Ivan Panin

Oriented Cohomology Theories of Algebraic Varieties

This article contains proofs of the results announced in [21] in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples. Mathematics Subject Classifications:

A purity theorem for the witt group

Abstract Let A be a regular local ring and K its field of fractions. We denote by W the Witt group functor that classifies quadratic spaces. We say that purity holds for A if W(A) is the intersection of all W(A p ) ⊂ W(K), as p runs over the height-one prime ideals of A. We prove purity for every regular local ring containing a field of characteristic ≠ 2. The question of purity and of the injectivity of W(A) into W(K) for arbitrary regular local rings is still open.

RIGIDITY FOR ORIENTABLE FUNCTORS

Abstract In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K -theory of algebraically closed fields. Besides K -theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.

Rationally trivial Hermitian spaces are locally trivial

Abstract. Let R be a regular local ring, K its field of fractions and A an Azumaya algebra with involution over R. Let h be an

Witt sheaves and the η-inverted sphere spectrum

\epsilon

On Grothendieck–Serre’s conjecture concerning principal -bundles over reductive group schemes: I

-hermitian space over A. We show that if

A variant of a theorem by Springer

\bold{h}\otimes_R K

On a Theorem of Grothendieck

is hyperbolic over

On the Grothendieck–Serre Conjecture Concerning Principal G-Bundles Over Semilocal Dedekind Domains

A\otimes_R K

T-spectra and Poincaré duality

, then h is hyperbolic over A.