In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multi-degrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolu...

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P1 × · · · × Pk . MSC 2000: 13D02, ...

Let S be a standard N-graded polynomial ring over a field k, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z-graded Smodule. We prove that the resolution regularity, a multigraded variant of CastelnuovoMumford regularity, of IM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to the multigraded situation.

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound...

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X . As a special case, we obtain a new proof of Gotzmann’s regularity the...

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theo...

Let S be a standard N k-graded polynomial ring over a field K, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I n M is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.

We show that the presentation of affine T-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexi...

This paper is concerned with the relationships between two concepts, vanishing of cohomology groups and the structure of free resolutions. In particular, we study the connection between vanishing theorems for the local cohomology of multigraded modules and the structure of their free multigraded resolutions.