Bounds for codimensions of Fitting ideals
Let \(R\) be a regular affine \(k\)-algebra where \(k\) is a field of characteristic 0, and \(M\) a finitely generated \(R\)-module such that the symmetric algebra \(\text{Sym}(M)\) of \(M\) is equidimensional. The main result of the paper says that \(\text{codim} Q\leq r(r-\text{rank} M)\) for every minimal prime ideal \(Q\) of the Fitting ideal \(\text{Fitt}_{r-1}(M)\) that is not associated to \(\text{Fitt}_r(M)\). The proof is based on the author's earlier results concerning minimal primes of tensor powers of \(\text{Sym}(M)\). In terms of an \(m\times n\) matrix \(A\) over \(R\) the bound yields \(\text{codim} I_t(A)\leq(m-t+1)(\text{rank} A-t+1)\) for the ideal of \(t\)-minors of \(A\), provided the cokernel of the linear map \(R^n\to R^m\) represented by \(A\) has an equidimensional symmetric algebra. In this case the bound is sharper than the Eagon-Northcott bound and other bounds given by the reviewer [\textit{W. Bruns}, Proc. Am. Math. Soc. 83, 19-26 (1981; Zbl 0478.13006)].
scientific article; zbMATH DE number 1066128