On Restricted Powers of Complete Intersections

TL;DR

This paper investigates the homological properties of restricted powers of complete intersections, providing explicit resolutions and showing these quotients are Golod rings with DG-algebra structures.

Contribution

It constructs explicit minimal free resolutions for restricted powers of complete intersections, generalizing the L-complex, and demonstrates their Golod property and DG-algebra structure.

Findings

Explicit minimal free resolutions for restricted powers

Restricted powers quotients are Golod rings for d ≥ 2

Resolutions admit an associative DG-algebra structure

Abstract

A restricted $d$th power of an ideal $I$ is obtained by restricting the exponent vectors allowed to appear on the "natural" generating set of $I^d$, for some integer $d$. In this paper, we study homological properties of restricted powers of complete intersections. We construct an explicit minimal free resolution for any restricted power of a complete intersection which generalizes the $L$-complex construction of Buchsbaum and Eisenbud. We use this resolution to compute an explicit basis for the Koszul homology which allows us to deduce that the quotient defined by any restricted $d$th power of a complete intersection is a Golod ring for $d \geq 2$. Finally, using techniques of Miller and Rahmati, we show that the minimal free resolution of the quotient defined by any restricted power of a complete intersection admits the structure of an associative DG-algebra.