We discuss the linearity of the minimal free resolution of a power of an edge ideal.

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...

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...

Let I be a square-free monomial ideal of projective dimension one. Starting with the Taylor complex on generators $$I^r$$ , we use discrete Morse theory to describe CW that supports minimal free resolution . To do so, concretely acyclic matching faces complex.