For $X$ a smooth scheme acted on by linear algebraic group $G$ and $p$ prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over Steenrod algebra. We compute Lannes's $T$-functor applied to \mathbb{F}_p$. As application, we localization of away from $n$-nilpotent modules algebra, affirming conjecture Totaro as special case. The case when point $n = 1$ generali...

The purpose of this paper is to forge a direct link between the hit problem for the action of the Steenrod algebra A on the polynomial algebra P(n) = F2[x1, . . . , xn], over the field F2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F2). The cohits Qd(n) = Pd(n)/Pd(n)∩A+(P(n)) form a modular representation o...

This paper provides analogues of the results of [16] for odd primes p . It is proved that for certain irreducible representations L(λ) of the full matrix semigroup Mn(Fp), the first occurrence of L(λ) as a composition factor in the polynomial algebra P = Fp[x1, . . . , xn] is linked by a Steenrod operation to the first occurrence of L(λ) as a submodule in P. This operation is given explicitly a...

We study homological algebra in the abelian category Γ̃, whose objects are functors from finite pointed sets to vector spaces over Fp. The full calculation of Tor ∗ -groups between functors of degree not exceeding p is presented. We compare our calculations with known results on homology of symmetric groups, Steenrod algebra and functor homology computations in the abelian category F of functors...

Let C be a coalgebra over a field k and A its dual algebra. The category of C-comodules is equivalent to a category of A-modules. We use this to interpret the cotensor product M N of two comodules in terms of the appropriate Hochschild cohomology of the A-bimodule M ⊗N , when A is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comod...

In this paper, we analyze the structure possessed by the homotopy groups of a simplicial abelian Hopf algebra over the field F2. Specifically, we review the higher-order structure that the homotopy groups of a simplicial commutative algebra and simplicial cocommutative coalgebra possess. We then demonstrate how these structures interact under the added conditions present in a Hopf algebra. Intr...

In 1993, Henn, Lannes and Schwartz established a very strong relation between the Steenrod algebra and the category F.p/ of functors from the category E of finite dimensional Fp –vector spaces to the category E of all Fp –vector spaces, where Fp is the prime field with p elements [5]. To be more precise, they study the category U of unstable modules over the Steenrod algebra localized away from...

Let p be an odd prime and A be the mod p Steenrod algebra. For computing the stable homotopy groups of spheres with the classical Adams spectral sequence, we must compute the E2 -term of the Adams spectral sequence, Ext A ( p, p) . In this paper we prove that in the cohomology of A , the product k0hnδ̃s+4 ∈ Ext s+7,t(s,n)+s A ( p, p) , is nontrivial for n ≥ 5, and trivial for n = 3, 4, where δ̃s+...

Let $A_1$ be any spectrum in the class of finite spectra whose mod-2 cohomology is isomorphic to $\mathcal{A}(1)$ as a module over subalgebra Steenrod algebra; let $tmf$ connective topological modular forms. In this paper, we prove that $tmf$-Hurewicz image surjective.