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-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power 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-Power ideal.

Distractions of Shakin rings

“…Our first result, Theorem 3.4, which we derive as a direct consequence of all the available results on embeddings of Hilbert functions [CaKu1,CaKu2,CaSb], states that Shakin rings are Macaulaylex and that they satisfy the properties (1),(2) and (3) mentioned above (with the exception that, to prove (2) when P = (0), we assume char(K) = 0).…”

“…The use of embeddings proved to be valuable to extend many significant results known for the polynomial ring to other standard graded K-algebras, [CaKu1,CaKu2,CaSb]. We are therefore interested in understanding for which ideals a such an ǫ exists, and if this is the case we say that the ring A/a has an embedding ǫ.…”

“…2.16], ǫ is the lex-embedding. Part (2) is a special case of [CaSb,Thm. 3.1], namely when there is only one ring.…”

“…, X n−1 ], and letS =Ā/b andR =Ā/ā be standard graded algebras such thatR has an embeddingǭ and HS ⊆ HR, so that, as in Section 2, we can consider the triplet (S,R,ǭ). In the proof of the following theorem we shall need a technical result about extension of embeddings we proved in [CaSb,Thm. 3.1]: if (S,R,ǭ) is cohomology extremal, then (S[X n ],R[X n ], ǫ) is cohomology extremal, where ǫ is the usual extension of the embeddingǭ tō R[X n ] which has been consider throughout this paper.…”

“…Let ǫD and ǫ denote the extensions of embeddingsǭD andǭ to the ringsRD[X n ] = RD andR[X n ] = R yielded by Theorem 2.3, respectively. Notice that ǫ is the lex-embedding, both (RD, ǫD) and (R, ǫ) are Betti extremal, cohomology extremal, and by [CaSb,Theorem 3.1], also (RD, R, ǫ) is cohomology extremal.…”