We prove that every quasi-complete intersection (q.c.i.) ideal is obtained from a pair of nested complete ideals by way flat base change. As by-product we establish rigidity statement for the minimal two-step Tate complex associated to an I in local ring R. Furthermore, define T each R; and result it. The exact if only q.c.i. ideal; this case, resolution R/I free R-modules.

The goal of this paper is to provide homological descriptions of monomial ideals. The key concepts in these descriptions are the minimal free resolution, the Koszul homology and the multigraded Betti numbers. These three objects are strongly related, being Tor modules a simple way to describe this relation. We introduce a new tool, Mayer-Vietoris trees which provides a good way to compute the h...

Fluorescence lifetime imaging microscopy (FLIM) can be used to probe concentration profiles of key marco-molecules within sub-cellular environments. However, current analyses for FLIM require a large number photons per pixel provide such quantitative picture life. In order acquire photons, we must either increase data acquisition time, which limits temporal resolution, or laser intensity, cause...

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS],[PS]. Introduction Given a field k, we consider the Laurent polynomial ring T = k[x 1 , . . . , x ±1 n ] as a module over the polynomial ring S = k[x1, . . ...

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of Macaulay [Mac] regularity is the number of rows in the diagram produced by the “betti” command. A Betti number βi,j 6= 0 will be called extremal if βl,r = 0 for all l ≥ i and r ≥ j + 1, that is if βi,j is ...