scholarly journals Unstable modules with only the top k Steenrod operations

2021 ◽  
Vol 21 (7) ◽  
pp. 3623-3662
Author(s):  
Zhulin Li
Keyword(s):  
Steenrod Operations

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 $p=2$ .

scholarly journals Steenrod operations and transfer

1968 ◽  
Vol 19 (6) ◽  
pp. 1387-1387 ◽  
Author(s):  
Leonard Evens
Keyword(s):  
Steenrod Operations

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.

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

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