Motivic realizations of singularity categories and vanishing cycles

Anthony Blanc; Marco Robalo; Bertrand Toën; Gabriele Vezzosi

doi:10.5802/jep.81

Motivic realizations of singularity categories and vanishing cycles
[Réalisations motiviques des catégories de singularités et cycles évanescents]

Anthony Blanc ¹ ; Marco Robalo ² ; Bertrand Toën ³ ; Gabriele Vezzosi ⁴

¹ Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265, 34136 Trieste, Italia
² Sorbonne Université, Faculté des sciences et ingénierie Pierre et Marie Curie, Institut de Mathématiques de Jussieu-PRG 4 place Jussieu, Case 247, 75252 Paris Cedex 05, France
³ CNRS, Université de Toulouse, Institut de Mathématiques de Toulouse (UMR 5219) 118 route de Narbonne, 31062 Toulouse Cedex 9, France
⁴ Dipartimento di Matematica ed Informatica “Ulisse Dini”, Università degli Studi di Firenze Viale Morgagni 67/a, 50134 Firenze, Italy

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 651-747.

Résumé
Abstract

Dans cet article, on démontre un théorème de comparaison entre la théorie des cycles évanescents à la SGA7 et la catégorie des singularités d’un modèle de Landau-Ginzburg définie sur un anneau de valuation discrète, complet. Dans une première partie, nous étendons au cadre infini-catégorique le théorème de comparaison d’Orlov entre catégories de singularités et catégories de factorisations matricielles. Dans une seconde partie nous démontrons l’énoncé de comparaison, à l’aide d’une notion de réalisations motiviques de catégories.

In this article we establish a precise comparison between vanishing cycles and the singularity category of Landau–Ginzburg models over an excellent Henselian discrete valuation ring. By using noncommutative motives, we first construct a motivic $ℓ$ -adic realization functor for dg-categories. Our main result, then asserts that, given a Landau–Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the $ℓ$ -adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and $\infty$ -categorical lax symmetric monoidal version of Orlov’s comparison theorem between the derived category of singularities and the derived category of matrix factorizations for a Landau–Ginzburg model over a Noetherian regular local ring.

Reçu le : 2017-09-07
Accepté le : 2018-09-03
Publié le : 2018-10-09

MR Zbl

DOI : 10.5802/jep.81

Classification : 14F42, 19E08, 32S30, 16S38
Keywords: Landau-Ginzburg model, dg-categories of singularities, matrix factorisations, vanishing cycles, nearby cycles, motives, noncommutative motives, motivic homotopy theory Morel-Voevodsky, motivic realisations, $\ell $-adic sheaves, algebraic K-theory
Mot clés : Modèles de Landau-Ginzburg, dg-catégories de singularités, factorisations matricielles, cycles évanescents, cycles proches, motifs, motifs non-commutatifs, théorie homotopique motivique des schémas, réalisations motiviques, faisceaux $\ell $-adiques, K-théorie algébrique

Affiliations des auteurs :

Anthony Blanc ¹ ; Marco Robalo ² ; Bertrand Toën ³ ; Gabriele Vezzosi ⁴

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Anthony Blanc; Marco Robalo; Bertrand Toën; Gabriele Vezzosi. Motivic realizations of singularity categories and vanishing cycles. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 651-747. doi : 10.5802/jep.81. https://jep.centre-mersenne.org/articles/10.5802/jep.81/

Bibliographie
Cité par

[AG15] D. Arinkin & D. Gaitsgory - “Singular support of coherent sheaves and the geometric Langlands conjecture”, Selecta Math. (N.S.) 21 (2015) no. 1, p. 1-199 | DOI | MR | Zbl

[AHW18] A. Asok, M. Hoyois & M. Wendt - “Affine representability results in $𝔸^{1}$ -homotopy theory: II: Principal bundles and homogeneous spaces”, Geom. Topol. 22 (2018) no. 2, p. 1181-1225 | DOI | MR | Zbl

[Ayo07a] J. Ayoub - Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque, vol. 314, Société Mathématique de France, Paris, 2007 | MR | Zbl

[Ayo07b] J. Ayoub - Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II, Astérisque, vol. 315, Société Mathématique de France, Paris, 2007 | MR | Zbl

[Ayo10] J. Ayoub - “Note sur les opérations de Grothendieck et la réalisation de Betti”, J. Inst. Math. Jussieu 9 (2010) no. 2, p. 225-263 | DOI | Zbl

[Ayo14] J. Ayoub - “La réalisation étale et les opérations de Grothendieck”, Ann. Sci. École Norm. Sup. (4) 47 (2014) no. 1, p. 1-145 | DOI | Zbl

[Bar15] C. Barwick - “On exact infinity-categories and the Theorem of the Heart”, Compositio Math. 151 (2015), p. 2160-2186, arXiv:1212.5232 | DOI | MR | Zbl

[BBD82] A. A. Beilinson, J. Bernstein & P. Deligne - “Faisceaux pervers”, in Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Astérisque, vol. 100, Société Mathématique de France, Paris, 1982, p. 5-171 | MR | Zbl

[BH17] T. Bachmann & M. Hoyois - “Norms in motivic homotopy theory” (2017), arXiv:1711.03061

[Bla13] A. Blanc - Invariants topologiques des espaces non commutatifs, Université Montpellier 2, 2013, PhD thesis

[Bla15] A. Blanc - “Topological K-theory of complex noncommutative spaces”, Compositio Math. (2015), p. 1-67 | MR

[Blo85] S. Bloch - “Cycles on arithmetic schemes and Euler characteristics of curves”, in Algebraic geometry (Bowdoin, 1985), Proc. Symp. Pure Math., vol. 46, part II, American Mathematical Society, Providence, 1985, p. 421-450 | MR

[BvdB03] A. Bondal & M. van den Bergh - “Generators and representability of functors in commutative and noncommutative geometry”, Moscow Math. J. 3 (2003) no. 1, p. 1-36, 258 | DOI | MR | Zbl

[BW12] J. Burke & M. E. Walker - “Matrix factorizations over projective schemes”, Homology Homotopy Appl. 14 (2012) no. 2, p. 37-61 | DOI | MR | Zbl

[BZFN10] D. Ben-Zvi, J. Francis & D. Nadler - “Integral transforms and Drinfeld centers in derived algebraic geometry”, J. Amer. Math. Soc. 23 (2010) no. 4, p. 909-966 | DOI | MR | Zbl

[CD12] D.-C. Cisinski & F. Déglise - “Triangulated categories of mixed motives” (2012), arXiv:0912.2110

[CD16] D.-C. Cisinski & F. Déglise - “Etale motives”, Compositio Math. 152 (2016), p. 556-666 | DOI | Zbl

[Cis13] D.-C. Cisinski - “Descente par éclatements en K-théorie invariante par homotopie”, Ann. of Math. (2) 177 (2013), p. 425-448 | DOI | Zbl

[Coh13] L. Cohn - “Differential graded categories are $k$ -linear stable infinity categories” (2013), arXiv:1308.2587

[CT11] D.-C. Cisinski & G. Tabuada - “Non-connective K-theory via universal invariants”, Compositio Math. 147 (2011) no. 4, p. 1281-1320 | DOI | MR | Zbl

[CT12] D.-C. Cisinski & G. Tabuada - “Symmetric monoidal structure on non-commutative motives”, J. K-Theory 9 (2012) no. 2, p. 201-268 | DOI | MR | Zbl

[CT13] A. Căldăraru & J. Tu - “Curved $A_{\infty}$ algebras and Landau-Ginzburg models”, New York J. Math. 19 (2013), p. 305-342 | MR | Zbl

[Del77] P. Deligne - Cohomologie étale, Lect. Notes in Math., vol. 569, Springer-Verlag, Berlin-New York, 1977, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 $_{1 / 2}$ , Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier

[Dri04] V. Drinfeld - “DG quotients of DG categories”, J. Algebra 272 (2004), p. 643-691 | DOI | MR | Zbl

[Dyc11] T. Dyckerhoff - “Compact generators in categories of matrix factorizations”, Duke Math. J. 159 (2011) no. 2, p. 223-274 | DOI | MR | Zbl

[Efi18] A. I. Efimov - “Cyclic homology of categories of matrix factorizations”, Internat. Math. Res. Notices (2018) no. 12, p. 3834-3869, arXiv:1212.2859 | DOI | MR

[Eke90] T. Ekedahl - “On the adic formalism”, in The Grothendieck Festschrift, Vol. II, Progress in Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, p. 197-218 | MR | Zbl

[EP15] A. I. Efimov & L. Positselski - “Coherent analogues of matrix factorizations and relative singularity categories”, Algebra Number Theory 9 (2015) no. 5, p. 1159-1292 | DOI | MR | Zbl

[Fuj02] K. Fujiwara - “A proof of the absolute purity conjecture (after Gabber)”, in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, p. 153-183 | MR | Zbl

[GR17] D. Gaitsgory & N. Rozenblyum - A study in derived algebraic geometry. Vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221, American Mathematical Society, Providence, RI, 2017 | MR | Zbl

[Gro65] A. Grothendieck - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II”, Publ. Math. Inst. Hautes Études Sci. 24 (1965), p. 5-231 | Zbl

[GS09] D. Gepner & V. Snaith - “On the motivic spectra representing algebraic cobordism and algebraic K-theory”, Documents Math. 14 (2009), p. 359-396 | MR | Zbl

[HLP14] D. Halpern-Leistner & A. Preygel - “Mapping stacks and categorical notions of properness” (2014), arXiv:1402.3204

[Hov99] M. Hovey - Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999 | MR | Zbl

[Ill94] L. Illusie - “Autour du théorème de monodromie locale”, in Périodes $p$ -adiques (Bures-sur-Yvette, 1988), vol. 223, Société Mathématique de France, Paris, 1994, p. 9-57

[ILO14] L. Illusie, Y. Laszlo & F. Orgogozo - “Introduction”, in Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque, vol. 363-364, Société Mathématique de France, Paris, 2014, p. xiii-xix | Zbl

[IS13] F. Ivorra & J. Sebag - “Nearby motives and motivic nearby cycles”, Selecta Math. (N.S.) 19 (2013) no. 4, p. 879-902 | DOI | MR | Zbl

[Kap91] M. Kapranov - “On DG-modules over the de Rham complex and the vanishing cycles functor”, in Algebraic geometry (Chicago, 1989), Lect. Notes in Math., vol. 1479, Springer, Berlin, 1991, p. 57-86 | DOI | MR | Zbl

[Kas83] M. Kashiwara - “Vanishing cycle sheaves and holonomic systems of differential equations”, in Algebraic geometry (Tokyo/Kyoto, 1982), Lect. Notes in Math., vol. 1016, Springer, Berlin, 1983, p. 134-142 | DOI | MR | Zbl

[Kha16a] A. A. Khan - “Brave new motivic homotopy theory I” (2016), arXiv:1610.06871

[Kha16b] A. A. Khan - Motivic homotopy theory in derived algebraic geometry, Universität Duisburg-Essen, 2016, PhD Thesis

[Kha18] A. A. Khan - “Virtual Cartier divisors and blow-ups” (2018), arXiv:1802.05702

[Kra08] H. Krause - “Localization theory for triangulated categories” (2008), arXiv:0806.1324

[KS05] K. Kato & T. Sato - “On the conductor formula of Bloch”, Publ. Math. Inst. Hautes Études Sci. 100 (2005), p. 5-151 | DOI

[KS11] M. Kontsevich & Y. Soibelman - “Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants”, Commun. Number Theory Phys. 5 (2011) no. 2, p. 231-352 | MR | Zbl

[LG14] J. Lurie & D. Gaitsgory - “Weil’s conjecture for function fields” (2014), tamagawa.pdf

[LP11] K. H. Lin & D. Pomerleano - “Global matrix factorizations” (2011), arXiv:1101.5847

[LS16] V. A. Lunts & O. Schnürer - “Motivic vanishing cycles as a motivic measure”, Pure Appl. Math. Q 12 (2016) no. 1, p. 33-74 | MR | Zbl

[Lur09] J. Lurie - Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009 | MR | Zbl

[Lur11a] J. Lurie - “Derived algebraic geometry VIII: Quasi-coherent sheaves and Tannaka duality theorems” (2011), DAG-VIII.pdf

[Lur11b] J. Lurie - “Derived algebraic geometry XIII: Rational and $p$ -adic homotopy theory” (2011), DAG-XIII.pdf

[Lur17] J. Lurie - “Higher algebra” (2017), higheralgebra.pdf

[Lur18] J. Lurie - “Spectral algebraic geometry” (2018), SAG-rootfile.pdf

[LZ12a] Y. Liu & W. Zheng - “Enhanced six operations and base change theorem for sheaves on Artin stacks” (2012), arXiv:1211.5948

[LZ12b] Y. Liu & W. Zheng - “Gluing restricted nerves of infinity-categories” (2012), arXiv:1211.5294

[MV99] F. Morel & V. Voevodsky - “ $A^{1}$ -homotopy theory of schemes”, Publ. Math. Inst. Hautes Études Sci. (1999) no. 90, p. 45-143 (2001) | DOI | MR

[NS17] T. Nikolaus & P. Scholze - “On topological cyclic homology” (2017), arXiv:1707.01799

[NSØ15] N. Naumann, M. Spitzweck & P. A. Østvær - “Existence and uniqueness of $E_{\infty}$ structures on motivic K-theory spectra”, J. Homotopy Relat. Struct. 10 (2015) no. 3, p. 333-346 | DOI | MR | Zbl

[Orl04] D. Orlov - “Triangulated categories of singularities and D-branes in Landau-Ginzburg models”, in Algebr. Geom. Metody, Svyazi i Prilozh., Trudy Mat. Inst. Steklov., vol. 246, 2004, p. 240-262 | Zbl

[Orl12] D. Orlov - “Matrix factorizations for nonaffine LG-models”, Math. Ann. 353 (2012) no. 1, p. 95-108 | MR | Zbl

[Pre11] A. Preygel - “Thom-Sebastiani and duality for matrix factorizations” (2011), arXiv:1101.5834

[Qui73] D. Quillen - “Higher algebraic K-theory. I”, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lect. Notes in Math., vol. 341, Springer, Berlin, 1973, p. 85-147 | MR | Zbl

[Rio10] J. Riou - “Algebraic K-theory, $A^{1}$ -homotopy and Riemann-Roch theorems”, J. Topology 3 (2010) no. 2, p. 229-264 | DOI | MR | Zbl

[Rob14] M. Robalo - Motivic homotopy theory of non-commutative spaces, Université Montpellier 2, 2014, PhD thesis

[Rob15] M. Robalo - “K-theory and the bridge from motives to noncommutative motives”, Adv. Math. 269 (2015), p. 399-550 | DOI | MR | Zbl

[RZ82] M. Rapoport & T. Zink - “Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik”, Invent. Math. 68 (1982) no. 1, p. 21-101 | Zbl

[Sab10] C. Sabbah - “On a twisted de Rham complex, II” (2010), arXiv:1012.3818

[Seg13] E. Segal - “The closed state space of affine Landau-Ginzburg B-models”, J. Noncommut. Geom. 7 (2013) no. 3, p. 857-883 | DOI | MR | Zbl

[Ser62] J.-P. Serre - Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., vol. 1296, Hermann, Paris, 1962 | Zbl

[Ser65] J.-P. Serre - Algèbre locale. Multiplicités, Lect. Notes in Math., vol. 11, Springer-Verlag, Berlin-New York, 1965 | Zbl

[SGA6] - Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lect. Notes in Math., vol. 225, Springer-Verlag, Berlin-New York, 1971, dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J.-P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J.-P. Serre

[SGA7 I] - Groupes de monodromie en géométrie algébrique. I, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lect. Notes in Math., vol. 288, Springer-Verlag, Berlin, 1972, dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim | Zbl

[SGA7 II] - Groupes de monodromie en géométrie algébrique. II, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lect. Notes in Math., vol. 340, Springer-Verlag, Berlin-New York, 1973, dirigé par P. Deligne et N. Katz | Zbl

[Shk14] D. Shklyarov - “Non-commutative Hodge structures: towards matching categorical and geometric examples”, Trans. Amer. Math. Soc. 366 (2014) no. 6, p. 2923-2974 | DOI | MR | Zbl

[SS14] C. Sabbah & M. Saito - “Kontsevich’s conjecture on an algebraic formula for vanishing cycles of local systems”, Algebraic Geom. 1 (2014) no. 1, p. 107-130 | MR | Zbl

[Tab05] G. Tabuada - “Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories”, Comptes Rendus Mathématique 340 (2005) no. 1, p. 15-19 | DOI | Zbl

[Tab08] G. Tabuada - “Higher K-theory via universal invariants”, Duke Math. J. 145 (2008) no. 1, p. 121-206 | DOI | MR | Zbl

[Toë07] B. Toën - “The homotopy theory of dg-categories and derived Morita theory”, Invent. Math. 167 (2007) no. 3, p. 615-667 | DOI | MR | Zbl

[Toë11] B. Toën - “Lectures on dg-categories”, in Topics in algebraic and topological K-theory, Lect. Notes in Math., vol. 2008, Springer, Berlin, 2011, p. 243-302 | MR | Zbl

[Toë12a] B. Toën - “Derived Azumaya algebras and generators for twisted derived categories”, Invent. Math. 189 (2012) no. 3, p. 581-652 | MR | Zbl

[Toë12b] B. Toën - “Proper local complete intersection morphisms preserve perfect complexes” (2012), arXiv:1210.2827

[Toë14] B. Toën - “Derived algebraic geometry”, EMS Surv. Math. Sci. 1 (2014) no. 2, p. 153-240 | DOI | MR | Zbl

[TT90] R. W. Thomason & T. Trobaugh - “Higher algebraic K-theory of schemes and of derived categories”, in The Grothendieck Festschrift, Vol. III, Progress in Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, p. 247-435 | DOI | MR | Zbl

[TV07] B. Toën & M. Vaquié - “Moduli of objects in dg-categories”, Ann. Sci. École Norm. Sup. (4) 40 (2007) no. 3, p. 387-444 | DOI | Numdam | MR | Zbl

[TV16] B. Toën & G. Vezzosi - “The $ℓ$ -adic trace formula for dg-categories and Bloch’s conductor conjecture” (2016), Research announcement, arXiv:1605.08941

[TV17] B. Toën & G. Vezzosi - “A non-commutative trace formula and Bloch’s conductor conjecture”, in preparation, 2017 | Zbl

[Voe96] V. Voevodsky - “Homology of schemes”, Selecta Math. (N.S.) 2 (1996) no. 1, p. 111-153 | DOI | MR | Zbl

[Wei89] C. A. Weibel - “Homotopy algebraic K-theory”, in Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, American Mathematical Society, Providence, RI, 1989, p. 461-488 | DOI | MR | Zbl

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