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Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

Part of: Higher algebraic $K$-theory (Co)homology theory Commutative algebra: Homological methods Homological methods

Published online by Cambridge University Press:  22 June 2021

Benjamin Antieau ,
Bhargav Bhatt  and
Akhil Mathew
Benjamin Antieau
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL60208; E-mail: [email protected]
Bhargav Bhatt
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109; E-mail: [email protected]
Akhil Mathew
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL60637; E-mail: [email protected]

Abstract

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We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

MSC classification

Primary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
Secondary: 14F40: de Rham cohomology 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e49
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Antieau, B., ‘Čech approximation to the Brown-Gersten spectral sequence’, Homology Homotopy Appl. 13(1) (2011), 319–348.CrossRefGoogle Scholar
Antieau, B., ‘Cohomological obstruction theory for Brauer classes and the period-index problem’, J. K-Theory 8(3) (2011), 419–435.CrossRefGoogle Scholar
Antieau, B., ‘Periodic cyclic homology and derived de Rham cohomology’, Ann. K-Theory 4(3) (2019), 505–519.CrossRefGoogle Scholar
Antieau, B. and Bragg, D., ‘Derived invariants from topological Hochschild homology’, Preprint, 2019, arXiv:1906.12267.Google Scholar
Antieau, B. and Gepner, D., ‘Brauer groups and étale cohomology in derived algebraic geometry’, Geom. Topol. 18(2) (2014), 1149–1244.CrossRefGoogle Scholar
Antieau, B., Mathew, A. and Nikolaus, T., ‘On the Blumberg-Mandell Künneth theorem for TP’, Selecta Math. (N.S.) 24(5) (2018), 4555–4576.CrossRefGoogle Scholar
Antieau, B. and Vezzosi, G., ‘A remark on the Hochschild-Kostant-Rosenberg theorem in characteristic $p$ $^{\prime }$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20(3) (2020), 1135–1145.Google Scholar
Bernardara, M., ‘A semiorthogonal decomposition for Brauer-Severi schemes’, Math. Nachr. 282(10) (2009), 1406–1413.CrossRefGoogle Scholar
Bhatt, B., ‘Completions and derived de Rham cohomology’, Preprint, 2012, arXiv:1207.6193.Google Scholar
Bhatt, B., ‘p-Adic derived de Rham cohomology’, Preprint, 2012, arXiv:1204.6560.Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., ‘Integral $p$-adic Hodge theory’, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397.CrossRefGoogle Scholar
Bhatt, B., Morrow, M. and Scholze, P., ‘Topological Hochschild homology and integral $p$-adic Hodge theory’, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310.CrossRefGoogle Scholar
Bombieri, E. and Mumford, D., ‘Enriques’ classification of surfaces in char. $p$. III’, Invent. Math. 35 (1976), 197–232.CrossRefGoogle Scholar
Cortiñas, G. and Weibel, C., ‘Homology of Azumaya algebras’, Proc. Amer. Math. Soc. 121(1) (1994), 53–55.CrossRefGoogle Scholar
Cossec, F. R. and Dolgachev, I. V., Enriques Surfaces. I, Progress in Mathematics, Vol. 76 (Birkhäuser, Boston, 1989).Google Scholar
de Jong, A. J., ‘The period-index problem for the Brauer group of an algebraic surface’, Duke Math. J. 123(1) (2004), 71–94.Google Scholar
Deligne, P. and Illusie, L., ‘Relèvements modulo ${p}^2$ et décomposition du complexe de de Rham’, Invent. Math. 89(2) (1987), 247–270.CrossRefGoogle Scholar
Deligne, P. and Katz, N.. II, Lecture Notes in Mathematics, Vol. 340 (Springer, Berlin, 1973).Google Scholar
Friedlander, E. M., ‘Lectures on the cohomology of finite group schemes’, in Rational Representations, the Steenrod Algebra and Functor Homology, Panor. Synthèses, Vol. 16 (Société Mathématique de France, Paris, 2003), pp. 27–53.Google Scholar
Gerstenhaber, M. and Schack, S. D., ‘A Hodge-type decomposition for commutative algebra cohomology’, J. Pure Appl. Algebra 48(3) (1987), 229–247.CrossRefGoogle Scholar
Illusie, L., Complexe cotangent et déformations . II, Lecture Notes in Mathematics, Vol. 283 (Springer, Berlin, 1972).CrossRefGoogle Scholar
Illusie, L., ‘Complexe de de Rham-Witt et cohomologie cristalline’, Ann. Sci. École Norm. Sup. (4) 12(4) (1979), 501–661.CrossRefGoogle Scholar
Illusie, L., ‘Frobenius et dégénérescence de Hodge’, in Introduction à la théorie de Hodge, Panor. Synthèses, Vol. 3 (Société Mathématique de France, Paris), 1996, pp. 113–168.Google Scholar
Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, Vol. 131 (Academic Press, Boston, 1987).Google Scholar
Kaledin, D., ‘Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie’, Pure Appl Math Q. 4(3) (2008), 785–876.CrossRefGoogle Scholar
Kaledin, D., ‘Spectral sequences for cyclic homology’, in Algebra, Geometry, and Physics in the 21st Century (Springer, Basel, 2017), pp. 99–129.Google Scholar
Lurie, J., Higher Topos Theory, Annals of Mathematics Studies, Vol. 170 (Princeton University Press, Princeton, NJ, 2009).Google Scholar
Lurie, J., ‘Spectral algebraic geometry’, (2018). URL: http://www.math.harvard.edu/~lurie/.Google Scholar
Mathew, A., ‘Kaledin’s degeneration theorem and topological Hochschild homology’, Geom. Topol. 24(6) (2020), 2675–2708.CrossRefGoogle Scholar
Nikolaus, T. and Scholze, P., ‘On topological cyclic homology’, Acta Math. 221(2) (2018), 203–409.CrossRefGoogle Scholar
Poonen, B., ‘Bertini theorems over finite fields’, Ann. Math. (2) 160(3) (2004), 1099–1127.CrossRefGoogle Scholar
Quillen, D., ‘On the (co-) homology of commutative rings’, in Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) (American Mathematical Society, Providence, RI, 1970), pp. 65–87.CrossRefGoogle Scholar
Quillen, D., ‘Higher algebraic $\text{K}$-theory. I’, in Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics (Berlin, Heidelberg), Vol. 341 (1973), pp. 85–147.Google Scholar
Rao, S., Yang, S., Yang, X. and Yu, X., ‘Hodge cohomology on blow-ups along subvarieties’, Preprint, 2019, arXiv:1907.13281.Google Scholar
Rosenberg, J., ‘Algebraic $\text{K}$-theory and its applications’, in Graduate Texts in Mathematics, Vol. 147 (Springer, New York, 1994).Google Scholar
Serre, J.-P., ‘Sur la topologie des variétés algébriques en caractéristique $p$’, in Symposium internacional de topología algebraica [International symposium on algebraic topology] (Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958), pp. 24–53.Google Scholar
Shklyarov, D., ‘Hirzebruch-Riemann-Roch-type formula for DG algebras’, Proc. Lond. Math. Soc. (3) 106(1) (2013), 1–32.CrossRefGoogle Scholar
The Stacks Project Authors, ‘The Stacks Project’, (2019). URL: https://stacks.math.columbia.edu.Google Scholar
Toën, B., ‘Derived Azumaya algebras and generators for twisted derived categories’, Invent. Math. 189(3) (2012), 581–652.CrossRefGoogle Scholar
Toën, B. and Vezzosi, G., ‘Algèbres simpliciales ${S}^1$-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs’, Compos. Math. 147(6) (2011), 1979–2000.CrossRefGoogle Scholar
Totaro, B., ‘The Chow ring of a classifying space’, in Algebraic $K$-Theory, Proc. Sympos. Pure Math., Vol. 67 (American Mathematical Society, Providence, RI, 1999), pp. 249–281.CrossRefGoogle Scholar
Totaro, B., ‘Hodge theory of classifying stacks’, Duke Math. J. 167(8) (2018), 1573–1621.CrossRefGoogle Scholar
Weibel, C., ‘The Hodge filtration and cyclic homology’,$K$-Theory 12(2) (1997), 145–164.CrossRefGoogle Scholar
Weibel, C. A. and Geller, S. C., ‘Étale descent for Hochschild and cyclic homology’, Comment. Math. Helv. 66(3) (1991), 368–388.CrossRefGoogle Scholar
Yekutieli, A., ‘The continuous Hochschild cochain complex of a scheme’, Can. J. Math. 54(6) (2002), 1319–1337.CrossRefGoogle Scholar