A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space.

Transcription

1 A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space. DIETER HAPPEL AND DAN ZACHARIA Abstract. Let P n be the projective n space over the complex numbers. In this note we show that an indecomposable rigid coherent shea on P n has a trivial endomorphism algebra. This generalizes a result o Drezet or n = 2. Let P n be the projective n-space over the complex numbers. Recall that an exceptional shea is a coherent shea E such that Ext i (E, E) = 0 or all i > 0 and End E = C. Drézet proved in [4] that i E is an indecomposable shea over P 2 such that Ext 1 (E, E) = 0 then its endomorphism ring is trivial, and also that Ext 2 (E, E) = 0. Moreover, the shea is locally ree. Partly motivated by this result, we prove in this short note that i E is an indecomposable coherent shea over the projective n-space such that Ext 1 (E, E) = 0, then we automatically get that End E = C. The proo involves reducing the problem to indecomposable linear modules without sel extensions over the polynomial algebra. In the second part o this article, we look at the Auslander-Reiten quiver o the derived category o coherent sheaves over the projective n-space. It is known ([10]) that i n > 1, then all the components are o type ZA. Then, using the Bernstein-Geland-Geland correspondence ([3]) we prove that each connected component contains at most one shea. We also show that in this case the shea lies on the boundary o the component. Throughout this article, S will denote the polynomial ring in n + 1 variables with coeicients in C. Its Koszul dual is the exterior algebra in n + 1 variables x 0, x 1,..., x n which we denote by R. It is well known that R is a graded, local, inite dimensional algebra over C. Moreover, R is a selinjective algebra, that is, the notions o ree and o injective R-modules coincide. Denote by Lin R and Lin S the categories o linear R-modules (linear S-modules respectively), that is, o the modules having a linear ree resolution. It is well known that both Lin R and Lin S are closed under extensions and cokernels o monomorphisms. We have mutual inverse Koszul dualities: Lin R E F Lin S given by E(M) = n 0 Extn R(M, C) with the obvious action o S on E(M), and with the inverse duality F deined in a similar way Mathematics Subject Classiication. Primary 14F05, 16E10. Secondary 16E65, 16G20, 16G70. Key words and phrases. Exterior algebra, projective space, coherent sheaves, vector bundles, exceptional objects. The second author is supported by the NSA grant H

2 2 DIETER HAPPEL AND DAN ZACHARIA Let mod Z R be the category o initely generated graded R-modules and graded (degree 0) homomorphisms, where we denote by Hom R (M, N) 0 the degree zero homomorphisms rom M to N. Denote by mod Z R the stable category o initely generated graded R-modules. The objects o mod Z R are the initely generated graded R-modules, and the morphisms are given by Hom R (M, N) 0 = Hom R (M, N) 0 /P(M, N) 0 where P(M, N) 0 is the space o graded homomorphisms rom M to N actoring through a ree R-module. The stable graded category mod Z R is a triangulated category where the shit unctor is given by the irst cosyzygy, that is, every distinguished triangle in the stable category is o the orm A B C Ω 1 A where Ω: mod Z R mod Z R denotes the syzygy unctor. Note also that the stable category mod Z R has a Serre duality: DHom R (X, Ω m Y ) 0 = HomR (Y, Ω m+1 X(n + 1)) 0 or each integer m and all R-modules X and Y, where X(n + 1) is the graded shit o X. It is immediate to see that we can derive the well-known Auslander-Reiten ormula rom the above in the case when m = 1. Let M = M i be a initely generated graded module over the exterior algebra R and let ξ be a homogeneous element o degree 1 in R. Following [3], we can view the let multiplication by ξ on M as inducing a complex o K-vector spaces L ξ M : M i 1 ξ M i ξ M i+1 We call the graded module M nice or proper, i the homology H i (L ξ M) = 0 or each i 0 and or each homogeneous element ξ 0 o degree 1. It is well-known that a graded module M is ree i and only i H i (L ξ M) = 0 or each i Z and or each nonzero homogeneous element ξ R 1. The derived category D b (coh P n ) o coherent sheaves over the projective space is also triangulated and, the BGG correspondence (see [3]) Φ: mod Z R D b (coh P n ) is an exact equivalence o triangulated categories, that assigns to each initely generated graded R-module the equivalence class o a bounded complex o vector bundles on P n. I M = M i, we use also M i to denote the trivial bundle with inite dimensional iber M i and we set: Φ(M):... O(i) M i δ i O(i + 1) M i+1... where the dierentials δ i are deined in the ollowing way. Let x 0, x 1,..., x n be a basis o V, and let ξ 0, ξ 1,..., ξ n denote the dual basis o V. Then δ i = n j=0 ξ j x j. Moreover or each nice R-module M, the complex Φ(M) is quasi-isomorphic to the stalk complex o a vector bundle over the projective n-space, and all the vector bundles can be obtained in this way rom the nice R-modules. Throughout this article i M is a graded module, then its graded shit, denoted by M(i), is the graded module given by M(i) n = M i+n or all integers n. I M is a graded module, we denote its truncation at the k-th level by M k = M k M k+1

3 A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space The endomorphism ring o a rigid shea. Let F be an indecomposable coherent shea over the projective n-space. Assume also that F is rigid, that is Ext 1 (F, F ) = 0. We prove in this section that the endomorphism ring o F is trivial. This is done by reducing the problem to indecomposable rigid linear modules over the polynomial algebra S in n + 1 variables. Let M be a initely generated, graded, indecomposable non projective S-module. We say that M has no graded sel-extensions, i every short exact sequence o graded S-modules 0 M X M 0 splits, that is Ext 1 S(M, M) 0 = 0. In order to prove our next result we recall the ollowing deinition and acts rom [5]. We say that a module M mod Z S has a linear presentation i it has a graded ree presentation P 1 P 0 M 0 where P 0 is generated in degree 0 and P 1 is generated in degree 1. Obviously such a linear presentation is always minimal. A stronger notion is that o a linear module. We say that a graded module M is linear, i it has a minimal graded ree resolution: P k P 1 P 0 M 0 where or each i 0, the ree module P i is generated in degree i. I we denote by L S the subcategory o mod Z S consisting o the modules with a linear presentation, then we have an exact equivalence o categories L S = LS/J 2 where J denotes the maximal homogeneous ideal o S, ([5]). This equivalence is given by assigning to each module M having a linear presentation over S, the module M/J 2 M. We start with the ollowing background result: Lemma 1.1. An exact sequence 0 M E N 0 in L S splits i and only i the induced sequence 0 M/J 2 M E/J 2 E N/J 2 N 0 splits too. Proo. To prove this, it is enough to show that a monomorphism M E in L S splits i and only i the induced map M/J 2 M E/J 2 E is also a splittable monomorphism, and one direction is trivial. For the other direction, recall that the category L S is equivalent to the category C whose objects are graded morphisms : P 1 P 0 between ree S-modules, where P 0 is generated in degree zero, P 1 is generated in degree one, and the degree one component o is a monomorphism rom the degree one component o P 1 to the degree one component o P 0. A morphism in C rom : P 1 P 0 to g : Q 1 Q 0 is a pair (u 1, u 0 ) o graded homomorphisms such that the diagram P 1 P 0 Q 1 u 1 g Q 0 u 0 P0 is commutative. Let M and E be two modules with linear presentations P 1 g and Q 1 Q0, and let h: M E be a homomorphismm such that the induced map h: M/J 2 M E/J 2 E is a splittable monomorphism. We have an exact

4 4 DIETER HAPPEL AND DAN ZACHARIA commutative diagram: P 1 /J 2 P 1 P 0 /J 2 P 0 π M M/J 2 M 0 h 1 Q 1 /J 2 Q 1 g h 0 Q 0 /J 2 Q 0 π E h E/J 2 E 0 q 1 P 1 /J 2 P 1 q 0 P 0 /J 2 P 0 π M M/J 2 M 0 q where q h = 1 M/J 2 M. Observe that the litings q 1 and q 0 also have the property that q 1 h1 = 1 P1/J 2 P 1 and q 0 h0 = 1 P0/J 2 P 0. By liting to L S we obtain the ollowing exact commutative diagram: P 1 P 0 π M M 0 Q 1 h 1 g Q 0 h 0 h π E E 0 P 1 q 1 q 0 P 0 π M M 0 and we have that q 1 h 1 = 1 P1 and q 0 h 0 = 1 P0. There exists an unique homomorphism l : E M such that lπ E = π M q 0, and it ollows immediately that lh = 1 M. We will also need the ollowing results about graded modules over the polynomial algebra S = C[x 0,... x n ]: Lemma 1.2. Let 0 A B g C 0 be an exact sequence in mod Z S where A is a linear module and C is semisimple generated in a single degree m 0. Then the sequence splits. m g m Proo. It suices to show that the sequence 0 A m B m C 0 splits in the category o graded S-modules since we can easily construct a right inverse to g rom a right inverse to g m. Each o the modules A m, B m, C is an m-shit o a linear S-module, so by Koszul duality we obtain an exact sequence over the exterior algebra R: 0 E(C) E(B m ) E(A m ) 0 But R is selinjective and E(C) is ree, so this sequence splits. The lemma ollows now immediately. Lemma 1.3. Let C be a linear S-module. Let i be a positive integer and assume that Ext 1 S(C i, C) 0 = 0. Then Ext 1 S(C i, C i ) 0 = 0.

5 A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space. 5 Proo. Assume to the contrary that Ext 1 S(C i, C i ) 0 0. We have the ollowing pushout diagram o graded modules: 0 C i α X β C i 0 0 C j γ Y h δ C i 0 where j is the inclusion map. The bottom sequence splits by assumption so we have homomorphisms q : C i Y and p: Y C such that δq = 1 and pγ = 1. But then each graded component o γ, δ splits so we obtain an induced split exact sequence 0 C i Y i C i 0 and an induced commutative diagram: 0 C i α X β C i 0 0 C i γ i Y i δ i C i 0 and since the two extensions represent the same element in Ext 1 they both split. The ollowing result holds or an arbitrary Koszul algebra over the complex numbers. Lemma 1.4. Let Γ be a Koszul algebra and let M L Γ be an indecomposable non projective Γ-module o Loewy length two having no graded sel-extensions. Then End Γ (M) 0 = C. Proo. The proo ollows the same idea as the one given in [7], but we present it here or the reader s convenience. Assume that there exists a non zero graded homomorphism : M M that is not an isomorphism, and let N denote its image and L its kernel. Furthermore, let T denote the cokernel o the inclusion map N M. Note that T is also generated in degree zero, and that L lives in degree 1 and possibly also in degree zero. Applying Hom Γ (T, ) to the exact sequence 0 L M N 0 and taking only the degree zero parts we obtain we get an exact sequence Ext 1 Γ(T, M) 0 Ext 1 Γ(T, N) 0 Ext 2 Γ(T, L) 0. But i P 2 P 1 P 0 T 0 is the beginning o a graded minimal projective resolution o T, the second term P 2 is generated in degrees 2 or higher, hence Hom Γ (P 2, L) 0 = 0 and thereore Ext 2 Γ(T, L) 0 vanishes too. We obtain a surjection Ext 1 Γ(T, M) 0 Ext 1 Γ(T, N) 0, and hence a commutative exact diagram o graded Γ-modules o Loewy length 2 and degree zero maps: 0 M X T 0 0 N M T 0 This pushout yields an exact sequence o graded Γ-modules 0 M N X M 0 h

6 6 DIETER HAPPEL AND DAN ZACHARIA Our assumption on M implies that this sequence must split. The Krull-Remak- Schmidt theorem yields a contradiction, since N 0, M is indecomposable and l(n) < l(m). We can proceed now with the proo o our main result: Theorem 1.5. Let F be an indecomposable coherent shea over the projective space P n such that Ext 1 (F, F ) = 0. Then, End F = C. Proo. By Serre s theorem, we may identiy the category o coherent sheaves over P n with the quotient category Q mod Z S o mod Z S. Recall that the objects o Q mod Z S are the objects o mod Z S modulo the graded modules o inite dimension. Let X denote the sheaiication o X, that is the image o X in the quotient category. I X and Y are two graded S-modules, then X = Ỹ in Q modz S i and only i or some integer k, their truncations X k and Y k are isomorphic as graded S-modules. Now, let F be an indecomposable shea with Ext 1 (F, F ) = 0. We may assume that F = X where X is a linear S-module up to some shit ([1]), and that X has no inite dimensional submodules. Then (see [9] or instance), End(F ) = lim Hom S (X k, X) 0 and Ext 1 (F, F ) = lim Ext 1 S(X k, X) 0. For each k, we have an exact sequence o graded S-modules 0 X k+1 X k S k 0 where S k denotes the semisimple module X k /X k+1 concentrated in degree k. Applying Hom S (, X) and taking degrees we obtain a long exact sequence 0 Hom S (X k, X) 0 Hom S (X k+1, X) 0 Ext 1 S(S k, X) 0 Ext 1 S(X k, X) 0 Ext 1 S(X k+1, X) 0 since Hom S (S k, X) 0 = 0 by our assumption on X. Ext 1 S(S k, X) 0 also vanishes by Lemma 1.2., since having a nontrivial extension 0 X Y S k 0 yields a nonsplit exact sequence o degree k shits o linear modules 0 X k Y k S k 0. Since or each k, Hom S (X k, X) 0 = End S (X k ), we see that we have End(F ) = End S (X) 0 = EndS (X k ) 0 or each k 0. At the same time we have a sequence o embeddings Ext 1 S(X, X) 0 Ext 1 S(X k, X) 0 Ext 1 S(X k+1, X) 0 Thus, the assumption that Ext 1 (F, F ) = 0 implies that or each k we must have Ext 1 S(X k, X) 0 = 0, in particular Ext 1 S(X, X) 0 = 0. X is a shit o a linear module so we may apply Lemma 1.1. We now apply 1.4. to conclude that the shea F has trivial endomorphism ring. 2. Auslander-Reiten components containing sheaves We start with the ollowing general observation: Lemma 2.1. Let A [ u1 u 2 ] [ v 1 v 2 ] B 1 B 2 C A[1] be a triangle in a Krull-Schmidt triangulated category T, and assume that C is indecomposable and both maps v 1, v 2 are nonzero. Then Hom T (A, C) 0. Proo. We prove that v 1 u 1 0. Assume that v 1 u 1 = 0. Then, since the composition [ v 1 0 ][ ] u 1 : A C is equal to 0, we have an induced homomorphism u 2

7 A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space. 7 ψ : C C such that ψ [ ] [ v 1 v 2 = v1 0 ]. Thus ψv 1 = v 1 and ψv 2 = 0. Since C is indecomposable, ψ is either an isomorphism or it is nilpotent. It cannot be nilpotent since v 1 0 and it cannot be an isomorphism since v 2 0, hence a contradiction. The BGG correspondence Φ is an exact equivalence o triangulated categories, hence or every graded R-module M, we have that Φ(ΩM) = Φ(M)[ 1]. An easy computation also shows that or each initely generated graded module M and integer i, we have that Φ(M(i)) O(i) = Φ(M)[i]. It turns out that the modules corresponding to sheaves are located on the boundary o the Auslander-Reiten component containing them. This and more ollows immediately rom the next result by playing with the BGG correspondence. Proposition 2.2. Let B and C be two R-modules corresponding to sheaves under the BGG correspondence. Then, or each i > 0 Hom R (τ i B, C) 0 = 0. Proo. Since taking syzygies and graded shits are sel-equivalences o mod Z R, we have by [10] that C is a weakly Koszul module, and so we can use the ormula τb = Ω 2 B(n + 1), and so τ i B = Ω 2i B(ni + i) or all i 1. We have then Φ(τ i B) = Φ(Ω 2i B(ni+i)) = Φ(Ω 2i B)[ni+i] O( ni i) = Φ(B)[ni i] O( ni i). Applying the BGG correspondence, we obtain: Hom R (τ i B, C) 0 = HomDb (coh(p n ))(Φ(τ i B), Φ(C)) = Hom D b (coh(p n ))(Φ(B)[ni i] O( ni i), Φ(C)) = Hom D b (coh(p n ))(Φ(B)( ni i)[ni i], Φ(C)) = 0 where the last equality holds since both Φ(C) and Φ(B)( ni i) are sheaves, n 2, so ni i 1. We have the ollowing immediate consequence by letting B = C and i = 1 in the previous proposition, and applying Lemma 2.1.: Corollary 2.3. Let C be an indecomposable R-module corresponding to a shea under the BGG correspondence, where n 2. Then the Auslander-Reiten sequence ending at C has indecomposable middle term. We end with the ollowing observation: Proposition 2.4. Let C be a connected component o the Auslander-Reiten quiver o D b (coh P n ) or n > 1. Then C contains at most one indecomposable coherent shea. Proo. By the previous results, i C contains a shea, then this shea must lie on the its boundary. Under the BGG correspondence, two sheaves in C must correspond to two modules o the orm C and τ i C or some i 1. But by applying the BGG correspondence we get stalk complexes concentrated in dierent degrees, hence we obtain a contradiction. Reerences [1] Avramov, L., Eisenbud, D, Regularity o modules over a Koszul algebra, Journal o Algebra 153 (1992),

8 8 DIETER HAPPEL AND DAN ZACHARIA [2] Auslander, M., Reiten, I., Smalø, S. O.; Representation Theory o Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, (1995). [3] Bernstein, I.N., Geland, I.M., Geland, S.I.; Algebraic bundles over P n and problems in linear algebra, Funct. Anal. Appl., 12, (1979), [4] Drézet, J.-M., Fibrés exceptionnels et suite spectrale de Beilinson généralisée sur P 2 (C). Math. Ann. 275 (1986), no. 1, [5] Green, E. L., Martínez-Villa, R., Reiten, I., Solberg, Ø., Zacharia, D.; On modules with linear presentations, J. Algebra 205, (1998), no. 2, [6] Happel, D., Triangulated categories in the representation theory o inite-dimensional algebras. London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, [7] Happel, D., Ringel, C,M,; Tilted Algebras, Transactions o the American Mathematical Society, 274, No. 2, 1982, pp [8] Huybrechts, D.; Fourier-Mukai Transorms in Algebraic Geometry, Oxord Mathematical Monographs. Clarendon Press, [9] Martínez-Villa, R.; Koszul algebras and sheaves over projective space arxiv math.rt/ [10] Martinez-Villa, R.; Zacharia, D. Approximations with modules having linear resolutions. J. Algebra 266 (2003) [11] Martinez-Villa, R.; Zacharia, D. Auslander-Reiten sequences, locally ree sheaves and Chebyshe polynomials. Compos. Math. 142 (2006), no. 2, Fakultät ür Mathematik, Technische Universität Chemnitz, Chemnitz, Germany. happel@mathematik.tu-chemnitz.de Department o Mathematics, Syracuse University, Syracuse, NY , USA. zacharia@syr.edu