Adela Vraciu

Special tight closure

We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.

Strong test ideals

Abstract We prove that the test ideal is a strong test ideal whenever it commutes with completion. Additional constructions of strong test ideals are considered.

The Weak Lefschetz Property for monomial complete intersection in positive characteristic

Let A = k[x1, . . . , xn]/(x1, . . . , x d n), where k is an infinite field. If k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, k has positive characteristic p. If n = 3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4 ≤ n. If 4 ≤ n and p = 2, then A has the WLP if and only if d = 1. If n = 4 and p is odd, then we prove that A has the WLP if and only if d = kq + r for integers k, q, r with 1 ≤ k ≤ p−1 2 , r ∈ { q−1 2 , q+1 2 } , and q = pe for some non-negative integer e. If 5 ≤ n, then we prove that A has the WLP if and only if ⌊ n(d−1)+3 2 ⌋ ≤ p. We first interpret the WLP for the ring k[x1, . . . , xn]/(x1 , . . . , x d n) in terms of the degrees of the non-Koszul relations on the elements x1 , . . . , x d n−1, (x1 + . . . + xn−1) d in the polynomial ring k[x1, . . . , xn−1]. We then exhibit a sufficient condition for k[x1, . . . , xn]/(x1, . . . , x d n) to have the WLP. This condition is expressed in terms of the non-vanishing in k of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on x1 , . . ., x d n−1, (x1 + . . .+ xn−1) d. From this we obtain a necessary condition for A to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in k.

When is tight closure determined by the test ideal

We characterize the rings in which the equality

Chains and Families of Tightly Closed Ideals

(\tau I:\tau)= I^*

A new version of

holds for every ideal

\mathfrak{a}

I \subset R

-tight closure

. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.

Poincaré series of compressed local Artinian rings with odd top socle degree

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

A formula for the *-core of an ideal

Hara and Yoshida introduced a notion of