Lattice homology of integrally closed submodules and Artin algebras

TL;DR

This paper develops a lattice homology framework for integrally closed submodules and Artin algebras, linking algebraic and geometric invariants through categorification.

Contribution

It introduces realizable submodules and proves invariance of lattice homology under different valuations, enabling categorification of numerical invariants in algebraic geometry.

Findings

Lattice homology modules are isomorphic for different valuation realizations.

Structural characterization of homological dimension for integrally closed monomial ideals.

Geometric invariants like delta invariant and geometric genus are categorified via lattice homology.

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 $\delta(C, o)$ 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 \'Agoston and the first author.