arXiv subjects

Ivan Panin

Publications and source records attributed to Ivan Panin.

47 records · Page 3Linked to original sources

arXiv2014

Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel. Certain arguments used in the present preprint do not work if the ring R contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring R contains an infinite field this result is proved in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the Grothendieck--Serre conjecture holds for regular local rings containing a field.

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arXiv2014

On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II

In three preprints [Pan1], [Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here $G$ is a reductive group scheme). The preprint [Pan1] contains main geometric presentation theorems which are necessary for that. The present preprint contains reduction of the Grothendieck--Serre's conjecture to the case of semi-simple simply-connected group schemes (see Theorem 1.0.1). The preprint [Pan3] contains a proof of that conjecture for regular semi-local domains R containing a finite field. The Grothendieck--Serre conjecture for the case of regular semi-local domains containing an infinite field is proven in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the conjecture holds for regular semi-local domains containing a field. The reduction is based on two purity results (Theorem 1.0.2 and Theorem 10.0.29).

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arXiv2014

The triangulated category of K-motives DK_(k)

For any perfect field k a triangulated category of K-motives DK_(k) is constructed in the style of Voevodsky's construction of the category DM_(k). To each smooth k-variety X the K-motive is associated in the category DK_(k). Also, it is shown that K_n(X)=DK_(k)(M_K(X)[n],M_K(pt)), where K(X) is Quillen's K-theory of X.

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arXiv2013

Principal bundles of reductive groups over affine schemes

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.

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arXiv2013

Rationally trivial quadratic spaces are locally trivial:III

Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let q be a quadratic space over R on a free rank n R-module P such that the projective quadric q=0 is smooth over R. It is proved that if the quadratic space q is isotropic over K, then there is a unimodular vector v in the free rank n R-module P such that q(v)=0. If characteristic of R is 2, then in the case of even n our assumption on q is equivalent to the one that q is a non-singular space in the sense of \cite{Kn} and in the case of odd n > 2 our assumption on q is equivalent to the one that q is a semi-regular in the sense of \cite{Kn}.

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arXiv2012

Rationally isotropic exceptional projective homogeneous varieties are locally isotropic

Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.

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arXiv2012

K-motives of algebraic varieties

A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point.

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arXiv2009

Grothendieck-Serre Conjecture I: Appendix

We prove here some supplementary statements that appeared without proof in I. Panin, A. Stavrova, N. Vavilov, On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:I, arXiv:0905.1418

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arXiv2008

Duality Theorem for Motives

Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodsky's duality to the case of arbitrary base field characteristic.

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arXiv2005

T-spectra and Poincaré Duality

Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring-spectrum and proved the Poincaré Duality theorem for this general case. In the current paper we consider oriented cohomology theories on algebraic varieties represented by multiplicative symmetric $T$-spectra and prove the Duality theorem, which mimics the result of Adams. This result is held, in particular, for Motivic Cohomology and Algebraic Cobordism of Voevodsky.

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arXiv2003

Purity for Similarity Factors

Two Azumaya algebras with involutions are considered over a regular local ring. It is proved that if they are isomorphic over the quotient field, then they are isomorphic too. In particular, if two quadratic spaces over such a ring are similar over its quotient field, then these two spaces are similar already over the ring. The result is a consequence of a purity theorem for similarity factors proved in this text and the known fact that rationally isomorphic hermitian spases are locally isomorphic.

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