On asymptotic vanishing behavior of local cohomology

Abstract

Let R be a standard graded algebra over a field k, with irrelevant maximal ideal m, and I a homogeneous R-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules {H i m (R/I n )} n∈N for i < dim R/I. We show that, when char k = 0, R/I is Cohen-Macaulay, and I is a complete intersection locally on Spec R \ {m}, the lowest degrees of the modules {H i m (R/I n )} n∈N are bounded by a linear function whose slope is controlled by the generating degrees of the dual of I/I 2 . Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field k, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.

An asymptotic vanishing theorem for the cohomology of thickenings

“…The hypotheses in [DM2] are somewhat different from those in Theorem 1.2 of the present paper, where there is no assumption on the characteristic, nor do we require the ring R/I to be Cohen-Macaulay.…”

“…Remark 3.5. In the recent paper [DM2], the authors prove the following result: Let R be a standard graded ring over a field of characteristic zero; let m denote the homogeneous maximal ideal of R. Suppose I is a homogeneous ideal such that R/I is Cohen-Macaulay and of dimension at least 2, and I is locally a complete intersection on SpecR {m}. Fix an integer k with k < dim R/I.…”

On asymptotic socle degrees of local cohomology modules

“…Similarly, without particular hypotheses on R and I, local cohomology modules H j m (R/I t ) may not be finitely generated and hence it may be the case that H j m (R/I t ) ℓ = 0 for all ℓ ≪ 0. Question 1.1 remains valid without any further hypotheses on R and I, and can certainly be viewed as a natural extension of the main results in [DMn20] and [BBL + 20]. The other source comes from the connection between the notion of gaugeboundedness and a linear lower bound of Soc(H d m (ω [p e ] )) (cf.…”

An asymptotic vanishing theorem for the cohomology of thickenings

“…In the recent paper [7], the authors prove the following result: let R be a standard graded ring over a field of characteristic zero; let m denote the homogeneous maximal ideal of R. Suppose I is a homogeneous ideal such that R/I is Cohen-Macaulay and of dimension at least 2, and I is locally a complete intersection on Spec R {m}. Fix an integer k with k < dim R/I .…”