Abstract
By a result of Gerstenhaber and Schack, the simplicial cohomology ring \(\operatorname{H}^{\bullet}({\mathcal C},k)\) of a poset \({\mathcal C}\) is isomorphic to the Hochschild cohomology ring \(\operatorname{HH}^{\bullet}(k{\mathcal C})\) of the category algebra \(k{\mathcal C}\), where the poset is viewed as a category and \(k\) is a field. Extending results of Mishchenko, under certain assumptions on a category \({\mathcal C}\), we construct a category \({\mathcal D}\) and a graded \(k\)-linear isomorphism \(\operatorname{HH}^{\bullet}(k{\mathcal C})\cong \operatorname{H}^{\bullet}({\mathcal D},k)\). Interpreting the degree one cohomology, we also show how the \(k\)-space of derivations on \(k{\mathcal C}\) graded by some semigroup corresponds to the \(k\)-space of characters on \({\mathcal D}\).
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References
M. Gerstenhaber and S. D. Schack, “Simplicial cohomology is Hochschild cohomology,” J. Pure Appl. Algebra. 30, 143–156 (1983).
M. Gerstenhaber and S. D. Schack, “Relative Hochschild cohomology, rigid algebras, and the Bockstein,” J. Pure Appl. Algebra. 43, 53–74 (1986).
A. S. Mishchenko, “Correlation between the Hochschild cohomology and the Eilenberg–MacLane cohomology of group algebras from a geometric point of view,” Russ. J. Math. Phys. 27 (2), 236–250 (2020).
F. Xu, “Hochschild and ordinary cohomology rings of small categories,” Adv. Math. 219 (6), 1872–1893 (2008).
N. Snashall and Ø. Solberg, “Support varieties and Hochschild cohomology rings,” Proc. London Math. Soc. 88 (3), 705–732 (2004).
P. Webb, An Introduction to the Representations and Cohomology of Categories, in: Group Representation Theory (EPFL Press, Lausanne, 2007), pp. 149–173.
M. V. Lawson and A. R. Wallis, “A categorical description of Bass–Serre theory,” ArXiv:1304.6854 [math. CT] (2014).
B. Steinberg, “A Theory of transformation monoids: combinatorics and representation theory,” Electron. J. Comb. 17 (R164), 1–56 (2010).
J. Lodder, “Hochschild and simplicial cohomology,” ArXiv:1802.03076 [math. AT] (2018).
A. A. Arutyunov and A. S. Mishchenko, “Smooth version of Johnson’s problem concerning derivations of group algebras,” Sb. Math. 210 (6), 756–782 (2019).
A. A. Arutyunov and L. M. Kosolapov, “Derivations of group rings for finite and FC groups,” Finite Fields Their Appl. 76, 101921 (2021).
A. A. Arutyunov and A. Alekseev, “Complex of \(n\)-categories and derivations in group algebras,” Topol. Appl. 275,, 107002 (2020).
C.-C. Todea, “BD algebras and group cohomology,” C. R. Math. 359 (8), 925–937 (2021).
S. J. Witherspoon, Hochschild Cohomology for Algebras. Graduate Studies in Mathematics (AMS, Providence, RI, 2019), Vol. 204.
D. Quillen, Higher Algebraic K-Theory I, in: Lecture Notes in Math. (Springer-Verlag, Berlin, 1973), Vol. 341, pp. 85–147.
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The authors are grateful for the remarks and suggestions of the referee.
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This work was supported by a grant of the Ministry of Research, Innovation, and Digitalization, CNCS/CCCDI–UEFISCDI, project number PN-III-P1-1.1-TE-2019-0136, within PNCDI III.
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Simion, II., Todea, CC. Hochschild Cohomology of Some Finite Category Algebras as Simplicial Cohomology. Math Notes 112, 741–754 (2022). https://doi.org/10.1134/S0001434622110104
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DOI: https://doi.org/10.1134/S0001434622110104