In this paper 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. Contents arXiv:1607.03012v4 [math.AG] 6 Sep 2018 2 ANTHONY BLANC, MARCO ROBALO, BERTRAND TÖEN, AND GABRIELE VEZZOSI 4.3. The action of the punctured disk η 77 4.4. -adic inertia-invariant vanishing cycles 87 4.5. Comparison between Vanishing Cycles and the singularity category 93 Appendix A. The formalism of six Operations in the Motivic Setting 99 References 104representing homotopy algebraic K-theory, i.e. to the commutative motive identified by the fact that BU S (Y ) is the spectrum of non-connective homotopy invariant algebraic K-theory of Y , for any smooth S-scheme Y . As a consequence, BU S is endowed with the structure of a commutative algebra in the symmetric monoidal ∞-category SH S . Therefore, M S actually factors, as a lax monoidal functor, M S : SHnc S → Mod BU S (SH S ) via the category of BU S -modules in SH S .The first main idea in this paper (see Section 3.2) is to modify the functor M S in order to obtain different informations, better suited to our goal. Instead of M S , we consider a somewhat dual versionIn particular, for p : X → S, with X quasi-compact and quasi-separated, we get (Proposition 3.13) an equivalence M ∨ S (Perf(X)) p * (BU X ) in Mod BU S (SH S ), where BU X denotes a relative version homotopy invariant algebraic K-theory, and (Proposition 3.30) an equivalence M ∨ S (Sing(S, 0 S )) BU S ⊕ BU S [1] in Mod BU S (SH S ). As consequence of this, the motive M ∨ S (Sing(X, f )) is a module over BU S ⊕ BU S [1], for any LG-pair (X, f ) over S.The second main idea in this paper (see Section 3.7) is to compose the functor M ∨ S : SHnc S → Mod BU S (SH S ) with the -adic realization functor R S : SH S → Sh Q (S) with values in the ∞-categorical version of Ind-constructible -adic sheaves on S with Q -coefficients of [Eke90].Building on results of Cisinski-Deglise and Riou, we prove (see Section 4.2) that one can refine R S to a functor, still denoted by the same symbol,where β denotes the algebraic Bott element. We then denote by R S the compositeWe are now in a position to state our main theorem comparing singularity categories and vanishing cycles (see Section 4). Let us take S = Spec A to be a henselian trait with Remark 1.1. The result of Theorem 4.39 is stated for the -adic reali...

The purpose of this work is to give a definition of a topological K-theory for dg-categories over C and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cylic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps : taking the topological realization of algebraic K-theory, and inverting the Bott element. The topological realization is the left Kan extension of the functor "space of complex points" to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum BU. For this we are led to prove a homotopical generalization of Deligne's cohomological proper descent, using Lurie's proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel's K\"unneth formula for periodic cyclic homology and once again the proper descent result. In the case of a dg-category of perfect complexes on a smooth scheme, we show that we recover the usual topological K-theory. Finally in the case of a finite dimensional associative algebra, we show that the lattice conjecture holds. This gives a formula for the periodic homology groups of a finite dimensional algebra in terms of the stack of projective modules of finite type.Comment: 72 pages. To appear in Compositio Mathematica. Corrected the assumptions in the comparison isomorphism for schemes (now stated for singular schemes, proposition 4.32) as well as some technical assumptions in proposition 3.2

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a k-linear ∞-category for a field k. Our main result states that if C is a k-linear ∞-category which has a compact generator whose groups of self extensions vanish for sufficiently high positive degrees, then every formal deformation of C has zero curvature and moreover admits a compact generator.

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