Finite group actions on free resolutions and modules arise naturally in many interesting examples. Understanding these amounts to describing the terms of a resolution or graded components module as representations which, non modular case, are completely determined by their characters. With this goal mind, we introduce Macaulay2 package for computing characters finite groups finitely generated o...

In this paper we study complete linear series on a hyperelliptic curve C of arithmetic genus g. Let A be the unique line bundle on C such that |A| is a g 2 , and let L be a line bundle on C of degree d. Then L can be factorized as L = A ⊗ B where m is the largest integer satisfying H(C,L⊗A) 6= 0. Let b = deg(B). We say that the factorization type of L is (m, b). Our main results in this paper a...

We study the structure of (minimal) free resolutions of monomial ideals over a polynomial ring. This has been a very active area of research, and a number of new ideas and approaches were introduced in the last decade. In this paper, we introduce three new notions: 1) We introduce the frame of a free resolution. The frame is a complex of vector spaces which encodes the structure of the resoluti...

The problem of explicitly finding a free resolution, minimal in some suitable sense, of a module over a polynomial ring is solved in principle by the algorithm of Hilbert [H]. However, this algorithm is of enormous computational difficulty. If the module happens to be finite dimensional over the ground field, and if the module structure is given by specifying the commuting linear transformation...

We prove that the minimal free resolution of secant variety a curve is asymptotically pure. As corollary, we show Betti numbers converge to normal distribution.

We characterize the lcm lattices that support a monomial ideal with a pure resolution. Given such a lattice, we provide a construction that yields a monomial ideal with that lcm lattice and whose minimal free resolution is pure.

Given a finite set, X, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is “as big as possible” inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals o...

When R is a commutative ring, the minimal free resolution of a map Ra → Rb and symmetric and skew-symmetric maps Ra → Ra, under suitable generality conditions, are well known and have been developed by a series of authors. See [2, A2.6] for an overview. In this note we do an analog for the exterior algebra E = ⊕ ∧i V on a finite dimensional vector space V and general graded maps Ea → E(1)b, and...

Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. In fact there are generalizations of a classical result of Dubreil. On the other hand there are investigations about the shifts and the dimensio...

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that ...