scholarly journals Motivic Steenrod operations in characteristic p

2020 ◽  
Vol 8 ◽  
Author(s):  
Eric Primozic
Keyword(s):  
Quadratic Forms ◽  
Chow Groups ◽  
Base Field ◽  
Characteristic P ◽  
Motivic Cohomology ◽  
Steenrod Operations ◽  
Adem Relations

Abstract For a prime p and a field k of characteristic $p,$ we define Steenrod operations $P^{n}_{k}$ on motivic cohomology with $\mathbb {F}_{p}$ -coefficients of smooth varieties defined over the base field $k.$ We show that $P^{n}_{k}$ is the pth power on $H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod p Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic $2,$ we obtain new results on quadratic forms.

scholarly journals Reduced Steenrod operations and resolution of singularities

2011 ◽  
Vol 9 (2) ◽  
pp. 269-290 ◽  
Author(s):  
Olivier Haution
Keyword(s):  
Prime Number ◽  
Quadratic Forms ◽  
Weak Form ◽  
Chow Groups ◽  
Resolution Of Singularities ◽  
Steenrod Operations ◽  
Projective Homogeneous Varieties ◽  
New Construction ◽  
Homogeneous Varieties

AbstractWe give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.

scholarly journals Product structures in motivic cohomology and higher Chow groups

2011 ◽  
Vol 215 (4) ◽  
pp. 511-522 ◽  
Author(s):  
Satoshi Kondo ◽  
Seidai Yasuda
Keyword(s):  
Chow Groups ◽  
Motivic Cohomology ◽  
Product Structures ◽  
Higher Chow Groups

scholarly journals THE 2-RANKS OF HYPERELLIPTIC CURVES WITH EXTRA AUTOMORPHISMS

2009 ◽  
Vol 05 (05) ◽  
pp. 897-910 ◽  
Author(s):  
DARREN GLASS
Keyword(s):  
Automorphism Group ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Closed Field ◽  
Base Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Carry Over ◽  
The Relationship

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

K-Theory ◽  
2004 ◽  
Vol 31 (4) ◽  
pp. 307-321 ◽  
Author(s):  
Oleg Pushin
Keyword(s):  
Chern Classes ◽  
Motivic Cohomology ◽  
Steenrod Operations

scholarly journals Jacques Tits motivic measure

2021 ◽  
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Quadratic Forms ◽  
Arbitrary Dimension ◽  
Base Field ◽  
Central Simple Algebras ◽  
Grothendieck Ring ◽  
Main Application ◽  
Simple Algebras ◽  
Central Simple

AbstractIn this article we construct a new motivic measure called the Jacques Tits motivic measure. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $$\{3, 4, 5, 6\}$$ { 3 , 4 , 5 , 6 } , have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field k with $$I^3(k)=0$$ I 3 ( k ) = 0 , have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.

Motivic cohomology and higher Chow groups

1998 ◽  
pp. 53-106
Author(s):  
Marc Levine
Keyword(s):  
Chow Groups ◽  
Motivic Cohomology ◽  
Higher Chow Groups

COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

2008 ◽  
Vol 07 (02) ◽  
pp. 147-166 ◽  
Author(s):  
ROBERTO LA SCALA ◽  
ALEXANDER ZUBKOV
Keyword(s):  
Null Space ◽  
General Linear ◽  
Base Field ◽  
Characteristic P ◽  
Schur Algebra ◽  
Arbitrary Characteristic ◽  
Linear Basis ◽  
General Linear Supergroup ◽  
Schur Superalgebra

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist (G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist (G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char (K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.

Variations on a theme of rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Nikita Karpenko
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Arbitrary Characteristic ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism ◽  
Variations On A Theme

AbstractWe prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

scholarly journals Cohomological Invariants in Positive Characteristic

2021 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Positive Characteristic ◽  
Differential Forms ◽  
Symmetric Groups ◽  
Characteristic P ◽  
Motivic Cohomology ◽  
Cohomological Invariants ◽  
Group Schemes ◽  
Characteristic 2 ◽  
Theory Of Fields ◽  
Mod P

Abstract We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are invariants of $G$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $p$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $p$ Milnor K-theory of fields.

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