Lattice homology of integrally closed submodules and Artin algebras

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 ${\mathbb{H}}_{\ast }$. 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 $N\le M$, then, although the corresponding lattices and weight functions might be different, the resulting lattice homology modules are isomorphic and have Euler characteristic ${\dim }_k(M\mathrm{/}N)$. In this way, we associate a well-defined lattice homology to any quotient of type $M\mathrm{/}N$, 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 $\delta (C,o)$ 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.