Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free resolution of S/M over S. The difficulty of this problem is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence [BH,Ho,St]) int...

Let X be a del Pezzo surface of degree one over an algebraically closed field, and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox(X) over a polynomial ring....

We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S = k[x1, x2, ..., xn]; this includes the case of powers of the homogeneous maximal ideal (x1, x2, ..., xn) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in...

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals. INTRODUCTION Let S = K[x1, . . . ,xn] denote the polynomial ring in n variables over a field K with each degxi = 1. Let I be a monomial ideal of S and G(I) = {u1, . . . ,us} its unique minimal system of monomial generators. The Ta...

We give a new method to construct minimal free resolutions of all monomial ideals. This relies on two concepts: one is the well-known lcm-lattice ideal; other concept called Taylor basis, which describes how resolution can be embedded in resolution. An approximation formula for ideals also obtained.

Let R be a standard graded K-algebra, that is, an algebra of the form R = K[x1, . . . , xn]/I where K[x1, . . . , xn] is a polynomial ring over the field K and I is a homogeneous ideal with respect to the grading deg(xi) = 1. Let M be a finitely generated graded R-module. Consider the (essentially unique) minimal graded Rfree resolution of M · · · → Ri → Ri−1 → · · · → R1 → R0 → M → 0 The rank ...

Let G be a finite group. A collection $$\Pi =\{H_1,\dots ,{H_r}\}$$ of subgroups G, where $$r>1$$ , is said non-trivial partition if every non-identity element belongs to one and only $$H_i$$ for some $$1\leqslant i\leqslant r$$ . We call group that does not admit any partition-free In this paper, we study whose all proper non-cyclic partitions.