HILBERT-KUNZ FUNCTIONS OF 2 × 2 DETERMINANTAL RINGS

摘要

Let κ be an arbitrary field (of arbitrary characteristic) and let X = [x_(i,j)] be a generic m×n matrix of variables. Denote by I_2(X) the ideal in κ[X] = κ[x(i,j) : i = 1,...,m; j = 1,... ,n] generated by the 2×2 minors of X. Using Grobner basis, we give a recursive formulation for the lengths of the κ[X]-module κ[X]/(I_2(X) + (x_(1,1)~q ,..., x_(m,n)~q)) as q varies over all positive integers. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We apply our method to give closed forms for these Hilbert-Kunz functions for cases m ≤ 2.