Vanishing of Avramov Obstructions for Products of Sequentially Transverse Ideals

TL;DR

This paper proves that Avramov obstructions vanish for products of sequentially transverse ideals in regular local rings, providing explicit resolutions and homological structures, and explores their algebraic properties.

Contribution

It demonstrates the vanishing of Avramov obstructions for such ideals and constructs explicit free resolutions and Massey operations.

Findings

Avramov obstructions are always zero for these ideals.

Explicit free resolutions and Koszul homology are computed.

Constructs a trivial Massey operation and minimal free resolution.

Abstract

Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.