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 -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 -adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and -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.
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DOI : 10.5802/jep.81
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
@article{JEP_2018__5__651_0, author = {Blanc, Anthony and Robalo, Marco and To\"en, Bertrand and Vezzosi, Gabriele}, title = {Motivic realizations of singularity~categories and vanishing~cycles}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {651--747}, publisher = {Ecole polytechnique}, volume = {5}, year = {2018}, doi = {10.5802/jep.81}, mrnumber = {3877165}, zbl = {06988591}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jep.81/} }
TY - JOUR AU - Blanc, Anthony AU - Robalo, Marco AU - Toën, Bertrand AU - Vezzosi, Gabriele TI - Motivic realizations of singularity categories and vanishing cycles JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 651 EP - 747 VL - 5 PB - Ecole polytechnique UR - http://archive.numdam.org/articles/10.5802/jep.81/ DO - 10.5802/jep.81 LA - en ID - JEP_2018__5__651_0 ER -
%0 Journal Article %A Blanc, Anthony %A Robalo, Marco %A Toën, Bertrand %A Vezzosi, Gabriele %T Motivic realizations of singularity categories and vanishing cycles %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 651-747 %V 5 %I Ecole polytechnique %U http://archive.numdam.org/articles/10.5802/jep.81/ %R 10.5802/jep.81 %G en %F JEP_2018__5__651_0
Blanc, Anthony; Robalo, Marco; Toën, Bertrand; Vezzosi, Gabriele. 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. http://archive.numdam.org/articles/10.5802/jep.81/
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