The purpose of this paper is to give definitions of some new Hopf algebras analogous to the mod-p Steenrod algebras. For example, we produce a Hopf algebra over the integers whose reduction mod-p is the mod-p Steenrod algebra. Just for fun, we “quantize” these algebras too! Further study of the examples of this paper will be done in part II. 1 Deformations of Hopf Algebras Suppose that k is a c...

We use the Steenrod algebra to study the Chow ring CH∗BG of the classifying space of an algebraic group G. We describe a localization property which relates a given G to its elementary abelian subgroups, and we study a number of particular cases, namely symmetric groups and Chevalley groups. It turns out that the Chow rings of these groups are completely determined by the abelian subgroups and ...

Article history: Received 20 November 2014 Received in revised form 17 February 2015 Communicated by C.A. Weibel MSC: 14F42; 55T15; 55S10 We compute the h1-localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the η-local motivic sphere. We compute some of the Adams differentials, a...

The hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebras over the field Fp, for p an arbitrary prime, and treat the equivalent problem of determining the set of A∗-primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action ...

We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg–Moore spectral sequence by a single use of the spectral sequence converging to H(ΩX;Z/2) obtained from the Goodwillie tower for ΣΩX. Much of the...

In this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as Steenrod algebra spectrum. For this, start by extending Hess–Shipley definition coTHH to ?$\infty$-categories, following Nikolaus–Scholze approach Hochschild (THH). Furthermore, prove that what call quasi-proper can be obtained from THH via Spanier–Whitehead duality which provides further in...