The purpose of this article is to study the minimal free resolution of homogeneous coordinate rings of elliptic ruled surfaces. Let X be an irreducible projective variety and L a very ample line bundle on X , whose complete linear series defines the morphism φL : X −→ P(H (L)) Let S = ⊕∞ m=0 S H(X,L) and R(L) ⊕∞ m=0 H (X,L). Since R(L) is a finitely generated graded module over S, it has a mini...

A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x1, . . . , xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the Nr-graded R[x1, . . ...

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is...

We establish the existence and uniqueness of finite free resolutions and their attendant Betti numbers for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example,...

Generalizing the concepts of Stanley–Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in Rd the Zd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen–Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Zd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences...

The Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the Betti numbers of a graded module M based on the codimension of M . We prove a special case of this conjecture via Boij-Söderberg theory. More specifically, we show that the conjecture holds for graded modules where the regularity of M is small relative to the minimal degree of a first syzygy of M . Our approach also y...

Suppose S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, E is a coherent OX -bimodule and I ⊂ T (E) is a graded ideal. We study the geometry of the functor Γn of flat families of truncated B = T (E)/I-point modules of length n + 1. We then use the results of our study to show that the point modules over B are parameterized by the closed points of P X2(E). When X = P , w...

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the (2, 2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding ‘rank 1’ supervector bundle over S2,2. The canonical connection ∇ = p ◦ d is used to compute the Chern numbers by me...

We determine the class of Hilbert series H so that if M is a finitely generated zero-dimensional R-graded module with the strong Lefschetz property, then M ⊗k k[y]/(y ) has the strong Lefschetz property for y an indeterminate and all positive integers m if and only if the Hilbert series of M is in H. Given two finite graded R-modules M and N with the strong Lefschetz property, we determine suff...

Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are give...