DG Structure on Length 3 Trimming Complexes and Applications to Tor Algebras

TL;DR

This paper studies the algebraic structure of length 3 complexes derived from trimming data, revealing how these structures behave under homology and applying findings to classify Tor-algebras of certain ideals.

Contribution

It provides explicit algebra structures for length 3 complexes and demonstrates how trimming preserves Tor-algebra classes in ideal realizations.

Findings

Explicit algebra structures for length 3 complexes derived from input data.

Many products become trivial after passing to homology.

Construction of new ideals realizing specific Tor-algebra classes.

Abstract

In this paper, we consider the iterated trimming complex associated to data yielding a complex of length $3$. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade $3$ perfect ideals, and show that under mild hypotheses, the process of "trimming" an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes $G(r)$ and $H(p,q)$ for a prescribed set of homological data.