Saturations of powers of certain determinantal ideals

Abstract

Let R be a Noetherian local ring and m a positive integer. Let I be the ideal of R generated by the maximal minors of an m × (m + 1) matrix M with entries in R . Assuming that the grade of the ideal generated by the k-minors of M is at least m − k + 2 for 1 ≤ ∀k ≤ m , we will study the associated primes of I n for ∀n > 0 . Moreover, we compute the saturation of I n for 1 ≤ ∀n ≤ m in the case where R is a Cohen-Macaulay ring and the entries of M are powers of elements that form an sop for R .

On the Symmetric and Rees Algebras of Certain Determinantal Ideals

“…Therefore, under the condition (2) of 1.2, we get a graded S-free resolution of R(I), and taking its homogeneous part of degree r ∈ Z, we get an R-free resolution of I r , from which we can deduce some homological properties of powers of I. In the subsequent paper [3], using the R-free resolution of I r constructed in this way, we study the associated prime ideals of R/I r and compute the saturation of I m . So, the author thinks that 1.2 is very convenient and it may have more application.…”

The $\ast$-transforms of Acyclic Complexes

“…we get η n ∈ Im(F n ⊗ ∂ 1 ) by (2) of 2.2. Finally we prove (2). Let us consider the following commutative diagram…”

“…This is useful when we compute the saturation of ideals. In fact, in the subsequent paper [2], using * -transform we compute the saturation of the m-th power of the ideal generated by the maximal minors of the following m × (m + 1) matrix        , where x 1 , x 2 , x 3 , . .…”