The DG Products of Peeva and Srinivasan Coincide

TL;DR

This paper proves that the differential graded algebra structures of the Buchsbaum-Eisenbud and Eliahou-Kervaire resolutions of a certain ideal are isomorphic, confirming Peeva's conjecture about their coincidence.

Contribution

It constructs an explicit isomorphism between the DG algebra structures of two known resolutions, affirmatively answering Peeva's question.

Findings

The DG algebra structures of the two resolutions are isomorphic.

An explicit algebra isomorphism between the complexes is constructed.

The result confirms the conjecture that these DG structures coincide.

Abstract

Consider the ideal $(x_1 , \dotsc , x_n)^d \subseteq k[x_1 , \dotsc , x_n]$, where $k$ is any field. This ideal can be resolved by both the $L$-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva's question.