The goal of this paper is to determine Gröbner bases of powers of determinantal ideals and to show that the Rees algebras of (products of) determinantal ideals are normal and CohenMacaulay if the characteristic of the base field is non-exceptional. Our main combinatorial result is a generalization of Schensted’s Theorem on the Knuth–Robinson–Schensted correspondence. Mathematics Subject Classif...

In this paper, we prove the following. Let (R,m) be a Cohen-Macaulay local ring of dimension d ≥ 2. Suppose that either R is not regular or R is regular with d ≥ 3. Let t ≥ 2 be a positive integer. If {α1, . . . , αd} is a regular sequence contained in m, then (α1, . . . , αd) : m t ⊆ m. This result gives an affirmative answer to a conjecture raised by Polini and Ulrich.

We discuss the linearity of the minimal free resolution of a power of an edge ideal.

Let R be a Noetherian ring and I an ideal. We prove that there exists an integer k such that for all n ≥ 1 there exists an irredundant primary decomposition I = q1 ∩ · · · ∩ ql such that √ qi nk ⊆ qi whenever ht (qi/I) ≤ 1. In particular, if R is a local ring with maximal ideal m and I is a prime ideal of dimension 1, then mI ⊆ I, where I denotes the n’th symbolic power of I . We study some asy...

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is homogeneous ideal in polynomial ring $S$, can be any convergent numerical function. also give positive answer to long-standing open question Ratliff on associated primes powers ideals.

Let $I(G)^{[k]}$ denote the $k$th squarefree power of edge ideal $G$. When $G$ is a forest, we provide sharp upper bound for regularity in terms $k$-admissable matching number For any positive integer $k$, classify all forests such that has linear resolution. We also give combinatorial formula $I(G)^{[2]}$ forest

Let G be a finite simple graph and J(G) denote its vertex cover ideal in polynomial ring over field. The k-th symbolic power of is denoted by \(J(G)^{(k)}\). In this paper, we give criterion for ideals decomposable graphs to have the property that all their powers are not componentwise linear. Also, necessary sufficient condition on so \(J(G)^{(k)}\) linear some (equivalently, all) \(k \ge 2\) ...