Algebraic K-theory of quasi-smooth blow-ups and cdh descent
Annales Henri Lebesgue, Volume 3 (2020), pp. 1091-1116.

We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason’s blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodsky’s cdh topology, which we use to give a direct proof of Cisinski’s theorem that Weibel’s homotopy invariant K-theory satisfies cdh descent.

Nous construisons une décomposition semi-orthogonale sur la catégorie des complexes parfaits sur l’éclaté d’un champ d’Artin dérivé le long d’un centre quasi-lisse. Ceci conduit à une généralisation de la formule d’éclatement de Thomason en K-théorie algébrique à une situation dérivée. Nous établissons aussi un nouveau critère de descente pour la topologie cdh de Voevodsky, que nous utilisons pour donner une démonstration directe d’un thèorème de Cisinski qui affirme que la K-théorie invariante par homotopie de Weibel satisfait la descente cdh.

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Accepted:
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DOI: 10.5802/ahl.55
Classification: 20C25, 14J99, 14A20
Keywords: derived algebraic geometry, semi-orthogonal decompositions, algebraic K-theory, cdh descent
Khan, Adeel A. 1

1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, (Germany)
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Khan, Adeel A. Algebraic K-theory of quasi-smooth blow-ups and cdh descent. Annales Henri Lebesgue, Volume 3 (2020), pp. 1091-1116. doi : 10.5802/ahl.55. http://archive.numdam.org/articles/10.5802/ahl.55/

[BGI71] Berthelot, Pierre; Grothendieck, Alexandre; Illusie, Luc Séminaire de Géométrie Algébrique du Bois Marie – 1966-67 – Théorie des intersections et théorème de Riemann–Roch – (SGA 6), Lecture Notes in Mathematics, 225, Springer, 1971 | Zbl

[BGT13] Blumberg, Andrew J.; Gepner, David; Tabuada, Gonçalo A universal characterization of higher algebraic K-theory, Geom. Topol., Volume 17 (2013) no. 2, pp. 733-838 | DOI | MR | Zbl

[BK89] Bondal, Aleksei Igorevich; Kapranov, Mikhail Mikhailovich Representable functors, Serre functors, and mutations, Izv. Ross. Akad. Nauk SSSR, Ser. Mat., Volume 53 (1989) no. 6, pp. 1183-1205

[BM12] Blumberg, Andrew J.; Mandell, Michael A. Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol., Volume 16 (2012) no. 2, pp. 1053-1120 | DOI | MR | Zbl

[BS20] Bergh, Daniel; Schnürer, Olaf M Conservative descent for semi-orthogonal decompositions, Adv. Math., Volume 360 (2020), 106882, p. 39 | MR | Zbl

[BVdB03] Bondal, Aleksei Igorevich; Van den Bergh, Michel Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., Volume 3 (2003) no. 1, pp. 1-36 | DOI | MR | Zbl

[Cis13] Cisinski, Denis-Charles Descente par éclatements en K-théorie invariante par homotopie, Ann. Math., Volume 177 (2013) no. 2, pp. 425-448 | DOI | Zbl

[GR17] Gaitsgory, Dennis; Rozenblyum, Nick A Study in Derived Algebraic Geometry: Volumes I and II, Mathematical Surveys and Monographs, 221.1-221.2, American Mathematical Society, 2017 | Zbl

[Hae04] Haesemeyer, Christian Descent properties of homotopy K-theory, Duke Math. J., Volume 125 (2004) no. 3, pp. 589-619 | DOI | MR | Zbl

[HK19] Hoyois, Marc; Krishna, Amalendu Vanishing theorems for the negative K-theory of stacks, Ann. K-Theory, Volume 4 (2019) no. 3, pp. 439-472 | DOI | MR | Zbl

[HLP14] Halpern-Leistner, Daniel; Preygel, Anatoly Mapping stacks and categorical notions of properness (2014) (https://arxiv.org/abs/1402.3204)

[Hoy16] Hoyois, Marc Cdh descent in equivariant homotopy K-theory (2016) (https://arxiv.org/abs/1604.06410) | Zbl

[HR17] Hall, Jack; Rydh, David Perfect complexes on algebraic stacks, Compos. Math., Volume 153 (2017) no. 11, pp. 2318-2367 | DOI | MR | Zbl

[Kha16] Khan, Adeel A. Motivic homotopy theory in derived algebraic geometry, Ph. D. Thesis, Universität Duisburg-Essen, Germany (2016) (Available at https://www.preschema.com/thesis/)

[Kha19] Khan, Adeel A. The Morel–Voevodsky localization theorem in spectral algebraic geometry, Geom. Topol., Volume 23 (2019) no. 7, pp. 3647-3685 | DOI | MR | Zbl

[KR18a] Khan, Adeel A.; Rydh, David Virtual Cartier divisors and blow-ups (2018) (https://arxiv.org/abs/1802.05702)

[KR18b] Krishna, Amalendu; Ravi, Charanya Algebraic K-theory of quotient stacks, Ann. K-Theory, Volume 3 (2018) no. 2, pp. 207-233 | DOI | MR | Zbl

[KST18] Kerz, Moritz; Strunk, Florian; Tamme, Georg Algebraic K-theory and descent for blow-ups, Invent. Math., Volume 211 (2018) no. 2, pp. 523-577 | DOI | MR | Zbl

[LT18] Land, Markus; Tamme, Georg On the K-theory of pullbacks (2018) (https://arxiv.org/abs/1808.05559) | Zbl

[Lur09] Lurie, Jacob Higher topos theory, Annals of Mathematics Studies, Princeton University Press, 2009 no. 170 | MR | Zbl

[Lur12] Lurie, Jacob Higher algebra, 2012 (Preprint, version of 2017-09-18 available at https://www.math.ias.edu/~lurie/papers/HA.pdf)

[Lur16] Lurie, Jacob Spectral algebraic geometry (2016) (Preprint, version of 2018-02-03, available at https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf)

[MV99] Morel, Fabien; Voevodsky, Vladimir A 1 -homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci., Volume 90 (1999) no. 1, pp. 45-143 | DOI | Numdam | Zbl

[Orl92] Orlov, Dmitri Olegovich Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 56 (1992) no. 4, pp. 852-862 | Zbl

[RG71] Raynaud, Michel; Gruson, Laurent Critères de platitude et de projectivité: Techniques de « platification » d’un module, Invent. Math., Volume 13 (1971) no. 1-2, pp. 1-89 | DOI | Zbl

[Tho87] Thomason, Robert Wayne Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., Volume 65 (1987) no. 1, pp. 16-34 | DOI | MR | Zbl

[Tho93a] Thomason, Robert Wayne Les K-groupes d’un fibré projectif, Algebraic K-theory and algebraic topology (Lake Louise. Proceedings of the NATO Advanced Study Institute, Lake Louise, Alberta, Canada, December 12–16, 1991 (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 407, Kluwer Academic Publishers, 1993, pp. 243-248 | MR

[Tho93b] Thomason, Robert Wayne Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math., Volume 112 (1993) no. 1, pp. 195-215 | DOI | Zbl

[TT90] Thomason, Robert Wayne; Trobaugh, Thomas Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift. Vol. III (Progress in Mathematics), Volume 88, Springer, 1990, pp. 247-435 | DOI | MR | Zbl

[TV08] Toën, Bertrand; Vezzosi, Gabriele Homotopical algebraic geometry II: Geometric stacks and applications, 902, American Mathematical Society, 2008

[Voe10a] Voevodsky, Vladimir Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra, Volume 214 (2010) no. 8, pp. 1384-1398 | DOI | MR | Zbl

[Voe10b] Voevodsky, Vladimir Unstable motivic homotopy categories in Nisnevich and cdh-topologies, J. Pure Appl. Algebra, Volume 214 (2010) no. 8, pp. 1399-1406 | DOI | MR | Zbl

[Wei89] Weibel, Charles A. Homotopy algebraic K-theory, Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987) (Contemporary Mathematics), Volume 83, American Mathematical Society, 1989, pp. 461-488 | DOI | MR | Zbl

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