general, given a positively graded locally finite -algebra Λ = j Λ j, with Λ semisimple, Λ i Λ j = Λ i+j and G a finite group of grading preserving -automorphisms of Λ, we associate to them two -algebras, the fixed ring Λ G = {λ Λ λ g = λ for all g G} and the skew group algebra Λ G defined as follows: 1

2 2 R. MARTÍNEZ-VILLA As a vector space Λ G = Λ G. For λ Λ and g G, we write λg instead of λ g, and multiplication is given by gλ = λ g g, where the element λ g Λ denotes the image of λ under the automorphism g. The algebra Λ G is the endomorphism ring of a finitely generated projective Λ G-module, explicitly: given the idempotent e = 1/ G g of Λ G there exists an isomorphisms e(λ G)e = Λ G [3]. Given a quiver Q and a field the preprojective algebra is the -algebra ˆQ/I with quiver ˆQ with vertices ˆQ = Q and arrows ˆQ 1 = Q 1 Q op 1, where Q op denotes the opposite quiver of Q. For any arrow α Q 1 write ˆα for the corresponding arrow in the opposite quiver. The ideal I is generated by relations α i ˆα i and ˆα i α i. We have the following: Theorem 1 (Lenzing [6], Reiten and Van den Bergh [1], Crawley -Boevey [4]). The preprojective algebras Λ = C ˆQ/I corresponding to an Euclidean quiver Q are Morita equivalent to the skew group algebras C[x, y] G, with G a polyhedral group. In the example of the cyclic group Z n acting on C[x, y] given above the skew group algebra C[x, y] Z n is the Mckay quiver [8] obtained as follows: Let {S 1, S 2,..., S n } be the irreducible representations of Z n, put a vertex v i for each simple S i and m ij arrows from v i to v j if V S i = m ij S j, C j where V is the two dimensional representation given by (x, y)/(x, y) 2. The Mckay quiver is: n 1 g G n. In this paper we consider generalized Auslander regular algebras, they constitute non commutative versions of the regular algebras and contain the preprojective algebras. We will prove that given a positively graded algebra Λ as above and a finite a group of automorphisms G, the skew group algebra Λ G is generalized Auslander regular if and only if Λ is so. Let M be a Λ G-module and V a G-module. The vector space M is a Λ G-module with action given by: λ.(m v) = λm v and g(m v) = gm gv, for λ Λ, m M and g G. Lemma 2. Let Λ = j Λ j be a positively graded, locally finite -algebra with Λ semisimple, Λ i Λ j = Λ i+j and an algebraically closed field. Let V

3 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 3 G be a finite grading preserving group of automorphisms of Λ such that the characteristic of does not divide the order of G and Λ G the skew group algebra. Then the Λ G-simple modules are of the form: S V with S a simple Λ G submodule of Λ and V an irreducible G-module. Proof. The radical of Λ G is J G and J G(S V ) = JS V =, then S V is a Λ G/J G = Λ G-module. By [9], Λ G is semisimple, therefore: S V is semisimple. We need to prove S V is indecomposable. We have natural isomorphisms: Hom Λ G (S V, S V ) = Hom Λ (S V, S V ) G = Hom (V, Hom Λ (S, S V )) G = Hom G (V, Hom Λ (S, S V )). We have also natural isomorphisms: Hom Λ (S, S V ) = Hom Λ/J (S, S V ) = Hom Λ/J (S, Λ/J) S Λ/J V = Hom Λ/J (S, S) V = Hom Λ (S, S) V. Then we have isomorphisms: Hom G (V, Hom Λ (S, S V )) = Hom G (V, V ) = =. Hom Λ (S, S) If S is a Λ G simple, then S is a Λ G/J G = Λ G-module. Therefore: S is isomorphic to a summand of Λ G = Λ G. Decomposing Λ = G = m V j we obtain S is isomorphic to some module S i V j. j=1 t i=1 S i,

4 4 R. MARTÍNEZ-VILLA Lemma 3. Let be a field, G a finite group with characteristic of not dividing the order of G. Let M be a -vector space with G-action. Denote by M G = {m M gm = m for all g G} the set of fixed points. Then the fixed point functor ( ) G : Mod G Mod G is exact. Proof. Let t : M M be a -linear map given by: t(m) = 1/ G gm. Then it is clear t(m) = M G. Let L j M π N be an exact sequence of G-modules and G-maps. Then we have an exact commutative diagram: L M which induces an exact diagram: t t N L M N, t g G M π N t M G t res π N G. Therefore, res π is a map onto N G. It is clear M G L = L G. Hence; L G M G N G is exact. Lemma 4. Let Λ be a positively graded -algebra, G a finite grading preserving group of automorphisms of Λ. Let L, M and N be Λ G-modules. Then the following statements hold: i) The abelian group Hom Λ (M, N) is a G-module with action g f(m) = gf(g 1 m). ii) There is an equality Hom Λ (M, N) G = Hom Λ G (M, N). iii) For all k there is a natural action of G on Ext k Λ (M, N). If x Ext k Λ (M, N) and y Exts Λ (N, L), then g(y.x) = (gy).(gx). Proof. i) Let g 1, g 2 be elements of G and f Hom Λ (M, N). We have identities: (g 1.g 2 ) f(m) = g 1 g 2 f(g2 1 g 1 1 m) = g 1(g 2 f(g2 1 ))(g 1 1 m) = g 1 (g 2 f(m)). ii) If f is an element of Hom Λ (M, N) G, then g f = f for all g G, in particular: g 1 f = f. It follows gf(m) = f(gm) and f Hom Λ G (M, N). If f is in Hom Λ G (M, N), then g 1 f(m) = f(g 1 m). Therefore f(m) = gf(g 1 m), this is g f = f for all g G, hence; f Hom Λ (M, N) G.

5 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 5 iii) Let g be an element of G and M a Λ-module. Define the Λ-module M g 1 as follows: M g 1 = M as -vector space and for λ Λ and m M we have λ m = λ g 1 m. If M is a G-module, then we have an isomorphism φ g 1 : M M g 1 given by φ g 1(m) = g 1 m. Then φ g 1(λm) = g 1 λm = λ g 1 g 1 m = λ φ g 1(m). If M and N are G-modules and f : M N is a Λ-map, then we have the following commutative diagram: M g 1 φ g 1 f g 1 N g 1 M g f N, φ g 1 where f g 1 (x) = f(x) and f g 1 (λ x) = f(λ g 1 x) = λ g 1 f(x) = λ f g 1 (x). Then φ g f g 1 φ g 1(m) = gf(g 1 m) = g f(m). Let x Ext k Λ (M, N) be the extension: Define g.x as the extension: N j f E k f 1 k Ek 1 E 1 M. N jg 1 φg 1 E g 1 k f g 1 k E g 1 k 1 Eg 1 1 φ g f g 1 1 M. Since ( ) g is an exact functor we have: x y if and only if gx gy. We have the following commutative diagram: φ h 1 N φ h 1 φ h 1 g 1 g 1 N N h 1 g 1. It follows (hg)(x) = h(gx). Hence; G acts on Ext k Λ (M, N). It is clear that if x Ext k Λ (M, L) and y Exts Λ (L, N), then g(yx) = (gy)(gx). Corollary 5. Let Λ be a positively graded -algebra, G a finite grading preserving group of automorphisms of Λ such that the characteristic of does not divide the order of the group G. Let L f M π N be an exact sequence of G-modules and G-maps. Then for any G-module X the

6 6 R. MARTÍNEZ-VILLA long exact sequences: Hom Λ (X, L) Hom Λ (X, M) Hom Λ (X, N) Ext 1 Λ(X, L) Ext 1 Λ(X, M) Ext k Λ(X, M) Hom Λ (N, X) Hom Λ (M, X) Hom Λ (L, X) Ext 1 Λ(N, X) Ext 1 Λ(M, X) Ext k Λ(M, X) are exact sequences of G-modules and G-maps. t k Proof. Let L j t E k Ek 1 E 1 1 X be an exact sequence and f : L M be a G-map. We have an induced exact sequence y obtained from the commutative diagram: x : L j t k E k t 1 E k 1 E 1 X f y : M i F k s k F k 1 F 1 s 1 X. Applying ( ) g 1 and composing with the natural isomorphisms we have a commutative exact diagram: 1 gx : L jφ g 1 E g 1 k g f gy : M F g 1 k t k E g 1 k 1 Eg 1 1 F g 1 k 1 F g 1 1 φ g t 1 X 1 φ g s 1 X. Then Ext k Λ (f, M)(x) = y. We have g Extk Λ (f, M)(x) = gy = Extk Λ (g f, M)(gx). Since g f = f, then Ext k Λ (f, M) is a G-map. Let δ : Ext k Λ (X, N) Extk+1 Λ (X, L) be the connecting map and x Ext k Λ (X, N), where x : N E k E k 1 E 1 X and z is the exact sequence: L f M π N. We have the following commutative exact diagram: L f M π N φ g 1 L g 1 φ g 1 f φ g 1 M g 1 π N g 1 with φ g 1 isomorphisms. This implies gz = z. Then δ(gx) = zgx = gzgx = g(zx) = gδ(x). Hence; δ is a G-map. The proof for the second long exact sequence is by dual arguments.

7 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 7 Lemma 6. Let Λ be a positively graded -algebra, G a finite grading preserving group of automorphisms of Λ such that the characteristic of does not divide the order of the group G. Let X be a finitely generated graded Λ G-module. Then X is projective if and only if X is projective as Λ- module. Proof. Assume the module X is projective as Λ G-module, this implies there exists a graded Λ G-module Q such that X Q = (Λ G) n = ( G Λ) n. Therefore: X is a projective Λ-module. Assume X is projective as Λ-module. Let A B C be an exact sequence of Λ G-modules. Then the sequence: Hom Λ (X, A) Hom Λ (X, B) Hom Λ (X, C) is an exact sequence of G-modules. have an exact sequence: Applying the fixed point functor we Hom Λ (X, A) G Hom Λ (X, B) G Hom Λ (X, C) G, which is isomorphic to the sequence: Hom Λ G (X, A) Hom Λ G (X, B) Hom Λ G (X, C). It follows X is projective. Corollary 7. If Λ and G are as in the lemma, then Λ is a projective Λ G- module. Lemma 8. Let G be a finite grading preserving group of automorphisms of the -algebra Λ and assume the characteristic of Λ does not divide the order of the group G. Let P be a graded finitely generated projective Λ G- module and N an arbitrary graded Λ G-module. Then we have a natural isomorphism: θ : Hom Λ (P, N) W Hom Λ G (P G, N W ) given by θ(f w)(p g) = g f(p) gw. Proof. We have a natural isomorphism of -vector spaces: ψ : Hom (G, Hom Λ (P, N W )) Hom Λ (P G, N W ) given by ψ(γ)(p g) = γ(g)(p). The map ψ is a G-map. We have equalities: ψ(h γ)(p g) = h γ(g)(p). But h γ(g) = (h.γ)(h 1 g). Then, [(h.γ)(h 1 g)](p) = h(h 1 g)(h 1 p) = h(ψ(γ)(h 1 p h 1 g) = h(ψ(γ))(h 1 (p g) = (h ψ(γ)(p g) = ψ(h γ)(p g).

8 8 R. MARTÍNEZ-VILLA Hence ψ(h γ) = h ψ(γ). It follows ψ induces an isomorphisms: ψ : Hom (G, Hom Λ (P, N Hence an isomorphism: W )) G Hom Λ (P G, N W ) G ψ : Hom G (G, Hom Λ (P, N W )) Hom Λ G (P G, N W ). Consider the natural isomorphisms: σ 1 : Hom Λ (P, N) W Hom Λ (P, Λ) Λ N W, σ 2 : Hom Λ (P, Λ) Λ N W Hom Λ (P, N W ), where f Hom Λ (P, N) we have the equality f(p) = f i (p)n i with f i Hom Λ (P, Λ) and n i N. Then, σ 1 (f w) = f i n i w and σ 2 ( f i n i w)(p) = f i (p)n i w = ( f i (p)n i ) w = f(p) w. Hence; σ(f w)(p) = σ 2 σ 1 (f w)(p) = f(p) w. The map α : Hom Λ (P, N W ) Hom G (G, Hom Λ (P, N W )) is the isomorphism α(f)(h) = h f. The natural isomorphism θ is the composition ψασ. Then we have a chain of equalities: θ(h w)(p g) = ψασ(h w)(p g) = ψ(ασ(h w))(p g) = ασ(h w)(g)(p) = g σ(h w)(g)(p) = gσ(h w)(g 1 p) = g[h(g 1 p) w] = ghg 1 p hw = g h(p) gw. Proposition 9. Let G be a finite group of grading preserving automorphisms of the -algebra Λ such that the characteristic of does not divide the order of G. Let M be a graded Λ G-module with a graded projective resolution consisting of finitely generated modules, N a graded Λ G-module and W a G-module. Then for all k we have a natural isomorphism: ˆθ : Ext k Λ(M, N) W Ext k Λ G (M G, N W ). f k f Proof. Let P k Pk 1 P M be a Λ G-projective resolution of M. By lemma 6, each P j is a finitely generated projective Λ-module. Tensoring with G we have an exact sequence of Λ G-modules: P k P k 1 G P G f 1 M G. G f k1

9 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 9 Each P k G is isomorphic to P k as Λ-module, hence projective as Λ- G module. By lemma 6, P k G is a finitely generated graded projective Λ G-module. We have a complex: Hom Λ (P, N) Hom Λ (P k, N) Hom Λ (P k+1, N). Tensoring with W we obtain a complex: Hom Λ (P, N) W Hom Λ (P k, N) W Hom Λ (P k+1, N) W, whose k-th homology is Ext k Λ (M, N) W. By the lemma, we have an isomorphism of complexes: Hom Λ (P, N) W Hom Λ (P G, N W ) θ Hom Λ (P k, N) W Hom Λ (P k+1, N) W θ Hom Λ (P k G, N W ) Hom Λ (P k+1 G, N W ), which induces an isomorphism ˆθ of the homologies, therefore an isomorphism: ˆθ : Ext k Λ(M, N) W Ext k Λ G (M θ G, N W ). Theorem 1. Let Λ be a positively graded -algebra, G a finite grading preserving group of automorphisms of Λ with characteristic of do not dividing the order of G. Let M be a graded Λ G-module with a graded projective resolution consisting of finitely generated modules Γ = Ext k Λ (M, M). k Then the skew group algebra Γ G is isomorphic as graded algebra to ˆΓ = Ext k Λ G (M G, M G). k Proof. We need to prove isomorphism ˆθ preserves multiplication. Let x Ext k Λ (M, N) and y Exts Λ (M, N): x = M E k E k 1 E 1 M, y = M F s F s 1 F 1 M,

10 1 R. MARTÍNEZ-VILLA where the multiplication x y is the Yoneda product: x y = y.x = M F s F s 1 F 1 E k E k 1 E 1 M. α t α Let P t Pt 1 P M be a Λ G-projective resolution of M and Ω t (M) = er α t. To the extension x corresponds a map f : Ω k (M) M, to the extension y a map h : Ω s (M) M and to the Yoneda product y.x corresponds the composition h.ω k (f). We have the following commutative exact diagrams: Ω k (M) P k 1 P M f f k M E k E 1 M, f 1 Ω k s (M) P k+s 1 P k Ω k (M) Ω s (f) Ω s (M) f k P s 1 P f M h h s M F s F 1 M. We need to prove the following diagram: h 1 1 Ext k Λ (M, M) G Ext s Λ (M, M) G ˆθ Ext k Λ G (M G, M G) Ext s Λ G (M G, M G) ν Ext k+s Λ (M, M) G ˆθ µ Ext k+s Λ G (M G, M G) with ν(x g, y t) = (x.g)(y.t) and µ the Yoneda product, commutes. We have the following equalities: (x.g)(y.t) = x.(gy.t) = x.gy gt = gy.x gt and the following correspondences under the natural isomorphisms: x g f g, y t h t. ˆθ

11 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 11 Then the following correspondences: ˆθ(xg) θ(f g), ˆθ(yt) θ(ht). The maps θ(f g), θ(h t) induce exact commutative diagrams: Ω s (M) G P s 1 G P G M G θ(ht) 1 M G ˆFs ˆF1 M, Ω k+s (M) G P k+s 1 G Ω s θ(fg) Ω s M G P s G P k G Ω k (M) G θ(fg) P G M G, where the bottom rows are ˆθ(y t) and ˆθ(x g). We have the following correspondence: ˆθ(y t)ˆθ(x g) = ˆθ(x g) ˆθ(y t) θ(h t)ω s θ(f g). We claim Ω s θ(f g) = θ(ω s f g). We have commutative squares: P k+s i α k+s i P k+s i 1 f k+s i f k+s i 1 P s i α s i P s i 1. It is enough to prove the following square commute: P k+s i G θ(f k+s i g) P s i G α k+s i 1 P k+s i 1 G θ(f k+s i 1 g) α s i 1 P s i 1 G. But we have the following chain of equalities: θ(f k+s i g)α k+s i 1(p t) = θ(f k+s i g)(α k+s i (p) t) = t f k+s i 1 (α k+s i (p) gt) = tf k+s i 1 (t 1 α k+s i (p) gt = tf k+s i 1 (α k+s i (t 1 p) gt = tα s i f k+s i (t 1 p) gt = α s i (tf k+s i (t 1 p) gt = α s i t f k+s i (p) gt = (α s i 1)(t f k+s i (p) gt) = (α s i 1)(θ(f k+s i g)(p t).

12 12 R. MARTÍNEZ-VILLA We also have equalities: θ(h t)ω s θ(f g) = θ(h t)θ(ω s f g)(m l) = θ(h t)(l Ω s f(m) lg) = lg h(l Ω s f(m) lgt) = lg h(lω s f(l 1 m) lgt) = lgh(g 1 l 1 lω s f(l 1 m) lgt) = l(gh(g 1 Ω s f(l 1 m) lgt) = l((g h)ω s f(m) lgt) = θ(g hω s f gt)(m l). We have the following commutative diagrams with exact rows: Ω s M ϕ 1 h Ω s M g 1 h M g 1 P s 1 P g 1 s 1 F g 1 s ϕ g M Fs g 1 P k P g 1 k F g 1 1 F g 1 1 M ϕ g 1 M g 1 1 M g 1 ϕ g M, where the last raw is y g. Since ϕ g hϕ g 1 = g h, then we have a correspondence g h y g, and ˆθ(x g) ˆθ(y t) = ˆθ(x y g, gt). Recall the following definitions from [7] and the references given there: Definition 11. A positively graded -algebra Λ = j Λ j such that each Λ j is finite dimensional over the field and for each pair of integers i j we have equalities Λ i Λ j = Λ i+j, will be called a graded quiver algebra. By J we denote the graded Jacobson radical J = j 1 Λ j. Let M be a finitely generated graded Λ-module, generated in highest degree zero, we say M is a oszul module if F (M) = Ext i Λ (M, Λ/J) is i generated in highest degree zero. We say Λ is oszul if all graded simples generated in degree zero are oszul. Definition 12. Let Λ be a graded quiver algebra, we say Λ is generalized Auslander regular if the following statements are true: i) The algebra Λ has finite, graded, small, global dimension n. ii) All graded simples have projective dimension n. iii) For any graded simple S, Ext k Λ (S, Λ) = for k < n. iv) The functor Ext n Λ (, Λ) induces a bijection between the Λ and the Λ op -graded simples.

13 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 13 Lemma 13. Let Λ be a positively graded -algebra, G a finite group of grading preserving automorphisms of Λ with characteristic of not dividing the order of G. Then Λ G is generalized Auslander regular if and only if Λ is generalized Auslander regular. Proof. Assume Λ is generalized Auslander regular, M a Λ G-module semisimple. Then Ext k Λ G (M G, Λ G) = Ext k Λ (M, Λ) G for all k n. Then by hypothesis, Ext n Λ (M, Λ) is a semisimple Λ G-module. Assume G = n V i, V i an irreducible G-module. Then i=1 Ext n Λ G(M G, Λ G) = m i=1 Ext n Λ G (M V i, Λ G) and each Ext n Λ G (M V i, Λ G) is a semisimple Λ G-module. Let T be a simple Λ G-module with minimal projective resolution: Dualizing by ( P n P n 1 P T. ) = Hom Λ G (, Λ G) we have a complex: ( ) P P 1 P n Ext n Λ G(T, Λ G) and Ext k Λ G (T G, Λ G) = Ext k Λ (T, Λ) G = m Ext k Λ G (T V i, Λ i=1 G) =, imply Ext k Λ G (T, Λ G) = for all k n and the sequence ( ) is exact. Let S = Ext n Λ G (T, Λ G). Dualizing, we have an exact diagram: P n P n 1 P T P n P n 1 P T. Therefore: Ext k Λ G (S, Λ G) = for all k n. If S 1 S2 = S, with S 1, S 2 non zero semisimple Λ G-modules, then Ext k Λ G (S i, Λ G) = for i = 1, 2 and all k n. If Ext n Λ G (S i, Λ G) =, then S i =, a contradiction. Therefore: T = Ext n Λ G (S 1, Λ G) Ext n Λ G (S 2, Λ G), contradicting T is simple. Now if T 1, T 2 are simple Λ G-modules with Ext n Λ G (T 1, Λ G) = Ext n Λ G (T 2, Λ G) and projective resolutions: P n P n 1 P T 1, P n P n 1 P T 2.

14 14 R. MARTÍNEZ-VILLA Dualizing, we have exact sequences: (P ) (P 1) (P n) Ext n Λ G(T 1, Λ G), (P ) (P 1 ) (P n ) Ext n Λ G(T 2, Λ G). From the fact Ext n Λ G (T 1, Λ G) = Ext n Λ G (T 2, Λ G) we have isomorphisms: (P j ) = (P j ), in particular (P ) = (P ). Then P = P and T 1 = T 2. Now assume Λ G is generalized Auslander regular, S Λ a simple Λ-module. The module T defined as T = gs is a simple Λ G-module. Then Ext k Λ G (T G, Λ G) = Ext k Λ G (T, Λ) G. Decomposing G = m V i where each V i is irreducible and using the fact Ext k Λ (T, Λ) G = i=1 m Ext k Λ G (T V i, Λ G), we obtain Ext k Λ G (T V i, Λ G) = if k n. i=1 Therefore: Ext k Λ (T, Λ) = for all k n. m Since Ext n Λ (T, Λ) G = Ext n Λ G (T V i, Λ G), where Ext n Λ G ( i=1 T V i, Λ G) is a semisimple Λ G-module, then Ext n Λ (T, Λ) G is a semisimple Λ-module. The module S is a submodule of the semisimple Λ-module T, hence; a summand. It follows Ext n Λ (S, Λ) Extn Λ (T, Λ), hence Ext n Λ (S, Λ) is a semisimple Λ-module. Let P n P n 1 P T be a minimal projective resolution of the Λ-module S, dualizing with respect to Λ we have an exact sequence: g G (P ) (P 1 ) (P n ) Ext n Λ(S, Λ). Dualizing again we have isomorphisms: P n P n P n 1 P n 1 P P S Ext n Λ (Extn Λ (S, Λ), Λ). It follows S = Ext n Λ (Extn Λ (S, Λ), Λ). If S is simple, then Extn Λ (S, Λ) = S 1 S2, S 1 S 2. This implies S = Ext n Λ (S 1, Λ) Ext n Λ (S 2, Λ). If Ext n Λ (S i, Λ) =, then Ext k Λ (S i, Λ) = for all i, a contradiction. It follows Ext n Λ (S, Λ) is simple. Theorem 14. Let G be a finite group of automorphisms of a graded - algebra Λ. Assume characteristic of the field does not divide the order of the group, let M be a finitely generated graded Λ G-module. Then M G

15 SEW GROUP ALGEBRAS AND THEIR YONEDA ALGEBRAS 15 is a oszul Λ G-module if and only if M is a oszul Λ-module. In particular Λ is oszul if and only if Λ G is oszul. Proof. We have a natural isomorphisms: Ext k Λ G(M G, Λ G) = Ext k Λ G (M, Λ) G and Ext k Λ G(M G, Λ G) Ext j Λ G (Λ G, Λ G) = (Ext k Λ(M, Λ ) G)(Ext j Λ (Λ, Λ ) G) = Ext k Λ(M, Λ ) Ext j Λ (Λ, Λ ) G. Therefore, Ext k Λ G (M G, Λ G) Ext j Λ G (Λ G, Λ G) = Ext k+j Λ G ( M G, Λ G) if and only if Ext k Λ (M, Λ ) Ext j Λ (Λ, Λ ) = Ext k+j Λ (M, Λ ). It follows M is oszul if and only if M G is oszul, in particular Λ is oszul if and only if Λ G is oszul. References [1] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. (2) 293(1986), [2] M. Auslander and I. Reiten, Almost split sequences for rational double points, Trans. Amer. Math. Soc. (1) 32(1987), [3] M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282(1984), [4] W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of leinian singularities, Duke Math. J. 92(1998), [5] F. lein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fümften Grade, Teubner, Leipzig, [6] H. Lenzing, Curve singularities arising from the representation theory of tame hereditary algebras, Rep. Theory I. Finite dim. Algebras , Lecture Notes in Math. 1177, Springer-Verlag, [7] R. Martínez-Villa, Serre duality for generalized Auslander regular algebras, Trends in Rep. Theory of finite dimensional Algebras, Contemporary Math. 229(1998), [8] J. Mcay, Graphs, singularities and finite groups, Proceedings of Symposia in Pure Mathematics 37(198), [9] I. Reiten and C. Riedtmann, Skew Group Algebras in the Representation Theory of Artin Algebras, J. of Algebra (1) 92(1985), [1] I. Reiten and M. Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc. 8, 1989.

16 16 R. MARTÍNEZ-VILLA Roberto Martínez-Villa Instituto de Matemáticas, UNAM Unidad Morelia Apartado Postal 61-3 Morelia 5889, Mich. México address: (Received July 13, 21 )

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES KENTA UEYAMA Abstract. Gorenstein isolated singularities play an essential role in representation theory of Cohen-Macaulay modules. In this article,