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Tangent complexes and the Diamond Lemma

Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France

E-mail Address: vdotsenko@unistra.fr

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and

Pedro Tamaroff

https://orcid.org/0000-0002-2047-5615

Institut für Mathematik, Johann von Neumann-Haus, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany

E-mail Address: tamarofp@hu-berlin.de

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https://doi.org/10.1142/S1664360723500133Cited by:0 (Source: Crossref)

Abstract

The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra; this leads to a new proof using multiplicative free resolutions. Specifically, our main result states that every such resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, where Bergman’s condition of “resolvable ambiguities” is precisely the first nontrivial component of the Maurer–Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing resolutions of algebras with monomial relations.

Communicated by Efim Zelmanov

Keywords:

AMSC: Primary: 16Z10, Secondary: 13P10, Secondary: 16E05, Secondary: 17A61, Secondary: 18N40

References

1. D. J. Anick , A counterexample to a conjecture of Serre, Ann. of Math. (2) 115(1) (1982) 1–33. Crossref, Google Scholar
2. D. J. Anick , Noncommutative graded algebras and their Hilbert series, J. Algebra 78(1) (1982) 120–140. Crossref, Google Scholar
3. D. J. Anick , On the homology of associative algebras, Trans. Amer. Math. Soc. 296(2) (1986) 641–659. Crossref, Google Scholar
4. J. Backelin , La série de Poincaré-Betti d’une algèbre graduée de type fini à une relation est rationnelle, C. R. Acad. Sci. Paris Sér. A-B 287(13) (1978) A843–A846. Google Scholar
5. M. J. Bardzell, Resolutions and cohomology of finite dimensional algebras, Ph.D. thesis, Virginia Polytechnic Institute and State University (1996), iv+65 pp. Google Scholar
6. M. J. Bardzell , Noncommutative Gröbner bases and Hochschild cohomology, in Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000), Contemporary Mathematics, Vol. 286 (American Mathematical Society, Providence, RI, 2001), pp. 227–240. Crossref, Google Scholar
7. S. Barmeier and Z. Wang, Deformations of path algebras of quivers with relations, preprint (2020), arXiv:2002.10001. Google Scholar
8. H. J. Baues and J.-M. Lemaire , Minimal models in homotopy theory, Math. Ann. 225(3) (1977) 219–242. Crossref, Google Scholar
9. A. Berglund , Poincaré series of monomial rings, J. Algebra 295(1) (2006) 211–230. Crossref, Google Scholar
10. A. Berglund , Homological perturbation theory for algebras over operads, Algebr. Geom. Topol. 14(5) (2014) 2511–2548. Crossref, Google Scholar
11. G. M. Bergman , The diamond lemma for ring theory, Adv. Math. 29(2) (1978) 178–218. Crossref, Google Scholar
12. L. A. Bokut , Imbeddings into simple associative algebras, Algebra Logika 15(2) (1976) 117–142, 245. Crossref, Google Scholar
13. L. A. Bokut and Y. Chen , Gröbner-Shirshov bases for Lie algebras: After A. I. Shirshov, Southeast Asian Bull. Math. 31(6) (2007) 1057–1076. Google Scholar
14. L. A. Bokut and Y. Chen , Gröbner-Shirshov bases and their calculation, Bull. Math. Sci. 4(3) (2014) 325–395. Crossref, Google Scholar
15. L. A. Bokut and Y, Chen , Gröbner-Shirshov bases for universal algebras, J. South China Normal Univ. Natur. Sci. Ed. 46(6) (2014) 1–9. Google Scholar
16. L. A. Bokut, Y. Chen and Y. Li , Gröbner-Shirshov bases Vinberg-Koszul-Gerstenhaber for right-symmetric algebras, Fundam. Prikl. Mat. 14(8) (2008) 55–67. Google Scholar
17. L. A. Bokut, Y. Chen and Y. Li , Anti-commutative Gröbner-Shirshov basis of a free Lie algebra, Sci. China Ser. A 52(2) (2009) 244–253. Crossref, Google Scholar
18. L. A. Bokut and G. P. Kukin , Algorithmic and Combinatorial Algebra, Mathematics and its Applications, Vol. 255 (Kluwer Academic Publishers Group, Dordrecht, 1994). Crossref, Google Scholar
19. L. A. Bokut and P. Malcolmson , Gröbner-Shirshov bases for relations of a Lie algebra and its enveloping algebra, in Algebras and Combinatorics (Hong Kong, 1997) (Springer, Singapore, 1999), pp. 47–54. Google Scholar
20. M. R. Bremner and V. Dotsenko , Algebraic Operads: An Algorithmic Companion (CRC Press, Boca Raton, FL, 2016). Crossref, Google Scholar
21. B. Buchberger , A criterion for detecting unnecessary reductions in the construction of Gröbner-bases, in Symbolic and Algebraic Computation (EUROSAM ’79, Int. Symp., Marseille, 1979), Lecture Notes in Computer Science, Vol. 72 (Springer, Berlin, 1979), pp. 3–21. Crossref, Google Scholar
22. B. Buchberger , An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41(3–4) (2006) 475–511, Translated from the 1965 German original by Michael P. Abramson. Crossref, Google Scholar
23. S. Chouhy and A. Solotar , Projective resolutions of associative algebras and ambiguities, J. Algebra 432 (2015) 22–61. Crossref, Google Scholar
24. A. Church and J. B. Rosser , Some properties of conversion, Trans. Amer. Math. Soc. 39(3) (1936) 472–482. Crossref, Google Scholar
25. M. G. Citterio , Classifying spaces of categories and term rewriting, Theory Appl. Categ. 9 (2001/2002) 92–105. Google Scholar
26. V. Dotsenko and A. Khoroshkin , Quillen homology for operads via Gröbner bases, Doc. Math. 18 (2013) 707–747. Crossref, Google Scholar
27. V. Dotsenko and A. Khoroshkin , Shuffle algebras, homology, and consecutive pattern avoidance, Algebra Number Theory 7(3) (2013) 673–700. Crossref, Google Scholar
28. C. Eder , Predicting zero reductions in gröbner basis computations, in Proc. 2014 Symp. Symbolic-Numeric Computation (Association for Computing Machinery, New York, 2014), pp. 109–110. Crossref, Google Scholar
29. L. Gerritzen , Tree polynomials and non-associative Gröbner bases, J. Symb. Comput. 41(3–4) (2006) 297–316. Crossref, Google Scholar
30. E. S. Golod , Standard bases and homology, in Algebra — Some Current Trends (Varna, 1986), Lecture Notes in Mathematics, Vol. 1352 (Springer, Berlin, 1988), pp. 88–95. Crossref, Google Scholar
31. E. S. Golod , On the homology algebra of the Shafarevich complex of a free algebra, Fundam. Prikl. Mat. 5(1) (1999) 97–100. Google Scholar
32. E. S. Golod and I. R. Šafarevič , On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 261–272. Google Scholar
33. E. L. Green, D. Happel and D. Zacharia , Projective resolutions over Artin algebras with zero relations, Illinois J. Math. 29(1) (1985) 180–190. Crossref, Google Scholar
34. L. Guetta, Homology of categories via polygraphic resolutions, preprint (2020), arXiv:2003.10734. Google Scholar
35. V. K. A. M. Gugenheim and L. A. Lambe , Perturbation theory in differential homological algebra. I, Illinois J. Math. 33(4) (1989) 566–582. Crossref, Google Scholar
36. V. K. A. M. Gugenheim, L. A. Lambe and J. D. Stasheff , Perturbation theory in differential homological algebra. II, Illinois J. Math. 35(3) (1991) 357–373. Crossref, Google Scholar
37. V. K. A. M. Gugenheim and J. D. Stasheff , On perturbations and $A_{\infty}$ $A_{\infty}$ -structures, Bull. Soc. Math. Belg. Sér. A 38 (1986) 237–246. Google Scholar
38. Y. Guiraud, E. Hoffbeck and P. Malbos , Convergent presentations and polygraphic resolutions of associative algebras, Math. Z. 293(1–2) (2019) 113–179. Crossref, Google Scholar
39. V. Hinich , Homological algebra of homotopy algebras, Commun. Algebra 25(10) (1997) 3291–3323. Crossref, Google Scholar
40. H. Hironaka , On resolution of singularities (characteristic zero), in Proc. Int. Congr. Mathematicians (Stockholm, 1962) (Institut Mittag-Leffler, Djursholm, 1963), pp. 507–521. Google Scholar
41. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964) 109–203; ibid. (2), 79 (1964) 205–326. Google Scholar
42. M. Hovey , Model Categories, Mathematical Surveys and Monographs, Vol. 63 (American Mathematical Society, Providence, RI, 1999). Google Scholar
43. P. Hu , Transfinite spectral sequences, in Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998), Contemporary Mathematics, Vol. 239 (American Mathematical Society, Providence, RI, 1999), pp. 197–216. Crossref, Google Scholar
44. J. Huebschmann and T. Kadeishvili , Small models for chain algebras, Math. Z. 207(2) (1991) 245–280. Crossref, Google Scholar
45. N. Iyudu and I. Vlassopoulos, Homologies of monomial operads and algebras, preprint (2020), arXiv:2008.00985. Google Scholar
46. J. F. Jardine , A closed model structure for differential graded algebras, in Cyclic Cohomology and Noncommutative Geometry (Waterloo, ON, 1995), Fields Institute Communications, Vol. 17 (American Mathematical Society, Providence, RI, 1997), pp. 55–58. Google Scholar
47. T. Józefiak , Tate resolutions for commutative graded algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969) 617–621. Google Scholar
48. B. Keller, Notes on minimal models (2003), https://webusers.imj-prg.fr/bernhard.keller/publ/minmod.pdf. Google Scholar
49. D. E. Knuth and P. B. Bendix , Simple word problems in universal algebras, in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) (Pergamon, Oxford, 1970), pp. 263–297. Google Scholar
50. C. Kollreider and B. Buchberger , An improved algorithmic construction of Gröbner-bases for polynomial ideals, SIGSAM Bull. 12(2) (1978) 27–36. Crossref, Google Scholar
51. M. Kontsevich and Y. Soibelman , Deformations of algebras over operads and the Deligne conjecture, in Conf. Moshé Flato 1999, Vol. I (Dijon), Mathematical Physics Studies, Vol. 21 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 255–307. Google Scholar
52. Y. Lafont , A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier), J. Pure Appl. Algebra 98(3) (1995) 229–244. Crossref, Google Scholar
53. Y. Lafont and F. Métayer , Polygraphic resolutions and homology of monoids, J. Pure Appl. Algebra 213(6) (2009) 947–968. Crossref, Google Scholar
54. Y. Lafont and A. Prouté , Church-Rosser property and homology of monoids, Math. Struct. Comput. Sci. 1(3) (1991) 297–326. Crossref, Google Scholar
55. L. A. Lambe , Homological perturbation theory, Hochschild homology, and formal groups, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemporary Mathematics, Vol. 134 (American Mathematical Society, Providence, RI, 1992), pp. 183–218. Crossref, Google Scholar
56. V. N. Latyshev , Combinatorial Ring Theory, Standard Bases (Izdatel’stvo Moskovskogo Gosudarstvennogo Universiteta, 1988). Google Scholar
57. V. N. Latyshev , General version of standard bases in linear structures, in Algebra (Moscow, 1998), (de Gruyter, Berlin, 2000), pp. 215–226. Crossref, Google Scholar
58. V. N. Latyshev , A general version of standard basis and its application to $T$ $T$ -ideals, Acta Appl. Math. 85(1–3) (2005) 219–223. Crossref, Google Scholar
59. V. N. Latyshev , A general version of a standard basis in associative algebras and their derivative constructions, Fundam. Prikl. Mat. 15(3) (2009) 183–203. Google Scholar
60. V. N. Latyšev, Algebras with identical relations , Dokl. Akad. Nauk SSSR 146 (1962) 1003–1006. Google Scholar
61. V. N. Latyšev , On the choice of basis in a $T$ $T$ -ideal, Sibirsk. Mat. Zh. 4 (1963) 1122–1127. Google Scholar
62. J.-M. Lemaire , “ Autopsie d’un meurtre” dans l’homologie d’une algèbre de chaînes, Ann. Sci. Éc. Norm. Supér. (4) 11(1) (1978) 93–100. Crossref, Google Scholar
63. Y. Li, Q. Mo and L. A. Bokut , Generalized anti-commutative Gröbner-Shirshov basis theory and free Sabinin algebras, Commun. Algebra 48(12) (2020) 5086–5109. Crossref, Google Scholar
64. J.-L. Loday and M. Ronco , Permutads, J. Comb. Theory Ser. A 120(2) (2013) 340–365. Crossref, Google Scholar
65. J. Lurie , Moduli problems for ring spectra, in Proc. Int. Congr. Mathematicians. Volume II (Hindustan Book Agency, New Delhi, 2010), pp. 1099–1125. Google Scholar
66. J. Lurie, Derived algebraic geometry X: Formal moduli problems (2011), https://www.math.ias.edu/ lurie/papers/DAG-X.pdf. Google Scholar
67. M. Manetti , Differential graded Lie algebras and formal deformation theory, in Algebraic Geometry — Seattle 2005. Part 2, Proceedings of Symposia in Pure Mathematics, Vol. 80 (American Mathematical Society, Providence, RI, 2009), pp. 785–810. Crossref, Google Scholar
68. M. Markl , Models for operads, Commun. Algebra 24(4) (1996) 1471–1500. Crossref, Google Scholar
69. F. Métayer , Resolutions by polygraphs, Theory Appl. Categ. 11(7) (2003) 148–184. Google Scholar
70. M. H. A. Newman , On theories with a combinatorial definition of “equivalence”, Ann. of Math. (2) 43 (1942) 223–243. Crossref, Google Scholar
71. V. P. Palamodov , Deformations of complex spaces, Uspehi Mat. Nauk 31(3) (1976) 129–194. Google Scholar
72. S. B. Priddy , Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970) 39–60. Crossref, Google Scholar
73. J. P. Pridham , Unifying derived deformation theories, Adv. Math. 224(3) (2010) 772–826. Crossref, Google Scholar
74. Z. Qi, Y. Xu, J. J. Zhang and X. Zhao, Growth of nonsymmetric operads, preprint (2020), arXiv:2010.12172. Google Scholar
75. D. G. Quillen , Homotopical Algebra, Lecture Notes in Mathematics, Vol. 43 (Springer-Verlag, Berlin, 1967). Crossref, Google Scholar
76. S. Rahmati, Theory of spectral sequences of exact couples: Applications to countably and transfinitely filtered modules, Ph.D thesis, University of Alberta (Canada) (2013), xi+209 pp. Google Scholar
77. M. J. Redondo and F. R. Bertone, $L_{\infty}$ $L_{\infty}$ -structure on Bardzell’s complex for monomial algebras, preprint (2020), arXiv:2008.08122. Google Scholar
78. M. Ronco , Shuffle bialgebras, Ann. Inst. Fourier (Grenoble) 61(3) (2011) 799–850. Crossref, Google Scholar
79. M. Schlessinger and J. Stasheff , The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra 38(2–3) (1985) 313–322. Crossref, Google Scholar
80. M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, preprint (2012), arXiv:1211.1647. Google Scholar
81. W. Specht , Gesetze in Ringen. I, Math. Z. 52 (1950) 557–589. Crossref, Google Scholar
82. C. C. Squier , Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49(1–2) (1987) 201–217. Crossref, Google Scholar
83. D. Sullivan , Infinitesimal computations in topology, Publ. Math. Inst. Hautes Études Sci. 47 (1978) 269–331. Crossref, Google Scholar
84. A. I. Širšov , Some algorithm problems for Lie algebras, Sibirsk. Mat. Zh. 3 (1962) 292–296. Google Scholar
85. P. Tamaroff , Minimal models for monomial algebras, to appear in Homology Homotopy Appl. 23(1) (2021) 341–366. Crossref, Google Scholar
86. J. Tate , Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957) 14–27. Crossref, Google Scholar
87. R. Tuniyaz, L. A. Bokut, M. Xiryazidin and A. Obul , Gröbner-Shirshov bases for free Gelfand-Dorfman-Novokov algebras and for right ideals of free right Leibniz algebras, Commun. Algebra 46(10) (2018) 4392–4402. Crossref, Google Scholar
88. V. A. Ufnarovskij , Combinatorial and asymptotic methods in algebra, in Algebra, VI, Encyclopaedia of Mathematical Sciences, Vol. 57 (Springer, Berlin, 1995), pp. 1–196. Crossref, Google Scholar
89. A. I. Žukov , Reduced systems of defining relations in non-associative algebras, Mat. Sb. (N.S.) 27(69) (1950) 267–280. Google Scholar

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Received 6 November 2020

Revised 19 September 2023

Accepted 6 October 2023

Published: 31 October 2023

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Tangent complexes and the Diamond Lemma

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