Lattice homology of integrally closed submodules and Artin algebras

Abstract

The general construction of lattice (co)homology assigns to a lattice $\mathbb{Z}^r$ and a weight function $w:\mathbb{Z}^r \to \mathbb{Z}$ a bigraded $\mathbb{Z}[U]$-module $\mathbb{H}_*$. The weight function $w$ is often obtained from some geometric data as the difference of two `height functions'. In this paper we consider the case when these height functions are Hilbert functions of valuative multifiltrations on a Noetherian $k$-algebra $\mathcal{O}$ and a finitely generated $\mathcal{O}$-module $M$. We introduce the notion of `realizable submodules' in $M$, the prime example of which are finite codimensional integrally closed submodules in the sense of Rees (or integrally closed ideals when $M=\mathcal{O}$). We prove, that whenever two sets of `extended' discrete valuations `realize' the same submodule $N \leq M$, then, although the corresponding lattices and weight functions might be different, the resulting lattice homology modules are isomorphic and have Euler characteristic $\dim_k(M/N)$. In this way, we associate a well-defined lattice homology to any quotient of type $M/N$, where $N$ is a realizable submodule of $M$. We also present some structural and computational results: e.g., we geometrically characterize the (lattice) homological dimension of integrally closed monomial ideals of $k[x,y]$. The main upshot of the paper, however, is the possibility of categorifying numerical invariants defined as codimensions of realizable submodules or integrally closed ideals. The geometric applications include: the delta invariant $δ(C, o)$ of a reduced curve singularity; the geometric genus $p_g(X, o)$, the irregularity $q(X, o)$ and the various plurigenera of higher dimensional isolated normal singularities. The corresponding categorifications generalize the analytic lattice homologies of Ágoston and the first author.