K-theory Preprint Archives
http://www.math.uiuc.edu/K-theory/
en-us15http://blogs.law.harvard.edu/tech/rssK-theory preprintsEnd of the Journal of K-theorybalmer@math.ucla.edub (Board Members of the Journal of K-theory)Thu, 06 Nov 2014 09:52:34 +00001025:
<p>
Editors of the Journal of K-Theory have provided a public statement about the situation in the journal, available at
<a href="JKT_final_announcement.pdf">JKT_final_announcement.pdf</a>.
</p>
<p>
PS:
The 2007 September 5 public statement, announcing the launch of JKT, can still be found at:
<a href="http://permalink.gmane.org/gmane.science.mathematics.categories/3885">http://permalink.gmane.org/gmane.science.mathematics.categories/3885</a>.
It complements the statement on page 1 of volume 1 of JKT:
<a href="http://dx.doi.org/10.1017/is008004016jkt061">http://dx.doi.org/10.1017/is008004016jkt061</a>.
</p>
http://www.math.uiuc.edu/K-theory/1025/
http://www.math.uiuc.edu/K-theory/1025/Principal bundles of reductive groups over affine schemespaniniv@gmail.com (Ivan Panin), anastasia.stavrova@gmail.com (Anastasia Stavrova)Sun, 08 Apr 2012 13:45:06 +00001024:
Let R be a semi-local regular domain containing an infinite perfect field k,
and let K be the field of fractions of R. Let G be a reductive semi-simple
simply connected R-group scheme such that each of its R-indecomposable factors
is isotropic. We prove that for any Noetherian affine scheme A over k, the
kernel of the map of etale cohomology sets H^1(A\times_k R,G) ->
H^1(A\times_k K,G), induced by the inclusion of R into K, is trivial. If R is
the semi-local ring of several points on a k-smooth scheme, then it suffices to
require that k is infinite and keep the same assumption concerning G. The
results extend the Serre--Grothendieck conjecture for such R and G, proved in
K-theory preprint 929.
http://www.math.uiuc.edu/K-theory/1024/
http://www.math.uiuc.edu/K-theory/1024/From exceptional collections to motivic decompositionsmatilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada)Fri, 09 Mar 2012 19:05:00 +00001023:
In this article we prove that the Chow motive of every smooth and proper
Deligne-Mumford stack, whose bounded derived category of coherent schemes
admits a full exceptional collection, decomposes into a direct sum of
tensor powers of the Lefschetz motive. Examples include projective spaces,
quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli
spaces. As an application we obtain explicit obstructions for the existence
of full exceptional collections and a simplification of Dubrovin's
conjecture.
http://www.math.uiuc.edu/K-theory/1023/
http://www.math.uiuc.edu/K-theory/1023/A comparison of motivic and classical stable homotopy theoriesmarc.levine@uni-due.de (Marc Levine)Sun, 01 Jan 2012 12:07:03 +00001022:
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 <font><b>Sm</b></font>/k. We show
that c is fully faithful. In particular, c induces an isomorphism
π<sub>n</sub>(E) → π<sub>n,0</sub>(c(E)) for all spectra E.
</p>
<p>
Fix an embedding of k into the complex numbers and let Re:SH(k) → SH be
the associated Betti realization. Let S<sub>k</sub> be the motivic sphere
spectrum. We show that the Tate-Postnikov tower for S<sub>k</sub> 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 E<sub>2</sub> agrees with the
E<sub>2</sub> terms in the Adams-Novikov spectral sequence.
http://www.math.uiuc.edu/K-theory/1022/
http://www.math.uiuc.edu/K-theory/1022/Convergence of Voevodsky's slice towermarc.levine@uni-due.de (Marc Levine)Sun, 01 Jan 2012 12:05:53 +00001021:
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<sub>a,b</sub>f<sub>n</sub>E
of the nth term in the slice tower is finite, exhaustive and separated at each
stalk. This partially verifies a conjecture of Voevodsky.
http://www.math.uiuc.edu/K-theory/1021/
http://www.math.uiuc.edu/K-theory/1021/Morita homotopy theory of C*-categoriesambrogio@math.uni-bielefeld.de (Ivo Dell'Ambrogio), tabuada@math.mit.edu (Goncalo Tabuada)Fri, 23 Dec 2011 17:02:18 +00001020:
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.
http://www.math.uiuc.edu/K-theory/1020/
http://www.math.uiuc.edu/K-theory/1020/Unconditional motivic Galois groups and Voevodsky's nilpotence conjecture in the noncommutative worldmatilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada)Thu, 22 Dec 2011 21:01:10 +00001019:
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.
http://www.math.uiuc.edu/K-theory/1019/
http://www.math.uiuc.edu/K-theory/1019/Quillen's work in algebraic K-theorydrg@illinois.edu (Daniel R. Grayson)Thu, 15 Mar 2012 19:07:31 +00001018:
[ The previous version, dated Dec 2, 2011, has been replaced, Mar 15, 2012. ]
<p>
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.
http://www.math.uiuc.edu/K-theory/1018/
http://www.math.uiuc.edu/K-theory/1018/Weight structure on noncommutative motivestabuada@math.mit.edu (Goncalo Tabuada)Wed, 30 Nov 2011 14:41:38 +00001017:
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.
http://www.math.uiuc.edu/K-theory/1017/
http://www.math.uiuc.edu/K-theory/1017/Algebraic analogue of Atiyah's theoremalknizel@gmail.com (Alisa Knizel), neshitov@yandex.ru (Alexander Neshitov)Sun, 20 Nov 2011 14:48:12 +00001016:
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.
http://www.math.uiuc.edu/K-theory/1016/
http://www.math.uiuc.edu/K-theory/1016/Remark on rigidity over several fieldsyagunov@gmail.com (Serge Yagunov)Sun, 06 Nov 2011 15:52:39 +00001015:
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.
<p>
The article is published and available online at <a
href="http://intlpress.com/HHA/v13/n2/a10/">Homology, Homotopy and Applications
13 (2011) 159-164</a>.
http://www.math.uiuc.edu/K-theory/1015/
http://www.math.uiuc.edu/K-theory/1015/Lefschetz and Hirzebruch-Riemann-Roch formulas via noncommutative motivesdenis-charles.cisinski@math.univ-toulouse.fr (Denis-Charles Cisinski), tabuada@math.mit.edu (Goncalo Tabuada)Wed, 02 Nov 2011 14:25:08 +00001014:
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.
http://www.math.uiuc.edu/K-theory/1014/
http://www.math.uiuc.edu/K-theory/1014/Noncommutative numerical motives, Tannakian structures, and motivic Galois groupsmatilde@caltech.edu (Matilde Marcolli), tabuada@math.mit.edu (Goncalo Tabuada)Tue, 11 Oct 2011 19:50:33 +00001013:
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.
http://www.math.uiuc.edu/K-theory/1013/
http://www.math.uiuc.edu/K-theory/1013/Motives of Deligne-Mumford Stacksutsav.choudhury@math.uzh.ch (Utsav Choudhury)Wed, 28 Sep 2011 12:25:13 +00001012:
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.
http://www.math.uiuc.edu/K-theory/1012/
http://www.math.uiuc.edu/K-theory/1012/Non-connective K-theory of exact categories with weak equivalencesmochi81@hotmail.com (Satoshi Mochizuki)Tue, 13 Sep 2011 15:51:12 +00001011:
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.
http://www.math.uiuc.edu/K-theory/1011/
http://www.math.uiuc.edu/K-theory/1011/Une version relative de la conjecture des périodes de Kontsevich-Zagierjoseph.ayoub@math.uzh.ch (Joseph Ayoub)Sun, 04 Sep 2011 12:51:22 +00001010:
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.
http://www.math.uiuc.edu/K-theory/1010/
http://www.math.uiuc.edu/K-theory/1010/The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in Hermitian K-theoryberrick@math.nus.edu.sg (A. J. Berrick), max.karoubi@gmail.com (M. Karoubi), m.schlichting@warwick.ac.uk (M. Schlichting), paularne@math.uio.no (P. A. Ostvaer)Thu, 25 Aug 2011 12:15:34 +00001009:
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.
<p>
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.
<p>
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.
<p>
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.
http://www.math.uiuc.edu/K-theory/1009/
http://www.math.uiuc.edu/K-theory/1009/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 +00001008:
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.
http://www.math.uiuc.edu/K-theory/1008/
http://www.math.uiuc.edu/K-theory/1008/A guided tour through the garden of noncommutative motivestabuada@math.mit.edu (Goncalo Tabuada)Thu, 18 Aug 2011 17:41:38 +00001007:
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.
http://www.math.uiuc.edu/K-theory/1007/
http://www.math.uiuc.edu/K-theory/1007/The K-theory associated to a finite field Inone (Daniel Quillen)Tue, 16 Aug 2011 15:09:56 +00001006:
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.
<p>
An alternative proof of a lemma is written in Kervaire's hand on the last page.
<p>
[ Thanks to Bruno Kahn for scanning the preprint; posted by Dan Grayson. The djvu
file comes with OCR text. ]
http://www.math.uiuc.edu/K-theory/1006/
http://www.math.uiuc.edu/K-theory/1006/