K-theory Preprint Archives

Une version du théorème d'Amer et Brumer pour les zéro-cycles

Tue, 24 Nov 2009 15:27:20 +0000

948: M. Amer and A. Brumer have shown that, for two homogeneous quadratic polynomials f and g in at least 3 variables over a field k of characteristic different from 2, the locus f=g=0 has non-trivial solution over k if and only if, for a variable t, the equation f+tg=0 has a non-trivial solution over k(t). We consider a modified version of this result, and show that the projective variety over k defined by f_0=...=f_r=0, where the f_i are homogeneous polynomials over k of the same degree d ≥ 2 in n+1 variables (with n+1 ≥ r+2), has a 0-cycle of degree 1 over k if and only if the generic hypersurface f_0+t_1f_1+...+t_rf_r=0 has a 0-cycle of degree 1 over k(t_1,...,t_r).

K-Theory of Azumaya Algebras over Schemes

r.hazrat@qub.ac.uk (R. Hazrat), rthccc@gmail.com (R. T. Hoobler) — Sat, 07 Nov 2009 19:26:22 +0000

947: Let X be a connected, noetherian scheme and A be a sheaf of Azumaya algebras on X which is a locally free O_X-module of rank a. We show that the kernel and cokernel of K_i(X) ----> K_i(A) are torsion groups with exponent a^m for some m and any i \geq 0, when X is regular or X is of dimension d with an ample sheaf (in this case m \leq d+1). As a consequence, K_i(X,Z/m) is isomorphic to K_i(A,Z/m), for any m relatively prime to a.

Étale duality for constructible sheaves on arithmetic schemes

Tue, 20 Oct 2009 12:25:15 +0000

946: In this note we relate 3 topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Bloch-Ogus-Kato complex.

On the first Steenrod square for Chow groups

olivier.haution(at)gmail.com (Olivier Haution) — Thu, 15 Oct 2009 13:44:27 +0000

945: We construct a weak version of the homological first Steenrod square, a natural transformation from the modulo two Chow group to the Chow group modulo two and two-torsion. No assumption is made on the characteristic of the base field. As an application, we generalize a theorem of Nikita Karpenko on the parity of the first Witt index of quadratic forms to the case of a base field of characteristic two.

Cohomological Obstruction Theory for Brauer Classes and the Period-Index Problem

antieau@math.uic.edu (Benjamin Antieau) — Fri, 11 Sep 2009 22:47:40 +0000

944: Let U be a noetherian, quasi-compact, and connected scheme. Let [a] be a class in Br(U). For each positive integer m, we use the K-theory of [a]-twisted sheaves to identify obstructions to [a] being representable by an Azumaya algebra of rank m^2. We define the spectral index of [a], denoted spi([a]), to be the least positive integer such that all of the associated obstructions vanish. Let per([a]) be the order of [a] in Br(U). We give an upper bound on the spectral index that depends on the etale cohomological dimension of U, the exponents of the stable homotopy groups of spheres, and the exponents of the stable homotopy groups of B(\mu_{per([a])}). As a corollary, we prove that when U is the spectrum of a field of finite cohomological dimension d=2c or d=2c+1, then spi([a])|per([a])^c whenever per([a]) is not divided by any primes that are small relative to d.

Computing Borel’s Regulator II

Tue, 15 Sep 2009 07:56:50 +0000

943: In our earlier article we described a power series formula for the Borel regulator evaluated on the odd-dimensional homology of the general linear group of a number field and, concentrating on dimension three for simplicity, described a computer algorithm which calculates the value to any chosen degree of accuracy. In this sequel we give an algorithm for the construction of the input homology classes and describe the results of one cyclotomic field computation.

Brown representability in A^1-homotopy theory

Sat, 12 Sep 2009 08:25:51 +0000

942: In this paper we prove a result of V. Voevodsky saying that if a finite dimensional Noetherian base scheme can be covered by spectra of countable rings then the compact objects in the stable motivic homotopy category over S form a countable category.

Computing Borel’s Regulator I

Wed, 26 Aug 2009 08:57:25 +0000

941: We expand Hamida’s integral for the Borel regulator in odd dimension as an inﬁnite series and give a computer algorithm for dimension 3.

Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems

joel.riou@math.u-psud.fr (Joël Riou) — Thu, 13 Aug 2009 08:54:07 +0000

940: In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.

The n-motivic t-structure for n = 0, 1 and 2

joseph.ayoub@math.uzh.ch (Joseph Ayoub) — Mon, 03 Aug 2009 17:15:29 +0000

939: For n = 0, 1 and 2, we define an n-motivic t-structure on Voevodsky's triangulated category of effective motives. The 0-motivic t-structure is taken to be the homotopy t-structure, but for n = 1 and 2, we get new t-structures. We show that the heart of the 1-motivic t-structure contains the category of Deligne's 1-motives. We then specify certain objects in the heart of the 2-motivic t-structure which we call mixed 2-motives. We also show that the category of mixed 2-motives is abelian.

L'invariant de Suslin en caractéristique positive

tim@wouters.in (Tim Wouters) — Fri, 31 Jul 2009 14:14:28 +0000

938: Pour une k-algèbre simple centrale A d'indice inversible dans k, Suslin a défini un invariant cohomologique de SK_1(A). Dans ce texte, nous généralisons cet invariant à toute k-algèbre simple centrale par un relèvement de la caractéristique positive à la caractéristique 0. Pour pouvoir définir cet invariant, on a besoin des groupes de cohomologie des différentielles logarithmiques de Kato.

For a central simple k-algebra A with index invertible in k, Suslin defined a cohomological invariant for SK_1(A). In this text, we generalise his invariant to any central simple k-algebra using a lift from positive characteristic to characteristic 0. To be able to define the invariant, we use Kato's cohomology of logarithmic differentials.

Motivic strict ring models for K-theory

Thu, 23 Jul 2009 19:47:16 +0000

937: It is shown that the K-theory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective `strict' is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic functors and motivic symmetric spectra furnish convenient frameworks for constructing the ring models. Analogous topological results follow by running the same type of arguments as in the motivic setting.

On the interior motive of certain Shimura varieties the case of Hilbert-Blumenthal varieties

wildesh@math.univ-paris13.fr (J. Wildeshaus) — Tue, 23 Jun 2009 15:56:20 +0000

936: Applying the main results of the author's previous work (paper no. 899), we construct a Hecke-equivariant Chow motive whose realizations equal interior cohomology of Hilbert-Blumenthal varieties with non-constant coefficients.

Hyperbolicity of orthogonal involutions

karpenko-remove@math.jussieu.fr (Nikita A. Karpenko) — Wed, 17 Jun 2009 14:39:29 +0000

935: We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic different from 2 remains non-hyperbolic over some splitting field of the algebra.

The Norm Residue Isomorphism Theorem

weibel@math.rutgers.edu (Charles A. Weibel) — Tue, 16 Jun 2009 04:06:43 +0000

934:

This is an expanded version of the preprint /K-theory/0844, "Patching the Norm Residue Isomorphism Theorem" (the word 'Patching' has been removed from the title). It will appear in the Journal of Topology.

Two new sections have been added. Section 2 (Motivic Model Structures) explains the passage between the various homotopy categories in terms of Quillen adjunctions, which is used to construct classifying spaces. Section 7 (χ Duality) is an exposition of the notion of duality for motives over the embedded scheme χ and is used in the definition of a Rost motive.

I would like to thank the referee for asking that this material be included.

Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

joseph.ayoub@math.uzh.ch (Joseph Ayoub), zucker@jhu.edu (Steven Zucker) — Tue, 09 Jun 2009 16:17:45 +0000

933: We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its weight-zero part as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. We then give an application to locally symmetric varieties. Namely, we prove that the Betti realization of E_X for X the Baily-Borel compactification is isomorphic to the cohomological direct image of the constant sheaf Q along the projection from the reductive Borel-Serre compactification to the Baily-Borel compactification.

K-theory of cones of smooth varieties

Thu, 28 May 2009 17:25:32 +0000

932: Let R be the homogeneous coordinate ring of a smooth projective variety X over a field k of characteristic 0. We calculate the K-theory of R in terms of the geometry of the projective embedding of X. In particular, if X is a curve then we give formulas for K₀(R) and K₁(R) in terms of the Zariski cohomology of twisted Kähler differentials on the variety. We also prove that K_-1(R) is the direct sum of the cohomology groups H¹(C,O(t)).

Grothendieck-Serre's conjecture for adjoint groups of types E_6 and E_7

Sun, 10 May 2009 04:53:48 +0000

931: In this preprint we prove cetain interesting results concerning principal G-bundles, where G is an adjoint reductive group schemesof types E_6 and E_7. Specifically, assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner adjoint simple algebraic group of type E_6 or E_7 over R. We prove that under the above assumptions every principal H-bundle P which has a K-rational point is itself trivial. This confirms a conjecture posed by Serre and Grothendieck.

On Grothendieck-Serre's conjecture concerning principal G-bundles over reductive group schemes II

panin-remove@pdmi.ras.ru (Ivan Panin) — Sun, 10 May 2009 04:53:45 +0000

930: In this paper we prove an interesting theorem concerning principal G-bundles, where G is a reductive group scheme. connected group scheme. Specifically, let R be a regular semi-local ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive R-group scheme satifying a mild "isotropy condition". Then each principal G-bundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Our proof is heavily based on two recent Theorems due to Panin--Stavrova--Vavilov, on a result due to Colliot-Thelene and Sansuc concerning the case of tori and on two purity theorems proven in the present preprint.

On Grothendieck-Serre's conjecture concerning principal G-bundles over reductive group schemes I

Sun, 10 May 2009 04:53:42 +0000

929: In this paper we prove an interesting theorem concerning principal G-bundles, where G is a semi-simple-simply connected group scheme. Specifically, let R be a semi-local regular domain containing an infinite perfect subfield and let K be its field of fractions. 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 under the above assumptions every principal G-bundle P which has a K-rational point is itself trivial. This confirms a conjecture posed by Serre and Grothendieck.