Free resolutions via Gr\"obner bases
Abstract
For associative algebras in many different categories, it is possible to develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining relations for an algebra of such a category provides a "monomial replacement" of this algebra. The main goal of this article is to demonstrate how this machinery can be used for the purposes of homological algebra. More precisely, our approach goes in three steps. First, we define a combinatorial resolution for the monomial replacement of an object. Second, we extract from those resolutions explicit representatives for homological classes. Finally, we explain how to "deform" the differential to handle the general case. For associative algebras, we recover a well known construction due to Anick. The other case we discuss in detail is that of operads, where we discover resolutions that haven't been known previously. We present various applications, including a proofs of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin--Vilkovisky algebras and of Rota--Baxter algebras.
Cite
@article{arxiv.0912.4895,
title = {Free resolutions via Gr\"obner bases},
author = {Vladimir Dotsenko and Anton Khoroshkin},
journal= {arXiv preprint arXiv:0912.4895},
year = {2011}
}
Comments
34 pages, 4 figures. v2: added references to the work of Drummond-Cole and Vallette. v3: added an explicit description of homology classes in the monomial case and more examples, re-structured the exposition to achieve more clarity. v4: changed the presentation of the main construction to make it clearer, added another example (a computation of the bar homology of Rota--Baxter algebras)