Description
T. Ágoston and A. Némethi recently initiated the study of lattice homology theories associated with `split' weight functions, obtained as the difference of two `height functions'. In this talk, we will be considering the case when the height functions come as Hilbert functions of some valuative multifiltrations on a Noetherian k-algebra O and a finitely generated module M over it. We introduce the notion of `realizable submodules' in M, the prime example of which are finite codimensional integrally closed submodules. Our Independence Theorem states that whenever two sets of (extended) discrete valuations `realize' the same submodule N in M, then the resulting lattice homology is the same; in fact, it is a well-defined invariant of the quotient M/N. Moreover, it has Euler characteristic dim(M/N).
The main output of the Independence Theorem is this possibility of categorifying numerical invariants defined as codimensions of realizable submodules: e.g., the delta invariant of reduced curve singularities; or the geometric genus, the irregularity, and the various plurigenera of higher-dimensional isolated normal singularities. We will also discuss structural bounds, symmetry properties, and relations with deformation theory.
The results come from ongoing joint work with A. Némethi.
ZOOM: https://us06web.zoom.us/j/86570145006?pwd=fqSKabceyPQszxT6xzP3bokpg5c4xT.1
Meeting ID: 865 7014 5006
Passcode: 287154