Multigraded minimal free resolutions of simplicial subclutters

Minimal Free Resolutions and Asymptotic Behavior of Multigraded Regularity

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.

On simple A-multigraded minimal resolutions

Let A be a semigroup whose only invertible element is 0. For an A-homogeneous ideal we discuss the notions of simple i-syzygies and simple minimal free resolutions of R/I. When I is a lattice ideal, the simple 0-syzygies of R/I are the binomials in I. We show that for an appropriate choice of bases every A-homogeneous minimal free resolution of R/I is simple. We introduce the gcd-complex ∆gcd(b...

Shifts in Resolutions of Multigraded Modules

Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the BackelinLescot rational expression for multigraded Poincaré series. Let K be a field and S = K[x1, . . . , xn] the polynomial ring with its natural n-grading. When I is an ideal generated by monomials in the variables x1, . . . , xn, th...

Finite Free Resolutions and 1-Skeletons of Simplicial Complexes

A technique of minimal free resolutions of Stanley–Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d − 1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d − 1 is d-connected.

Minimal Free Resolutions of Homogenized D-modules