scholarly journals Steenrod Operations, Degree Formulas and Algebraic Cobordism

2007 ◽  
Vol 3 (1) ◽  
pp. 283-306 ◽  
Author(s):  
Marc Levine
Keyword(s):  
Steenrod Operations ◽  
Algebraic Cobordism

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 Symmetric operations for all primes and Steenrod operations in algebraic cobordism

2015 ◽  
Vol 152 (5) ◽  
pp. 1052-1070 ◽  
Author(s):  
Alexander Vishik
Keyword(s):  
Chow Group ◽  
Characteristic Numbers ◽  
Steenrod Operations ◽  
Algebraic Cobordism ◽  
Important Structure

In this article we construct symmetric operations for all primes (previously known only for $p=2$). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all $p$-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map $\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$, providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.

scholarly journals Around rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Raphaël Fino
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism

AbstractWe prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

Steenrod operators, the Coulomb branch and the Frobenius twist

2021 ◽  
Vol 157 (11) ◽  
pp. 2494-2552
Author(s):  
Gus Lonergan
Keyword(s):  
Dual Group ◽  
Coulomb Branch ◽  
Affine Grassmannian ◽  
Steenrod Operations ◽  
Fundamental Relationship ◽  
Theoretic Version

Abstract We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$ -theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

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