FROBENIUS MORPHISM AND VECTOR BUNDLES ON CYCLES OF PROJECTIVE LINES IGOR BURBAN Abstract. In this paper we describe the action of the Frobenius morphism on the indecomposable vector bundles on cycles of projective lines. This gives an answer on a question of Paul Monsky, which appeared in his study of the Hilbert Kunz theory for plane cubic curves. This note arose as a answer on a question posed by Paul Monsky in his study of the Hilbert Kunz theory for plane cubic curves [5]. Let k be a field of characteristic p > 0 and E be a plane rational nodal curve or a cycle of projective lines over k. Our goal is to describe the action of the Frobenius morphism on the set of indecomposable vector bundles on E. We start with recalling the general technique used in a study of vector bundles on singular projective curves, see [3, 1, 2]. Let X be a reduced singular projective) curve over k, π : X X its normalization and I := HomO π O X), O ) = Ann O π O X)/O ) the conductor ideal sheaf. Denote by η : Z = V I) X the closed artinian subspace defined by I its topological support is precisely the singular locus of X) and by η : Z X its preimage in X, defined by the Cartesian diagram 1) π Z η X π Z η X. Definition 1. The category of triples TriX) is defined as follows. Its objects are triples F, V, m ), where F VB X), V VBZ) and m : π V η F is an isomorphism of O Z-modules, called the gluing map. The set of morphisms Hom TriX) F1, V 1, m 1 ), F 2, V 2, m 2 ) ) consists of all pairs F, f), where F : F 1 F 2 and f : V 1 V 2 are morphisms of vector bundles 2000 Mathematics Subject Classification. Primary 14H60, 14G17. 1
2 IGOR BURBAN such that the following diagram is commutative π V 1 m 1 η F1 π f) π V 2 m 2 η F2. Theorem 2 Lemma 2.4 in [3] and Theorem 1.3 in [2]). Let X be a reduced curve. Then the functor F : VBX) TriX) assigning to a vector bundle F the triple π F, η F, m F ), where m F : π η F) η π F) is the onical isomorphism, is an equivalence of categories. Remark 3. In the case when X is a configuration of projective lines intersecting transversally, the above theorem also follows from a more general result of Lunts [4]. For a ringed space Y, O Y ) over the field k we denote by ϕ Y the Frobenius morphism Y, O Y ) Y, O Y ). Then for an open set U Y the algebra homomorphism ϕ Y U) : O Y U) O Y U) is given by the formula ϕ Y f) = f p, f O Y U). For a sake of simplicity, we shall omit the subscript in the notation of the Frobenius map. Definition 4. The endofunctor P : TriX) TriX) is defined as follows. Let T = F, V, m) be an object of the category TriX). Then PT ) := ϕ F, ϕ V, m ϕ ), where the gluing map m ϕ is determined via the commutative diagram η F ) ϕ π V π ϕ V ϕ m) m ϕ ϕ η F η ϕ F, where the vertical maps are onical isomorphisms. Lemma 5. Consider the following diagram of categories and functors: VBX) T VBX) F F TriX) P TriX), where for a vector bundle F on X we set: TF) = ϕ X F). Then there exists an isomorphism of functors P F F T.
FROBENIUS MORPHISM AND VECTOR BUNDLES 3 Proof. Let F be a vector bundle on X. Then the onical isomorphisms ϕ η F η ϕ F and ϕ π F π ϕ F induce the commutative diagram π ϕ η F m ϕ F η ϕ π F π η ϕ F m ϕ F η π ϕ F, which yields the desired isomorphism of functors. Next, we need a description of the action of the Frobenius map on the vector bundles on a projective line. Let z 0, z 1 ) be coordinates on V = C 2. They induce homogeneous coordinates z 0 : z 1 ) on P 1 = P 1 V ) = V \ {0} ) /, where v λv for all v V and λ C. We set U 0 = { z 0 : z 1 ) z 0 0 } and U = { z 0 : z 1 ) z 1 0 } and put 0 := 1 : 0), := 0 : 1), z = z 1 /z 0 and w = z 0 /z 1. So, z is a coordinate in a neighbourhood of 0. If U = U 0 U and w = 1/z is used as a coordinate on U, then the transition function of the line bundle O P 1n) is 2) U 0 C U C z,v) 1 z, v z n ) U C U C. Using the formula 2), the proof of the following lemma is straightforward. Lemma 6. For any n Z we have: ϕ O P 1n) ) = OP 1np). Next, recall the following classical result on vector bundles on a projective line. Theorem 7 Birkhoff Grothendieck). Any vector bundle F on P 1 splits into a direct sum of line bundles: 3) F = O P 1l) m l. l Z Now assume that E is an irreducible plane nodal cubic curve. Theorem 7 implies that for an object F, V, m) of the category of triples TriE) with rk F) = n, we have F = O P 1l) m l and V = OZ, n where m l = n. l Z l Z Note that V is in fact free, because Z is artinian. From now on we shall always fix a decomposition of F as above. In order to describe the morphism m in the terms of matrices, some additional choices have to be done. Recall that the vector bundle O P 1 1) is isomorphic to the sheaf of sections of the so-called tautological line bundle { } l, v) v l P 1 V ) V = O 2 P 1.
4 IGOR BURBAN The choice of coordinates on P 1 fixes two distinguished elements, z 0 and z 1, in the space Hom P 1 OP 1 1), O P 1) : P 1 C 2 O P 1 1) P 1 z i P 1 C where z i maps l, v 0, v 1 ) ) to l, v i ) for i = 0, 1. It is clear that the section z 0 vanishes at and z 1 vanishes at 0. After having made this choice, we may write Hom P 1 OP 1n), O P 1m) ) = C[z 0, z 1 ] m n := z0 m n, z0 m n 1 z 1,..., z1 m n In what follows we shall assume that the coordinates on the normalization Ẽ are chosen in such a way that Speck k) = Z = π 1 Z) = {0, }. Definition 8. For any l Z we define the isomorphism ξ l : η O P 1l) ) O Z by the formula ξ l p) = p z l 0 0), p z l 1 ) ), where p is an arbitrary local section of the line bundle O P 1l). Hence, for any vector bundle F of rank n on P 1 given by the formula 3), we have the induced isomorphism ξ F : η F O ñ Z. Let F, OZ n, m) be a object of the category of triples TriE). Then the morphism m be presented by a matrix Mm) via the following commutative diagram 4) π O n Z O ñ Z m Mm) η F ξ F O ñ Z, where the first vertical map is the onical isomorphism. Hence, the morphism Mm) is given by a pair of invertible n n) matrices M0) and M ) over the field k. Applying to 4) the functor ϕ, we get the following commutative diagram: C. π ϕ O n Z m ϕ η ϕ F ϕ π O n Z ϕ m) ϕ η F 5) ϕ O ñ Z ) ϕ Mm)) ϕ O ñ Z ϕ ξ F ) ) ξ ϕ F O ñ Z Mm ϕ ) O ñ Z.
FROBENIUS MORPHISM AND VECTOR BUNDLES 5 Corollary 9. Let E be an irreducible nodal cubic curve over a field k of characteristic p > 0 and F be a vector bundle corresponding to the triple F, OZ n, m), where F = l Z O P 1l) m l and m is given by a pair of matrices a 11 a 12... a 1n b 11 b 12... b 1n a M0) = 21 a 22... a 2n b and M ) = 21 b 22... b 2n. a n1 a n2... a nn b n1 b n2... b nn Then the vector bundle ϕ F is given by the triple ϕ F, O n Z, m ϕ ), where ϕ F = l Z O P 1lp) m l and m ϕ corresponds to the pair of matrices a p 11 a p 12... a p 1n b p a p 21 a p 22... a p 11 b p 12... b p 1n 2n and b p 21 b p 22... b p 2n. a p n1 a p n2... a p nn b p n1 b p n2... b p nn Corollary 10. Indecomposable vector bundles on an irreducible nodal cubic curve E over an algebraically closed field k are described by a non-periodic sequence of integers d 1,..., d l ), a positive integer m and a continuous parameter λ k, see [3, 1, 2]. The normalization of the corresponding vector bundle B d 1, d 2,..., d l ), m, λ ) is l i=1o P 1d i ) m. The gluing map m is described by a pair of matrices M0) and M ), which are given by the explicit formulae written in [1, Section 3.1]. In these notations, the action of ϕ on the indecomposable vector bundles on E takes the following form: 6) ϕ B d 1, d 2,..., d l ), m, λ ) = B pd1, pd 2,..., pd l ), m, λ p). Remark 11. The same argument applies literally to the case, when E is a cycle of projective lines. In particular, the formula 6) holds in that case, too. However, we have to stress that the parameter λ k arising in the formula 6) has no intrinsic meaning as an invariant of a vector bundle and its value actually depends on the choice of the trivializations { ξ l, fixed in Definition 8. }l Z References [1] L. Bodnarchuk, I. Burban, Yu. Drozd and G.-M. Greuel, Vector bundles and torsion free sheaves on degenerations of elliptic curves, Global Aspects of Complex Geometry, 83 129, Springer 2006). [2] I. Burban, Abgeleitete Kategorien und Matrixprobleme, PhD Thesis, Kaiserslautern 2003, available at http://deposit.ddb.de/cgi-bin/dokserv?idn=968798276. [3] Yu. Drozd, G.-M. Greuel, Tame and wild projective curves and classification of vector bundles, J. Algebra 246 2001), no. 1, 1 54. [4] V. Lunts, Coherent sheaves on configuration schemes, J. Algebra 244 2001), no. 2, 379 406. [5] P. Monsky, Hilbert Kunz theory for nodal cubics, via sheaves, preprint.
6 IGOR BURBAN Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany E-mail address: burban@math.uni-bonn.de