FRAMES and DEGENERATIONS of MONOMIAL RESOLUTIONS

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 resolution entirely. The key idea in the paper is that the problem of constructing a minimal monomial free resolution is equivalent to the problem of building its frame. There are three main known sources of frames: homology complexes from algebraic topology (which yield cellular resolutions), dehomogenization of resolutions (see 2), and in some cases it is possible to construct frames directly (see Theorems 6.1 and 7.1). 2) Homogenization and dehomogenization of ideals are widely used (for example, to relate the defining ideals of affine and projective varieties). We introduce homogenization and dehomogenization of complexes. 3) We introduce degenerations of a monomial free resolution. Starting from a free resolution of a monomial ideal, a degeneration yields (under certain conditions) a free resolution of another monomial ideal.