Free Resolutions via Gröbner Bases

Dotsenko, Vladimir and Khoroshkin, Anton (2009) Free Resolutions via Gröbner Bases. (Preprint)

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Abstract

In many different settings (associative algebras, commutative algebras, operads, dioperads), it is possible to develop the machinery of Gröbner bases; it allows to find a “monomial replacement” for every object in the corresponding category. The main goal of this article is to demonstrate how this machinery can be used for the purposes of homo-logical algebra. More precisely, we define combinatorial resolutions in the monomial case and then show how they can be adjusted to be used in the general homogeneous case. We also discuss a way to make our monomial resolutions minimal. For associative algebras, we recover a well known construction due to Anick. Various applications of these results are presented, including a new proof of Hoffbeck’s PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operad of Batalin–Vilkovisky algebras.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 19 Jun 2018 13:47
Last Modified: 13 Jul 2018 11:14
URI: http://dair.dias.ie/id/eprint/515

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