K-theory Preprint Archives

A comparison of motivic and classical stable homotopy theories

marc.levine@uni-due.de (Marc Levine) — Sun, 01 Jan 2012 12:07:03 +0000

1022: Let k be an algebraically closed field of characteristic zero. Let SH(k) be the motivic stable homotopy category of T-spectra over k, SH the classical stable homotopy category and let c:SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism π_n(E) → π_n,0(c(E)) for all spectra E.

Fix an embedding of k into the complex numbers and let Re:SH(k) → SH be the associated Betti realization. Let S_k be the motivic sphere spectrum. We show that the Tate-Postnikov tower for S_k has Betti realization which is strongly convergent. This gives a spectral sequence of algebro-geometric origin converging to the homotopy groups of the classical sphere spectrum; this spectral sequence at E₂ agrees with the E₂ terms in the Adams-Novikov spectral sequence.

Convergence of Voevodsky's slice tower

marc.levine@uni-due.de (Marc Levine) — Sun, 01 Jan 2012 12:05:53 +0000

1021: We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category SH(k) over a perfect field k. In case k has finite cohomological dimension (in characteristic two, we also require that k is infinite), we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves π_a,bf_nE of the nth term in the slice tower is finite, exhaustive and separated at each stalk. This partially verifies a conjecture of Voevodsky.

Morita homotopy theory of C*-categories

Fri, 23 Dec 2011 17:02:18 +0000

1020: In this article we establish the foundations of the Morita homotopy theory of C*-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure M_Mor on the category C*cat1 of small unital C*-categories. The weak equivalences are the Morita equivalences and the cofibrations are the *-functors which are injective on objects. As an application, we obtain an elegant description of the Brown-Green-Rieffel Picard group in the associated Morita homotopy category Ho(M_Mor). We then prove that the Morita homotopy category is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(M_Mor)^{-1} and a canonical functor C*cat1 --> Ho(M_Mor)^{-1} which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in Ho(M_Mor)^{-1} by the tensor unit object.

Unconditional motivic Galois groups and Voevodsky's nilpotence conjecture in the noncommutative world

matilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada) — Thu, 22 Dec 2011 21:01:10 +0000

1019: In this article we further the study of noncommutative pure motives. We construct unconditional noncommutative motivic Galois groups and relate them to the unconditional motivic Galois groups developed originally by Andre-Kahn. Then, we introduce the correct noncommutative analogue of Voevodsky's nilpotence conjecture and explore its interaction with the finite dimensionality of noncommutative Chow motives as well as with Voevodsky's original conjecture.

Quillen's work in algebraic K-theory

drg@illinois.edu (Daniel R. Grayson) — Fri, 02 Dec 2011 00:07:52 +0000

1018: This paper is dedicated to the memory of Daniel Quillen. In it, we examine his brilliant disovery of higher algebraic K-theory, including its roots in and genesis from topological K-theory and ideas connected with the proof of the Adams conjecture, and his development of the field into a complete theory in just a few short years. We provide a few references to further developments, including motivic cohomology.

Weight structure on noncommutative motives

tabuada@math.mit.edu (Goncalo Tabuada) — Wed, 30 Nov 2011 14:41:38 +0000

1017: In this note we endow Kontsevich's category KMM of noncommutative mixed motives with a non-degenerate weight structure in the sense of Bondarko. As an application we obtain a convergent weight spectral sequence for every additive invariant (e.g., algebraic K-theory, cyclic homology, topological Hochschild homology, etc.), and a ring isomorphism between the Grothendieck ring of KMM and the Grothendieck ring of the category of noncommutative Chow motives.

Algebraic analogue of Atiyah's theorem

alknizel@gmail.com (Alisa Knizel), neshitov@yandex.ru (Alexander Neshitov) — Sun, 20 Nov 2011 14:48:12 +0000

1016: In this paper we prove an algebraic analogue of Atiyah's theorem, concerning K-theory of classifying space of algebraic group. Also we extend this result for higher K-groups.

Remark on rigidity over several fields

yagunov@gmail.com (Serge Yagunov) — Sun, 06 Nov 2011 15:52:39 +0000

1015: It is shown that T-spectrum representable cohomology theories on smooth algebraic varieties satisfy normalization condition over nonreal fields. As a consequence, one can see that the rigidity property holds for all representable theories over considered fields.

The article is published and available online at Homology, Homotopy and Applications 13 (2011) 159-164.

Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motives

Wed, 02 Nov 2011 14:25:08 +0000

1014: V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this note, making use of the theory of noncommutative motives, we show how these beautiful theorems can be understood as instantiations of more general results.

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

matilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada) — Tue, 11 Oct 2011 19:50:33 +0000

1013: In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our main previous results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_NC and D_NC of Grothendieck's standard conjectures C and D. Assuming C_NC, we prove that NNum(k)_F can be made into a Tannakian category NNum'(k)_F by modifying its symmetry isomorphism constraints. By further assuming D_NC, we neutralize the Tannakian category NNum'(k)_F using HP. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic (super-)Galois groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit homomorphisms relating these new noncommutative motivic (super-)Galois groups with the classical ones.

Motives of Deligne-Mumford Stacks

utsav.choudhury@math.uzh.ch (Utsav Choudhury) — Wed, 28 Sep 2011 12:25:13 +0000

1012: For every smooth and separated Deligne-Mumford stack F, we associate a motive M(F) in Voevodsky's category of mixed motives with rational coefficients. When F is proper over a field of characteristic 0, we compare M(F) with the Chow motive associated to F by Toen. Without the properness condition we show that M(F) is a direct summand of the motive of a smooth quasi-projective variety.

Non-connective K-theory of exact categories with weak equivalences

mochi81@hotmail.com (Satoshi Mochizuki) — Tue, 13 Sep 2011 15:51:12 +0000

1011: The main objective of this paper is to extend a domain variables of non-connective K-theory to a wide class of exact categories with weak equivalences which need not satisfy the factorization axiom in general and develop fundamental properties of non-connective K-theory. The main application is to study the topological filtrations of non-connective K-theory of a noetherian commutative ring with unit in terms of Koszul cubes.

Une version relative de la conjecture des périodes de Kontsevich-Zagier

joseph.ayoub@math.uzh.ch (Joseph Ayoub) — Sun, 04 Sep 2011 12:51:22 +0000

1010: We start with a Laurent series F whose coefficients are given by holomorphic functions on an open neighborhood of the closed polydisc of radius 1 and dimension n. We assume furthermore that F is algebraic in an appropriate sense. Integrating the coefficients on the real unit cube of dimension n yields a Laurent series with complex coefficients. We are interested in knowing when the resulting series is zero. Our main result is reminiscent to the Kontsevich-Zagier conjecture on periods in a modified form.

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in Hermitian K-theory

Thu, 25 Aug 2011 12:15:34 +0000

1009: Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields.

We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field.

We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory.

Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.

Kontsevich's category of noncommutative numerical motives.

matilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada) — Thu, 18 Aug 2011 17:41:38 +0000

1008: In this note we prove that Kontsevich's category NCnum of noncommutative numerical motives is equivalent to the one constructed by the authors. As a consequence, we conclude that NCnum is abelian semi-simple as conjectured by Kontsevich.

A guided tour through the garden of noncommutative motives

tabuada@math.mit.edu (Goncalo Tabuada) — Thu, 18 Aug 2011 17:41:38 +0000

1007: These are the extended notes of a survey talk on noncommutative motives given at the 3era Escuela de Inverno Luis Santalo - CIMPA Research School: Topics in Noncommutative Geometry, Buenos Aires, July 26 to August 6, 2010.

The K-theory associated to a finite field I

none (Daniel Quillen) — Tue, 16 Aug 2011 15:09:56 +0000

1006: This preprint of Quillen's was a preliminary unpublished version of the work that was eventually published as "On the cohomology and K-theory of the general linear groups over a finite field", Ann. of Math. 96 (1972) 552-586. The main result concerns the homology of general linear groups over a finite field with finite coefficients prime to the characteristic, with an eye toward the computation of the K-groups of finite fields, to be accomplished in later papers. The direct sum representation ring appears side by side with the exact sequence representation ring in some of the arguments.

An alternative proof of a lemma is written in Kervaire's hand on the last page.

[ Thanks to Bruno Kahn for scanning the preprint; posted by Dan Grayson. The djvu file comes with OCR text. ]

Somekawa's K-groups and Voevodsky's Hom groups

kahn@math.jussieu.fr (Bruno Kahn), ytakao@math.tohoku.ac.jp (Takao Yamazaki) — Fri, 12 Aug 2011 17:02:54 +0000

1005: We construct an isomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic complexes.

Around Quillen's theorem A

kahn@math.jussieu.fr (Bruno Kahn) — Wed, 10 Aug 2011 15:14:49 +0000

1004: We reformulate some exact sequences of Quillen into a spectral sequence converging to the homology of certain K-theory spaces.

Letter to Segal, July 25, 1972

none (Daniel Quillen) — Mon, 08 Aug 2011 16:31:22 +0000

1003: This letter is about: (1) the image of J, the stable homotopy groups of spheres, and the K-groups of the integers; (2) the Q-construction and the S-construction, which give definitions of higher algebraic K-theory.

[Posted by Dan Grayson .]