Venue
Sala Riunioni (Dip. Matematica).
Abstract
(This comes from a joint work with Thanh Vu.) Fix a field k. For any finite simple graph G with vertex set {x_1,…,x_n}, there is a so-called edge ideal associated to G, denoted by I(G), defined as follows: I(G) lives in the polynomial ring k[x_1,…,x_n] and has as generators the monomials x_ix_j such that {x_i,x_j} is an edge of G. The algebraic study of the edge ideal I(G) yields interesting information about the combinatorics of the graph G, and vice versa. In this talk, I will concentrate on the following problem: Characterize the free resolution of I(G) combinatorially in terms of G. A classical result in this problem is Fröberg’s theorem, which says that I(G) has “the most compact possible” (in a precise sense) resolution if and only if the complement graph of G has only “the smallest possible holes” (also in a precise sense). Our main tool is the still elusive notion of linearity defect.