On asymptotic vanishing behavior of local cohomology

Abstract

Let R be a standard graded algebra over a field k, with irrelevant maximal ideal $\mathfrak {m}$, and I a homogeneous R-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$ for $i<\dim R/I$. We show that, when ${{\,\mathrm{{{char}}}\,}}k= 0$, R / I is Cohen–Macaulay, and I is a complete intersection locally on ${{\,\mathrm{{Spec}}\,}}R {\setminus }\{\mathfrak {m}\}$, the lowest degrees of the modules $\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$ 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.