Length of local cohomology of powers of ideals

Abstract

Let R be a polynomial ring over a field k with irrelevant ideal m and dimension d. Let I be a homogeneous ideal in R. We study the asymptotic behavior of the length of the modules H i m (R/I n ) for n ≫ 0. We show that for a fixed number α ∈ Z, lim sup n→∞< ∞ when X = Proj R/I is locally a complete intersection (lci) and i dim X. We also establish that the actual limit exists and is rational for certain classes of monomial ideals I such that the lengths of local cohomology of I n are eventually finite. Our proofs use Gröbner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that lim inf n→∞ λ(H i m (R/I n )) n d > 0 under certain conditions when R/I is either F -pure or lci.2010 Mathematics Subject Classification. 13D45, 13A30, 14B05, 05E40.

Asymptotic growth of symbolic multi-Rees algebras

“…A family of ideals I = {I n } n∈N r is said to be a Noetherian N r -graded family if I 0 = R and n∈N r I n is a Noetherian N r -graded R-algebra. The subsequent lemma is a modification of [11,Proposition 3.5] by Dao and Montaño. Lemma 6.5.…”

“…We use a projection argument to obtain the first assertion of our theorem. For proving the second assertion we use Presburger arithmetic and the techniques are motivated by the proofs in [11] due to Dao and Montaño. In Example 6.1 we strengthen Theorem 1.2 when the number of variables involved is two.…”

On asymptotic vanishing behavior of local cohomology

“…In this paper we study asymptotic behavior of the lowest degree of the local cohomology modules {H i m (R/I n )} n∈N , provided that they are finite. As we make clear below, such behavior can be viewed as an "asymptotic Kodaira vanishing for thickenings" phenomenon, and have recently appeared in various works such as [2,7,16].…”

“…In fact, since H i m (M) is an Artinian module, we have that indeg H i m (M) > −∞ if and only if H i m (M) is Noetherian. Our work is guided by the following questions raised in [7]. Question 1.1.…”

“…Our initial interest in the question came from [7] where we need an affirmative answer to part (1) of Question 1.1 to obtain efficient bounds on the lengths of local cohomology modules of powers. Robert Lazarsfeld pointed out to us that this is indeed the case when X = Proj(R/I) is a l.c.i variety, k is of characteristic zero, and i is at most the dimension of X.…”