Quillen homology for operads via Gröbner bases

Details Quillen homology for operads via Gröbner bases Quillen homology for operads via Gröbner bases

2012-03-22

arxiv.org/abs/1203.5053v1

Mathematics

K-Theory and Homology

25 pages, this paper supersedes our previous preprint arXiv:0907.4895

Scientific paper

The main goal of this paper is to present a way to compute Quillen homology of a shuffle operad with a known Gr\"obner basis. Similar to the strategy taken in a celebrated paper of David Anick, our approach goes in several steps. We define a combinatorial resolution for the "monomial replacement" of a shuffle operad, explain how to "deform"' the differential to handle the general case, and find explicit representatives of Quillen homology for a large class of operads with monomial relations. We present various applications, 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 operads of Batalin-Vilkovisky algebras and of Rota-Baxter algebras.

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