We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXXIV (1993), 1-48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giusti's proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer-Stillman criterion for detecting regularity.

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Given a homogeneous ideal I of a polynomial ring A = K[X 1 , . . . , X n ] and\ud a monomial order τ , we construct a new monomial ideal of A associated with I . We call it the zero-generic initial ideal of I with respect to τ and denote it with gin0 (I ). When char K = 0, a zero-generic initial ideal is the usual generic initial ideal. We show that gin0 (I ) is endowed with many interesting properties and, quite surprisingly, it also satisfies Green’s Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0

a b s t r a c tWe characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity. IntroductionLet X = (X ij ) be a skew-symmetric n × n matrix of indeterminates. By [11], the polynomial ring R = K [X] := K [X ij : 1 ≤ i < j ≤ n], K being a field, is an algebra with a straightening law (ASL for short) on the poset P(X ) of all Pfaffians of X with respect to the natural partial order defined in [11]. Given any subset of P(X ), the ideal of R that it generates is called a Pfaffian ideal. The special case of Pfaffian ideals I 2r (X) generated by the subset P 2r (X) of P(X ) consisting of all Pfaffians of size 2r has been studied extensively [1,18,21]. These ideals belong to a wider family of Pfaffian ideals called one-cogenerated or simply cogenerated. A cogenerated Pfaffian ideal of R is an ideal generated by all Pfaffians of P(X ) of any size which are not bigger than or equal to a fixed Pfaffian α. We denote it by I α (X) := (β ∈ P(X ) : β ̸ ≥ α). Clearly, if the size of α is 2t, then all Pfaffians of size bigger than 2t are in I α (X). The ring R α (X) = K [X]/I α (X) inherits the ASL structure from K [X], by means of which one is able to prove that R α (X) is a Cohen-Macaulay normal domain, and characterise Gorensteinness, as was done in [8]. In [9] a formula for the a-invariant of R α (X) is also given.Our attention will focus on the properties of cogenerated Pfaffian ideals and their Gröbner bases (G-bases for short) w.r.t. anti-diagonal term orders, which are natural in this setting. By [19, Theorem 4.14] and, independently, by [15, Theorem 5.1], the set P 2r (X) is a G-basis for the ideal I 2r (X). In a subsequent remark the authors ask whether their result can be extended to any cogenerated Pfaffian ideal. This question is very natural, and in the analogous cases of ideals of minors of a generic matrix and of a symmetric matrix the answer is affirmative, as proved respectively in [15,7]. Quite surprisingly the answer is negative (see Example 2.1) and that settles the starting point of our investigation. The aim of this paper is to characterise cogenerated Pfaffians ideals whose natural generators are a G-basis w.r.t. any anti-diagonal term order in terms of their cogenerator. We call such ideals G-Pfaffian ideals. In Section 1 we set some notation, recall some basic notions of standard monomial theory (cf. [6]), among which that of standard tableau, and describe the Knuth-Robinson-Schensted correspondence (KRS for short) introduced and studied in [17], since this is the main tool used to prove results of this kind. KRS was first used by Sturmfels [24] to compute G-bases of determinantal ideals (see also [2,3]) and it has been applied in [15] to the study of Pfaffian ideals of fixed size. It turns out that the original KRS is not ...

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.

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The relationships between the invariants and the homological properties of I, Gin(I) and I lex have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves.

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