Voir la notice de l'article provenant de la source Mathematical Sciences Publishers
We give a short proof of Kaledin’s theorem on the degeneration of the noncommutative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschild homology and the theory of cyclotomic spectra. As a consequence, we also obtain relative versions of the degeneration theorem, both in characteristic zero and for regular bases in characteristic .
Mathew, Akhil 1
@article{GT_2020_24_6_a0,
author = {Mathew, Akhil},
title = {Kaledin{\textquoteright}s degeneration theorem and topological {Hochschild} homology},
journal = {Geometry & topology},
pages = {2675--2708},
publisher = {mathdoc},
volume = {24},
number = {6},
year = {2020},
url = {http://geodesic.mathdoc.fr/item/GT_2020_24_6_a0/}
}
Mathew, Akhil. Kaledin’s degeneration theorem and topological Hochschild homology. Geometry & topology, Tome 24 (2020) no. 6, pp. 2675-2708. http://geodesic.mathdoc.fr/item/GT_2020_24_6_a0/
[1] , , , , , , Topological cyclic homology via the norm, Doc. Math. 23 (2018) 2101 | DOI
[2] , Periodic cyclic homology and derived de Rham cohomology, Ann. –Theory 4 (2019) 505 | DOI
[3] , , Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol. 18 (2014) 1149 | DOI
[4] , , , On the Blumberg–Mandell Künneth theorem for TP, Selecta Math. 24 (2018) 4555 | DOI
[5] , , A remark on the Hochschild–Kostant–Rosenberg theorem in characteristic p, preprint (2017)
[6] , , , The geometry of the cyclotomic trace, preprint (2017)
[7] , , , The Stacks project, electronic reference (2005–)
[8] , , , Topological Hochschild homology and integral p–adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019) 199 | DOI
[9] , , , A universal characterization of higher algebraic K–theory, Geom. Topol. 17 (2013) 733 | DOI
[10] , , , Uniqueness of the multiplicative cyclotomic trace, Adv. Math. 260 (2014) 191 | DOI
[11] , , Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012) 1053 | DOI
[12] , , The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015) 3105 | DOI
[13] , , The strong Künneth theorem for topological periodic cyclic homology, preprint (2017)
[14] , Topological Hochschild homology of Z∕pn, J. Pure Appl. Algebra 148 (2000) 29 | DOI
[15] , Differential graded categories are k–linear stable ∞–categories, preprint (2013)
[16] , Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Inst. Hautes Études Sci. Publ. Math. 35 (1968) 259
[17] , , Relèvements modulo p2 et décomposition du complexe de de Rham, Invent. Math. 89 (1987) 247 | DOI
[18] , , Finite generation and continuity of topological Hochschild and cyclic homology, Ann. Sci. École Norm. Sup. 50 (2017) 201 | DOI
[19] , , An introduction to the Riemann–Hilbert correspondence for unit F–crystals, from: "Geometric aspects of Dwork theory, II" (editors A Adolphson, F Baldassarri, P Berthelot, N Katz, F Loeser), de Gruyter (2004) 677 | DOI
[20] , Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology, from: "Quantum deformations of algebras and their representations" (editors A Joseph, S Shnider), Israel Math. Conf. Proc. 7, Bar-Ilan Univ. (1993) 65
[21] , Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187 | DOI
[22] , Relative algebraic K–theory and cyclic homology, Ann. of Math. 124 (1986) 347 | DOI
[23] , On the p–typical curves in Quillen’s K–theory, Acta Math. 177 (1996) 1 | DOI
[24] , Topological Hochschild homology and the Hasse–Weil zeta function, from: "An alpine bouquet of algebraic topology" (editors C Ausoni, K Hess, B Johnson, I Moerdijk, J Scherer), Contemp. Math. 708, Amer. Math. Soc. (2018) 157 | DOI
[25] , , On the K–theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29 | DOI
[26] , , , Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 | DOI
[27] , The homotopy fixed points of the circle action on Hochschild homology, preprint (2015)
[28] , , , Higher traces, noncommutative motives, and the categorified Chern character, Adv. Math. 309 (2017) 97 | DOI
[29] , Complexe cotangent et déformations, I, 239, Springer (1971) | DOI
[30] , Réduction semi-stable et décomposition de complexes de de Rham à coefficients, Duke Math. J. 60 (1990) 139 | DOI
[31] , Non-commutative Hodge-to-de Rham degeneration via the method of Deligne–Illusie, Pure Appl. Math. Q. 4 (2008) 785 | DOI
[32] , Spectral sequences for cyclic homology, from: "Algebra, geometry, and physics in the 21st century" (editors D Auroux, L Katzarkov, T Pantev, Y Soibelman, Y Tschinkel), Progr. Math. 324, Birkhäuser (2017) 99 | DOI
[33] , , Notes on A∞–algebras, A∞–categories and non-commutative geometry, from: "Homological mirror symmetry" (editors A Kapustin, M Kreuzer, K G Schlesinger), Lecture Notes in Phys. 757, Springer (2009) 153
[34] , Spectral algebraic geometry, preprint (2016)
[35] , Higher algebra, preprint (2017)
[36] , , Equivariant orthogonal spectra and S–modules, 755, Amer. Math. Soc. (2002) | DOI
[37] , Residue fields for a class of rational E∞–rings and applications, J. Pure Appl. Algebra 221 (2017) 707 | DOI
[38] , , , Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017) 994 | DOI
[39] , , , THH(R)≅R ⊗ S1 for E∞ ring spectra, J. Pure Appl. Algebra 121 (1997) 137 | DOI
[40] , , On topological cyclic homology, Acta Math. 221 (2018) 203 | DOI
[41] , , , The Gauss–Manin connection on the periodic cyclic homology, Selecta Math. 24 (2018) 531 | DOI
[42] , On Serre duality for compact homologically smooth DG algebras, preprint (2007)
[43] , Homotopy theory of spectral categories, Adv. Math. 221 (2009) 1122 | DOI
[44] , Noncommutative motives, 63, Amer. Math. Soc. (2015) | DOI
[45] , Anneaux de définition des dg-algèbres propres et lisses, Bull. Lond. Math. Soc. 40 (2008) 642 | DOI