We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring. In this case both are characterizations of the so-called Castelnuovo–Mumford regularity. Moreover, we can characterize a standard graded polynomial ring as a K-algebra with extremal properties with...

Let ρC be the regularity of the Hilbert function of a projective curve C in P K over an algebraically closed field K and α1, . . . , αn−1 be minimal degrees for which there exists a complete intersection of type (α1, . . . , αn−1) containing the curve C. Then the Castelnuovo-Mumford regularity of C is upper bounded by max{ρC +1, α1 + . . .+αn−1 − (n− 2)}. We study and, for space curves, refine ...

We study the equations defining a projective embedding of a toric variety X using multigraded Castelnuovo-Mumford regularity. Consider globally generated line bundles B1, . . . , Bl and an ample line bundle L := B ⊗m1 1 ⊗ B2 2 ⊗ · · · ⊗ Bl l on X . This article gives sufficient conditions on mi ∈ N to guarantee that the homogeneous ideal I of X in P := P (

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.

We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on m-regularity for graded modules over polynomial algebras.

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...

We establish bounds for the Castelnuovo-Mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations. We extend to modules (and improve) the known bounds for homogenous ideal in a polynomial ring established by Galligo, Guisti, Caviglia and Sbarra.