Related papers: On excess filtration on the Steenrod algebra

In the paper "The Steenrod algebra and its dual", J.Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme $G_p$ represented by the dual…

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool…

In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of…

The algebra ${\mathsf A}_q$ of Steenrod $q$th powers, where $q = p^e$ is a power of a prime $p$, is isomorphic to a subalgebra ${\mathsf A}'_q$ of the algebra of Steenrod $p$th powers ${\mathsf A}_p$. The filtration of ${\mathsf A}_p$ by…

We consider a theory of noncommutative Gr\"obner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gr\"obner bases to this context. As an important…

Necessary and sufficient conditions are obtained for an ambiskew polynomial algebra A over a Hopf k-algebra R to possess the structure of a Hopf algebra extending that of R, in which the added variables X+ and X- are skew primitive. The…

A new and natural description of the category of unstable modules over the Steenrod algebra as a category of comodules over a bialgebra is given; the theory extends and unifies the work of Carlsson, Kuhn, Lannes, Miller, Schwartz, Zarati…

We extend Wood's graph theoretic interpretation of certain quotients of the mod $2$ dual Steenrod algebra to quotients of the mod $p$ dual Steenrod algebra where $p$ is an odd prime and to quotients of the $C_2$-equivariant dual Steenrod…

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…

In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded…

Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the…

Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors.…

Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of…

We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In particular we…

We study criteria for freeness and for the existence of a vanishing line for modules over certain Hopf subalgebras of the motivic Steenrod algebra over $\mathrm{Spec}(\mathbb{C})$ at the prime 2. These turn out to be determined by the…

The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On…

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for…

Given a unital $*$-algebra $\mathscr{A}$ together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C$^*$-algebra $A_0$ with a dense two-sided ideal $A_c$ such that $\mathscr{A}$ maps…

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf…