Convergent presentations and polygraphic resolutions of associative algebras

被引:14
作者
Guiraud, Yves [1 ]
Hoffbeck, Eric [2 ]
Malbos, Philippe [3 ]
机构
[1] Univ Paris 07, INRIA, Case 7014, F-75205 Paris 13, France
[2] Univ Paris 13, CNRS UMR 7539, LAGA, Sorbonne Paris Cite, 99 Ave Jean Baptiste, F-93430 Villetaneuse, France
[3] Univ Claude Bernard Lyon 1, Univ Lyon, CNRS UMR 5208, Inst Camille Jordan, F-69622 Villeurbanne, France
关键词
Higher-dimensional associative algebras; Confluence and termination; Linear rewriting; Polygraphs; Free resolutions; Koszulness; FINITENESS CONDITION; GROBNER BASES; WORD-PROBLEMS; HOMOLOGY; SINGULARITIES; KOSZUL; RING;
D O I
10.1007/s00209-018-2185-z
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Several constructive homological methods based on noncommutative Grobner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Grobner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Grobner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property.
引用
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页码:113 / 179
页数:67
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