Decomposition of affine semigroup rings with applications to the Eisenbud-Goto conjecture

Let $K$ be a field and let $A\subseteq B\subseteq {\mathbb{N}}^d$ be affine semigroups such that the corresponding cones are equal, \mbox{i.\,e.}, $C(A)=C(B)$. We will discuss how $K[B]$ can be decomposed into a direct sum of monomial ideals in $K[A]$; this is a generalization of an idea by Hoa and Stückrad. In case that $B$ is homogeneous, we can use this to compute the Castelnuovo-Mumford regularity of $K[B]$. Since the decomposition can be computed very effectively by our Macaulay2 package MonomialAlgebras.m2, the regularity computation with our package in high codimension is much faster than the usual computation by using its free resolution. This enables us to test the Eisenbud-Goto conjecture for affine semigroup rings with high codimension. This is a joint work with Janko Böhm and David Eisenbud.

Moreover, I will explain how this decomposition has led to positive answers for the E-G conjecture in case that $B$ is simplicial, including a new combinatorial proof for arbitrary monomial curves, the seminormal case, and the case of at most two elements.