Related papers: Iterated doubles of the Joker and their realisabil…

The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $\mathcal A$. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the…

The algebraic Joker module was originally described in the 1970s by Adams and Priddy and is a $5$-dimensional module over the subHopf algebra $\mathcal{A}(1)$ of the mod $2$ Steenrod algebra. It is a self-dual endotrivial module, i.e., an…

The well known Joker $\mathcal{A}(1)$-module of Adams and Priddy is known to be realisable as the cohomology of a $1$-connected space. By attaching an extra cell we obtain an $8$-dimensional Poincar\'e duality space whose mod~$2$ cohomology…

We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes…

In this paper, we show that the finite subalgebra $\mathcal{A}^{\mathbb{R}}(1)$, generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$, of the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$ can be given $128$ different…

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and…

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple $(A,\mathcal{H},M)$ consisting of a Hopf algebra $\mathcal{H}$, an $\mathcal{H}$-comodule algebra $A$, an $\mathcal{H}$-module $M$, and a…

Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause…

Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq^1, Sq^2, and Sq^4. The method of calculation is a motivic version of…

Let ${\mathcal A}$ and ${\frak A}$ be Banach algebras such that ${\mathcal A}$ is a Banach ${\frak A}$-bimodule with compatible actions. We define the product ${\cal A}\rtimes{\frak A}$, which is a strongly splitting Banach algebra…

In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra $\mathcal{H}$ is bijective. We show that the cyclic module associated to the diagonal…

Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2…

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod $p$ singular cohomology of a space, for $p$ a prime. The first extends and refines earlier work of the second and fourth…

The $\mathbb{R}$-motivic cohomology of an $\mathbb{R}$-motivic spectrum is a module over the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$. In this paper, we describe how to recover the $\mathbb{R}$-motivic cohomology of…

We compute the cohomology of the quotient algebra $\mathcal{A}(2)$ of the $\mathbb{R}$-motivic dual Steenrod algebra. We do so by running a $\rho$-Bockstein spectral sequence whose input is the cohomology of $\mathbb{C}$-motivic…

Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf module algebras. If $A$ is an algebra with an action by an affine Hopf algebra $H$, then…

We compute the Hochschild homology and cohomology of $A(1)$, the subalgebra of the $2$-primary Steenrod algebra generated by the first two Steenrod squares, $Sq^1$ and $Sq^2$. The computation is accomplished using several May-type spectral…

Let $A$ be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over $A$.…

We define and investigate pairings of multiplier Hopf algebras. It is shown that two dually paired regular multiplier Hopf ($*$-)algebras $A$ and $B$ yield a quantum double multiplier Hopf ($*$-)algebra which is again regular. Integrals on…

In this note, we study some properties of the filtration of the Steenrod algebra defined from the excess of admissible monomials. We give several conditions on a cocommutative graded Hopf algebra A^* which enable us to develop the theory of…