Local cohomology with support in ideals of maximal minors

TL;DR

This paper investigates the local cohomology modules supported on ideals generated by maximal minors of a matrix, identifying vanishing indices, computing the highest nonvanishing module, and characterizing all nonzero modules in terms of indecomposable injectives.

Contribution

It provides explicit descriptions of local cohomology modules for ideals of maximal minors, extending understanding through group action techniques and indecomposable injective modules.

Findings

Identifies vanishing indices for local cohomology modules.

Computes the highest nonvanishing local cohomology module.

Characterizes all nonzero modules as submodules of indecomposable injectives.

Abstract

Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings.