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.

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 $p=2$ .

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

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.

We 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.

Let $\rho:G\hookrightarrow\GL(n,\F)$ be a representation of a finite group $G$ over a finite field $\F$ and $f_1,\dots,f_n\in\F[V]^G$ such that the ring of invariants is a polynomial algebra $\F[f_1,\dots,f_n]$. It is known that in this case the algebra of coinvariants $\F[V]_G$ is a Poincaré duality algebra, and if, moreover, the order of $G$ is invertible in $\F$, that a fundamental class is represented by the Jacobian determinant $\mathchoice{\det\biggl[\frac{\partial f_i}{\partial z_j}\biggr]}{\det[\partial f_i/\partial z_j]}{\det[\partial f_i/\partial z_j]} {\det[\partial f_i/\partial z_j]}$, and is therefore a $\det^{-1}$-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

AMS 2000 Mathematics subject classification: Primary 13A50; 55S10

The classical Schubert cells on a flag manifold $G/P$ give a cell decomposition for $G/P$ whose Kronecker duals (known as Schubert classes) form an additive base for the integral cohomology $H^{\ast}(G/P)$. We present a formula that expresses Steenrod mod-$p$ operations on Schubert classes in $G/P$ in terms of Cartan numbers of $G$.