Macaulay Constants and Vanishing of Cohomology

Consider any finitely generated graded module F over a polynomial ring S = K[X] over any field K with finite variable set X and standard grading, i.e., all the variables of S have degree one. A cone C in F is a graded K-subspace of the form C = hK[U ], where h ∈ F is homogeneous and U is a subset of X. For a graded subspace P of F , any finite direct sum of cones

is called a cone decomposition of P . Dube realized the importance of special cone decompositions, called q-exact, satisfying additional properties (see Theorem 2.2). Extending a result of Dubé, Liang showed in [12,Theorem 20] that certain graded subspaces of F admit a q-exact cone decomposition for a suitable integer q.

Denote by M any nonzero finitely generated graded S-module. We introduce the Macaulay constants b 0 (M ), . . . , b d+1 (M ) of M , where d is the Krull dimension of M , by

The author was partially supported by Simons Foundation grant #636513.

reading them off from suitable q-exact cone decompositions (see Section 2). It turns out that these integers satisfy

where e + (M ) denotes the maximum degree of a minimal generator of M (see Theorem 3.10).

The Macaulay constants of M determine the Hilbert polynomial of M when it is written in a particular form. We use this form to obtain an explicit characterization of Hilbert polynomials of finitely generated graded S-modules (see Theorem 3.5).

The Castelnuovo-Mumford regularity reg(M ) is an important and much-investigated complexity measure of M . It can be defined using a graded minimal free resolution of M over S or the local cohomology modules H i m (M ) of M with support in the homogeneous maximal ideal m of S. Taking into account the latter prospective, one defines, following [18], the k-regularity reg k (M ) of M as reg k (M ) = min{m ∈ Z | [H i m (M )] j = 0 whenever j > m -i and i ≥ k}. Thus, upper bounds on reg k (M ) correspond to vanishing results for local cohomology modules. Note that

The following statement partially summarizes our cohomological vanishing results (see Theorem 4.12 and Theorem 4.14 for more details).

Theorem. If M ̸ = 0 is any finitely generated graded S-module then one has It is worth stressing two special cases of this statement. For k = 0, one gets reg(M ) ≤ b 0 (M ). The estimate reg 1 (M ) ≤ b 1 (M ) may be interpreted as an extension of Gotzmann’s Regularity Theorem in [9] from ideals to modules (see Theorem 4.6). Our results also imply an upper bound on the Castelnuovo-Mumford regularity of a coherent sheaf on P n . It is attained for structure sheaves of certain projective subschemes (see Theorem 4.15). This paper is organized as follows. In Section 2, we recall the concept of a cone decomposition and use special cone decompositions to define the Macaulay constants of a finitely graded module M . We also establish some of their basic properties (see Theorem 2.10). If the grading of M is shifted its Macaulay constants change, but in a controlled way.

In Section 3, we use Macaulay constants to present the Hilbert polynomial of a finitely graded module M in a particular form. This leads to an explicit characterization of Hilbert polynomials of finitely graded modules. Subsequently, these results are used to establish further properties of Macaulay constants as, for example, their behavior under truncation and comparisons of Macaulay constants (see Theorem 3.10). The latter result also relies on Theorem 3.7. It guarantees that the Hilbert function and the Hilbert polynomial of M are the same in any degree j ≥ b 0 (M ).

Our vanishing results on local cohomology modules are established in Section 4. We begin by interpreting [17,Theorem 7.5] on saturated ideals using a Macaulay constant and generalizing the statement to modules. The latter may be viewed as an extension of Gotzmann Regularity Theorem. Using additional comparison results for Macaulay constants (see Theorem 4.10), we establish the above theorem on k-regularities and explicitly describe modules for which the bounds are simultaneously sharp (see Theorem 4.12 and Theorem 4.14). We conclude with an application to coherent sheaves on projective space (see Theorem 4.15).

Dubé introduced cone decompositions for certain graded subspaces of a polynomial ring S and Liang extended the concept to subspaces of a graded finitely generated free S-module. Our goal is to use cone decompositions for describing properties of finitely generated graded S-modules. We review cone decompositions from this point of view and introduce the Macaulay constants of a finitely generated graded module.

We begin by fixing some notation. We denote by K any field. A (Z-)graded K-vector space is a K-

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