Nikita A. Karpenko

The Algebraic and Geometric Theory of Quadratic Forms

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions

Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties

u

Criteria of motivic equivalence for quadratic forms and central simple algebras

-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

Characterization of minimal Pfister neighbors via Rost projectors

Abstract Let G be a semisimple affine algebraic group of inner type over a field F. We write 𝔛G for the class of all finite direct products of projective G-homogeneous F-varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in 𝔛G. More precisely, it is known that the motive of any variety in 𝔛G decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group PGL A of automorphisms of a given central simple F-algebra A, for any variety in the class 𝔛G (which includes the generalized Severi–Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in 𝔛G are p-incompressible. If A is a division algebra of degree pn for some n ≧ 0, then the list of p-incompressible varieties includes the generalized Severi–Brauer variety X(pm; A) of ideals of reduced dimension pm for m = 0, 1, …, n.

Isotropy of orthogonal involutions

Abstract. We give a short, elementary, and characteristic independent proof of the criterion for motivic isomorphism of two projective quadrics discovered by A. Vishik [24]. We also give a criterion for motivic isomorphism of two Severi-Brauer varieties.

Upper Motives of Outer Algebraic Groups

Let φ be an anisotropic nine-dimensional quadratic form over a field (of characteristic ≠2). We show that the projective quadric given by φ possesses a Rost projector if and only if φ is a Pfister neighbor. The following consequence of this result gives the initial step in the construction [10] of the field with the u-invariant 9: if φ is not a Pfister neighbor and the Schur index of its even Clifford algebra is at least 4, then φ stays anisotropic over the function field of any 9-dimensional form nonsimilar to φ.

Torsion in CH2 of Severi-Brauer varieties and indecomposability of generic algebras

An orthogonal involution on a central simple algebra becoming isotropic over any splitting field of the algebra, becomes isotropic over a finite odd degree extension of the base field (provided that the characteristic of the base field is not

Weil transfer of algebraic cycles

2

Incompressibility of quadratic Weil transfer of generalized Severi–Brauer varieties

). The proof makes use of a structure theorem for Chow motives with finite coefficients of projective homogeneous varieties, of incompressibility of certain generalized Severi-Brauer varieties, and of Steenrod operations.

Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a finite field of characteristic p of the projective G-homogeneous varieties. The complete motivic decomposition of any such variety contains one specific summand, which is the most understandable among the others and which we call the upper indecomposable summand of the variety. We show that every indecomposable motivic summand of any projective G-homogeneous variety is isomorphic to a shift of the upper summand of some (other) projective G-homogeneous variety. This result is already known (and has applications) in the case of G of inner type and is new for G of outer type (over F).