有限亞丁模上的龐曲爾根分解

Abstract

We want to study the structure of vector bundles after tenser product. But by cite{mukai}'s result, to investigate tensor product of vector bundles is equivalent to investigate Pontryagin product on the corresponding objects. And if the vector bundles are unipotent, cite{mukai} presents the equivalence between {Unipotent vector bundles on X} and {Coherent sheaves on hat X supported by hat0}={Artinian R'-modules}, where R' is a local ring. So we are going to study modules over Artinian local ring in order to know the structure on unipotent vector bundles. By Krull-Schmidt Theorem (cf. cite{jacobson} p115), , we know the decomposition is unique up to isomorphism. We first restrict our attention on the case of one dimensional local ring. In this case, let F be a field, R:=F[x], R':=R_{(x)} be the one dimensional local ring. We know that the indecomposable Artinian R'-module is of the form F[x]/(x^n) i.e. the decomposition of Artinian R'-module is the direct sum of this form. In particular, the decomposition of Pontryagin product is the direct sum of this form. We will see the decomposition in exact form, and the result is compatible with cite{atiyah}'s result in decomposition of tensor product of indecomposable unipotent vector bundles via the Fourier-Mukai transform. Secondly, we want to know the two dimensional local ring case. Let R=F[t_1,t_2], R'=R_{(t_1,t_2)}. In this case, indecomposable Artinian R'-modules could be subtle. We don't even understand in general how to decide if a R'-module is indecomposable. In order to know this structure more clearly, We first transform this problem into decomposition of Artinian R-module of the form M/N with M a finitely generated free R-module and I_{Max}^cMsubset N for some (large) integer c, where I_{Max}=(t_1,t_2) is the maximal ideal of R. To decompose R-module of the form M/N, we have the following result: N is `strongly' decomposable (see definition ef{st}), iff M/N is decomposable. So we transform our problem into `strongly' decomposition of N who satisfy I_{Max}^cMsubset N for some c. To `strongly' decompose N is equivalent to decompose a vector field M_0=V_1oplus V_2 s.t. N=(Ncap RV_1)oplus (Ncap RV_2) (see notation ef{noa}). But for some given N, we still don't know how to decompose M_0 in general, only some easy cases can be handled.