The Elementary Transformation of Vector Bundles on Regular Schemes

We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves. 0. Introduction An elementary transformation is the theory on ruled surfaces, which enables us to construct a new ruled surface from one given through birational geometry. In [Mar], Maruyama generalized this method to apply to the construction theory of vector bundles. By using his idea and theory, we can construct a lot of indecomposable vector bundles on schemes, especially those on low dimensional projective varieties. On the other hand in [Su-2] and [Su-3], Sumihiro gave an another definition of an elementary transformation of vector bundles on schemes, which is closely related to the geometric characterization of the elementary transformation of ruled surfaces. Let us review them shortly. The definition of an elementary transformation given by Maruyama is, roughly speaking, to give a surjection from the given vector bundle to the vector bundle whose support is a divisor. This is very useful to construct vector bundles and has a lot of applications and examples. However, on higher dimensional varieties, there is a disadvantage that not all vector bundles can be constructed from trivial bundles by this method. The definition given by Sumihiro can be applied to the vector bundle construction on higher dimensional varieties, i.e., by using his theory, we can construct, up to tensoring line bundles, all vector bundles on any dimensional nonsingular quasi-projective varieties over an algebraically closed field from trivial bundles. To apply this elementary transformation, we need some geometric data. Note that the explicit relation between these two was not clear. In this article, we give a new definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on their divisors. This is a natural generalization of Maruyama’s definition, and in the special case, it can be interpreted into Sumihiro’s definition, i.e., by this theory, Received by the editors July 16, 2004 and, in revised form, July 23, 2005. 2000 Mathematics Subject Classification. Primary 14F05. c ©2007 American Mathematical Society Reverts to public domain 28 years from publication