We prove an analogue of the Approximation Theorem of L-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.

Let I⊆R=K[x1,…,xn] be a toric ideal, i.e., binomial prime ideal. We investigate when the ideal I can “split” into sum of two smaller ideals. For general I, we give sufficient condition for this splitting in terms integer matrix that defines I. When I=IG is finite simple graph G, additional splittings IG related to subgraphs G. there exists I=I1+I2 show some cases describe (multi-)graded Betti n...

Betti numbers can be used as a means for feature detection to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once an isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data...

We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of l in its Jacobian. These statements can be viewed as analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of ...

This paper produces a recursive formula of the Betti numbers of certain StanleyReisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs. 0. Introduction Throughout this paper...