On the ideal theory of graphs

Abstract

We study algebras defined by finite sets G = {M1, ..., Mq} of monomials of a polynomial ring R. There are two basic algebras: (i) k[G] = k[M1, ..., Mq], the k-subalgebra of R spanned by the Mi, and (ii) the quotient ring R/I(G), where I(G) = (M1, ..., Mq). They come together in the construction of the Rees algebra View the MathML source(I(G)) of the ideal I(G). The emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs. The main results are assertions about the Cohen-Macaulay behaviour of the Koszul homology of I(G), and how normality or Cohen-Macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra.