Tangent complexes and the Diamond Lemma: homotopical methods for terms rewriting

Abstract: Term rewriting has been an indispensable tool to approach various computational problems involving associative algebras and algebraic operads, their homology theories and their deformation theory [1, 2, 3, 5, 6, 8, 9]. One of the cornerstones of the theory, the celebrated Diamond Lemma [4], gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras.

In joint work with V. Dotsenko [7], we presented a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of "resolvable ambiguities" becomes the first non-trivial component of the Maurer–Cartan equation in the corresponding tangent complex. Our approach works for many other algebraic structures, such as algebraic operads, highlighting the importance of computing multiplicative resolutions of algebras presented by monomial relations, as it was done in [10].

For those whose intuition comes from homotopical algebra, our work presents a conceptual explanation of useful (but seemingly technical) criteria of "resolvable ambiguities" for uniqueness of normal forms. For those with a background in Gröbner bases or term rewriting, our work offers intuition behind both the Diamond Lemma and its optimisations, as well as precise guidance on how to generalise those for other algebraic structures. Specifically, our work means that computing models of algebras with monomial relations explicitly helps both to state the relevant Diamond Lemmas and to optimise them.

With the aim of bringing the effective methods of term rewriting closer to the powerful methods of homotopical algebra and higher structures, prior knowledge of the techniques involved in our work is not assumed: they will be explained along the way, with an emphasis in providing a working dictionary to go from term rewriting to deformation theory and homotopical algebra, and back.

References:

[1] D. J. Anick. On the homology of associative algebras. Trans. Amer. Math. Soc., 296(2):641–659, 1986.

[2] J. Backelin. La série de Poincaré–Betti d'une algèbre graduée de type fini à une relation est rationnelle. C. R. Acad. Sci. Paris Ser. A-B, 287(13): A843–A846, 1978.

[3] M. J. Bardzell. Noncommutative Gröbner bases and Hochschild cohomology. In: Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), vol. 286 of Contemp. Math., pp. 227–240. Amer. Math. Soc., Providence, RI, 2001.

[4] G. M. Bergman. The diamond lemma for ring theory. Adv. Math., 29(2):178–218, 1978.

[5] V. Dotsenko, V. Gelinas, P. Tamaroff. Finite generation for Hochschild cohomology of Gorenstein monomial algebras. arXiv:1909.00487.

[6] V. Dotsenko and A. Khoroshkin. Quillen homology for operads via Gröbner bases. Doc. Math., 18:707–747, 2013.

[7] V. Dotsenko and P. Tamaroff. Tangent complexes and the Diamond Lemma. arXiv:2010.14792.

[8] E. S. Golod. Standard bases and homology. In Algebra – some current trends (Varna, 1986), vol. 1352 of Lecture Notes in Math., pp. 88–95. Springer, Berlin, 1988.

[9] E. L. Green, D. Happel, and D. Zacharia. Projective resolutions over Artin algebras with zero relations. Illinois J. Math., 29(1):180–190, 1985.

[10] P. Tamaroff. Minimal models for monomial algebras. Homology Homotopy Appl., 23(1): 341–366, 2021.

Mathematics

Audience: researchers in the topic

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Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)

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