Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

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.

Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such always have finite Macaulayfication. We show how characterize Buchsbaumness and Castelnuovo-Mumford regularity these means their Macaulayfication, we use this method study estimate large classes non-smooth terms given monomials.