The lex-plus-powers inequality for local cohomology modules

Abstract

We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans’ lex-plus-powers conjecture for Betti numbers. This results implies some cases of the classical lex-plus-powers conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud–Green–Harris conjecture is currently known, the projective dimension and the Castelnuovo–Mumford regularity of a graded ideal do not decrease by passing to the corresponding lex-plus-powers ideal.