Cohomology of non-commutative Hilbert schemes

TL;DR

This paper studies the structure and properties of non-commutative Hilbert schemes, focusing on cell decompositions, combinatorial parametrizations, and asymptotic behaviors of their Poincare polynomials.

Contribution

It introduces cell decompositions for non-commutative Hilbert schemes and analyzes their combinatorial and asymptotic properties, extending understanding of these algebraic varieties.

Findings

Cell decompositions parametrized by forests

Asymptotic formulas for Poincare polynomials

Properties of generating functions of these invariants

Abstract

Non-commutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincare polynomials and properties of their generating functions are discussed.