×

Trace and Künneth formulas for singularity categories and applications. (English) Zbl 07541902

Summary: We present an \(\ell\)-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch’s conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
18G90 Other (co)homology theories (category-theoretic aspects)
19L10 Riemann-Roch theorems, Chern characters
11S15 Ramification and extension theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beilinson, A. A. and Bernstein, J., A proof of Jantzen conjectures, in I. M. Gel’fand seminar, , vol. 16, eds Gel’fand, S., Gindikin, S. (American Mathematical Society, Providence, RI, 1993), 1-50. · Zbl 0790.22007
[2] Blanc, A., Topological K-theory of complex noncommutative spaces, Compos. Math. 152 (2016), 489-555. · Zbl 1343.14003
[3] Blanc, A., Robalo, M., Toën, B. and Vezzosi, G., Motivic realizations of singularity categories and vanishing cycles, J. Éc. Polytech. 5 (2018), 651-747. · Zbl 1423.14151
[4] Bloch, S., Cycles on arithmetic schemes and Euler characteristics of curves, in Algebraic geometry-Bowdoin 1985, Part 2, , vol. 46 (American Mathematical Society, Providence, RI, 1987), 421-450. · Zbl 0654.14004
[5] Gaitsgory, D. and Lurie, J., Weil’s conjecture over function fields, Preprint, http://www.math.harvard.edu/ lurie/papers/tamagawa.pdf. · Zbl 1439.14006
[6] Haugseng, R., The higher Morita category of \(E_n\)-algebras, Geom. Topol. 21 (2017), 1631-1730. · Zbl 1395.18011
[7] Hovey, M., Model categories, , vol. 63 (American Mathematical Society, Providence, RI, 1999). · Zbl 0909.55001
[8] Illusie, L., Autour du théorème de monodromie locale, in Périodes p-adiques, Séminaire de Bures, 1988, , vol. 223 (Société Mathématique de France, 1994), 9-57. · Zbl 0837.14013
[9] Illusie, L., Around the Thom-Sebastiani theorem, with an appendix by Weizhe Zheng, Manuscripta Math. 152 (2017), 61-125. · Zbl 1395.14010
[10] Illusie, L., Laszlo, Y. and Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie et́ale des schémas quasi-excellents, , vol. 363-364 (Société Mathématique de France, 2014). · Zbl 1297.14003
[11] Kapranov, M., On DG-modules over the de Rham complex and the vanishing cycles functor, in Algebraic geometry (Chicago, IL, 1989), , vol. 1479 (Springer, Berlin, 1991), 57-86. · Zbl 0757.14010
[12] Kato, K. and Saito, T., On the conductor formula of Bloch, Publ. Math. Inst. Hautes Études Sci. 100 (2005), 5-151. · Zbl 1099.14009
[13] Lurie, J., Higher Algebra, Preprint (2016), http://www.math.harvard.edu/ lurie/papers/HA.pdf.
[14] Lurie, J., On the classification of topological field theories, , vol. 2008 (International Press, Somerville, MA, 2009), 129-280; MR2555928 (2010k:57064). · Zbl 1180.81122
[15] Lurie, J., Higher topos theory, , vol. 170 (Princeton University Press, Princeton, 2009). · Zbl 1175.18001
[16] Neukirch, J., Algebraic number theory, , vol. 322 (Springer, Berlin, 1999). · Zbl 0956.11021
[17] Orgogozo, F., Conjecture de Bloch et nombres de Milnor, Ann. Inst. Fourier (Grenoble)53 (2003), 1739-1754. · Zbl 1065.14005
[18] Preygel, A., Thom-Sebastiani & duality for matrix factorizations, Thesis, Preprint (2011), arXiv:1101.5834.
[19] Robalo, M., K-theory and the bridge from motives to non-commutative motives, Adv. Math. 269 (2015), 399-550. · Zbl 1315.14030
[20] Saito, T., Characteristic cycles and the conductor of direct image, Preprint (2017), arXiv:1704.04832.
[21] Serre, J.-P., Linear representations of finite groups, , vol. 42 (Springer, Berlin, 1977). · Zbl 0355.20006
[22] Artin, M., Grothendieck, A. and Verdier, J.-L. (eds), Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3, , vol. 305 (Springer, Berlin, New York, 1972). · Zbl 0234.00007
[23] Grothendieck, A., Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1, , vol. 288 (Springer, Berlin, New York, 1972). · Zbl 0237.00013
[24] Deligne, P. and Katz, N. (eds), Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 2, , vol. 340 (Springer, Berlin, New York, 1973).
[25] Toën, B., DG-categories and derived Morita theory, Invent. Math. 167 (2007), 615-667. · Zbl 1118.18010
[26] Toën, B., Lectures on dg-categories, in Topics in algebraic and topological K-theory, , vol. 2008 (Springer, Berlin, 2011), 243-302. · Zbl 1216.18013
[27] Toën, B., Derived Azumaya algebras and generators for twisted derived categories, Invent. Math. 189 (2012), 581-652. · Zbl 1275.14017
[28] Toën, B. and Vaquié, M., Moduli of objects in dg-categories, Ann. Sci. Éc. Norm. Supér. (4)40 (2007), 387-444. · Zbl 1140.18005
[29] Toën, B. and Vezzosi, G., Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Selecta Math. (N.S.)21 (2015), 449-554. · Zbl 1333.19006
[30] Toën, B. and Vezzosi, G., Géométrie non-commutative, formule des traces et conducteur de Bloch, in Actes du 1er congrès national de la SMF, , vol. 31, ed. Lecouvrey, C. (Société Mathématique de France, 2018), 77-107. · Zbl 1404.14023
[31] Voevodsky, V., \( \mathbb{A}^1 \)-homotopy theory, in Proceedings of the International Congress of Mathematicians, Documenta Mathematica, Extra Volume ICM 1998, vol. I (Deutsche Mathematiker-Vereinigung, Berlin, 1998), 579-604. · Zbl 0907.19002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.