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 Hopf algebra of the mod $p$ Steenrod algebra. Then, $G_p$ assigns a graded commutative algebra $A_*$ over a prime field of finite characteristic $p$ to a set of isomorphisms of the additive formal group law over $A_*$, whose group structure is given by the composition of formal power series. The aim of this paper is to show some group theoretic properties of $G_p$ by making use of this presentation of $G_p(A_*)$. We give a decreasing filtration of subgroup schemes of $G_p$ which we use for estimating the length of the lower central series of finite subgroup schemes of $G_p$. We also give a successive quotient maps $G_p\xrightarrow{ρ_0}G_p^{\langle1\rangle}\xrightarrow{ρ_1}G_p^{\langle2\rangle}\xrightarrow{ρ_2}\cdots\xrightarrow{ρ_{k-1}} G_p^{\langle k\rangle}\xrightarrow{ρ_k}G_p^{\langle k+1\rangle}\xrightarrow{ρ_{k+1}}\cdots$ of affine group schemes over a prime field ${\boldsymbol F}_p$ such that the kernel of $ρ_k$ is a maximal abelian subgroup.
We describe an algorithm that allows to compute a minimal resolution of the Steenrod algebra. The algorithm has built-in knowledge about vanishing lines for the cohomology of sub Hopf algebras of the Steenrod algebra which makes it both faster and more economical than the generic approach.
A non-connected neither of finite type Hopf algebra $\mathcal{F}_{0}$ is defined over $\mathbb{Z}/ 2\mathbb{Z}$ and its hom dual turns out to be a tensor product of polynomial algebras. Certain quotient Hopf algebras include the Steenrod and Dyer-Lashof algebras. This setting provides a map between the Steenrod coalgebra and a direct limit of Dyer-Lashof coalgebras.
We introduce a new model for the secondary Steenrod algebra at the prime 2 which is both smaller and more accessible than the original construction of H.-J. Baues. We also explain how BP can be used to define a variant of the secondary Steenrod algebra at odd primes.
The notion of quadratic self-duality for coalgebras is developed with applications to algebraic structures which arise naturally in algebraic topology, related to the universal Steenrod algebra via an appropriate form of duality. This explains and unifies results of Lomonaco and Singer.
The aim of this paper is to study sub-algebras of the $\mathbb{Z}/2$-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor $\mathbb{F}_2$) which come from quotient Hopf algebroids of the $\mathbb{Z}/2$-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical $\mathcal{A}(n)$ and $\mathcal{E}(n)$ and show that the Steenrod algebra is free as a module over these.
The Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of `Quillen stratification' theorem for the Singer transfer and its consequences.
We describe algebraic obstruction theories for realizing an abstract coalgebra K_* over the mod p Steenrod algebra as the homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K_*.
The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology operations. The Steenrod algebra is presented as a subalgebra of the algebra of endomorphisms of a functor. The functor in question assigns to a vector space over a Galois field the algebra of polynomial functions on that vector space: the subalgebra of the endomorphisms of this functor that turns out to be the Steenrod algebra if the ground field is the prime field, is generated by the homogeneous components of a variant of the Frobenius map.
This two-page note gives a non-computational derivation of the dual Steenrod algebra as the automorphisms of the formal additive group. Instead of relying on computational tools like spectral sequences and Steenrod operations, the argument uses a few simple universal properties of certain cohomology theories.
We compute the action of the Steenrod algebra on generators of algebras of invariants of special linear group ${SL_n=SL(n,\mathbb Z/p)}$ in the polynomial algebra with $ p$ an odd prime number.
In this very short note, we expand the Hu-Kriz computation of the $G$-equivariant Borel dual Steenrod algebra in characteristic $2$, from the group $G=C_2$ to all power-$2$ cyclic groups $G=C_{2^n}$.
The relations of Barratt and Miller are shown to include all relations among the elements $P^iχP^{n-i}$ in the mod $p$ Steenrod algebra, and a minimal set of relations is given.
We study the Mahowald operator $M = \langle g_2,h_0^3, - \rangle$ in the cohomology of the Steenrod algebra. We show that the operator interacts well with the cohomology of $A(2)$, in both the classical and $\mathbb{C}$-motivic contexts. This generalizes previous work of Margolis, Priddy, and Tangora.
We compute the $C_p$-equivariant dual Steenrod algebras associated to the constant Mackey functors $\underline{\mathbb{F}}_p$ and $\underline{\mathbb{Z}}_{(p)}$, as $\underline{\mathbb{Z}}_{(p)}$-modules. The $C_p$-spectrum $\underline{\mathbb{F}}_p \otimes \underline{\mathbb{F}}_p$ is not a direct sum of $RO(C_p)$-graded suspensions of $\underline{\mathbb{F}}_p$ when $p$ is odd, in contrast with the classical and $C_2$-equivariant dual Steenrod algebras.
Unlike the p = 2 case, the universal Steenrod Algebra Q(p) at odd primes does not have a fractal structure that preserves the length of monomials. Nevertheless, when p is odd we detect inside Q(p) two different families of nested subalgebras each isomorphic (as length-graded algebras) to the respective starting element of the sequence
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 unstable A^*-modules.
A minimal resolution of the mod 2 Steenrod algebra in the range $0 \leq s \leq 128$, $0 \leq t \leq 200$, together with chain maps for each cocycle in that range and for the squaring operation $Sq^0$ in the cohomology of the Steenrod algebra.
The purpose of this article is to compute the cohomology of the motivic Steenrod algebra over Spec C through the geometric 70-stem. The main computational tool is the motivic May spectral sequence. Everywhere in this article, we are working only over Spec C, and we are always computing at the prime p = 2.
Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of the finite sub-Hopf algebras $A(n)$ of the mod 2 Steenrod algebra that correspond to generators in the homotopy rings of certain periodic spectra. The purpose of this note is to present a proof of the conjecture.
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod…
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 vanishing of certain Margolis homology groups in the quotient Hopf algebra $\mathcal{A}/τ$.
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 instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory ($2$-compact groups, topological modular forms) a…
Let $p$ be an odd prime number. We study the problem of determining the module structure over the mod $p$ Steenrod algebra $\mathcal A(p)$ of the Dickson algebra $D_n$ consisting of all modular invariants of general linear group $GL(n,\mathbb F_p)$. Here $\mathbb F_p$ denotes the prime field of $p$ elements. In this paper, we give an explicit answer for $n=2$. More precisely, we explicitly compute the action of the Steenrod-Milnor operations $St^{S,R}$ on the generators of $D_n$ for $n=2$ and for either $S=\emptyset, R=(i)$ or $S=(s), R=(i)$ with $s,i$ arbitrary nonnegative integers.
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 and others. Related categories of comodules are studied, which shed light upon the structure of the category of unstable modules at odd primes. In particular, a category of bigraded unstable modules is introduced; this is related to the study of modules over the motivic Steenrod algebra.
We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for p=2, in particular in the nature and consequences of the defining Adem relations.
The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Vi…
Motivated by the construction of Steenrod cup-$i$ products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential g…
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 posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to …
We compute the action of the primitive Steenrod-Milnor operations on generators of algebras of invariants of subgroups of general linear group GL_n=GL(n,F_p) in the polynomial algebra with p an odd prime number.
Let $P_k:= \mathbb F_2[x_1,x_2,\ldots,x_k]$ be the polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $k$ variables $x_1, x_2, \ldots, x_k$, each of degree 1. We are interested in the Peterson hit problem of finding a minimal set of generators for $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study the hit problem in degree $(k-1)(2^d-1)$ with $d$ a positive integer. Our result implies the one of Mothebe [4,5].
The mod 2 Steenrod algebra $\mathcal{A}_2$ can be defined as the quotient of the mod 2 Leibniz--Hopf algebra $\mathcal{F}_2$ by the Adem relations. Dually, the mod 2 dual Steenrod algebra $\mathcal{A}_2^*$ can be thought of as a sub-Hopf algebra of the mod 2 dual Leibniz--Hopf algebra $\mathcal{F}_2^*$. We study $\mathcal{A}_2^*$ and $\mathcal{F}_2^*$ from this viewpoint and give generalisations of some classical results in the literature.
First, by inspiration of the results of Wood \cite{differential,problems}, but with the methods of non-commutative geometry and different approach, we extend the coefficients of the Steenrod squaring operations from the filed $\mathbb{F}_2$ to the dyadic integers $\mathbb{Z}_2$ and call the resulted operations the dyadic Steenrod squares, denoted by $Jq^k$. The derivation-like operations $Jq^k$ generate a graded algebra, called the dyadic Steenrod algebra, denoted by $\mathcal{J}_2$ acting on the polynomials $\mathbb{Z}_2[ξ_1, \dots, ξ_n]$. Being $\mathcal{J}_2$ an Ore domain, enable us to loc…
Denote by $P_k$ the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the Peterson hit problem of determining a minimal set of generators for $P_k$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_k$ in the case $k=5$ and the degree $4(2^d - 1)$ with $d$ an arbitrary positive integer.
Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theor…
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 improve upon a known connection between the graph theoretic interpretation of the mod 2 dual Steenrod algebra and its structure as a Hopf algebra.
We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem is to determine the set of $\mathcal {A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each $k \geq 0$, the set of {\it $k$ partially $\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i…
We present a descent style, Bockstein spectral sequence computing Ext over the motivic Steenrod algebra over $\R$ and related sub-Hopf algebras. We demonstrate the workings of this spectral sequence in several examples, providing motivic analogues to the classical computations related to BP<n> and ko.
Working over the prime field of characteristic two, consequences of the Koszul duality between the Steenrod algebra and the big Dyer-Lashof algebra are studied, with an emphasis on the interplay between instability for the Steenrod algebra action and that for the Dyer-Lashof operations. The central algebraic framework is the category of length-graded modules over the Steenrod algebra equipped with an unstable action of the Dyer-Lashof algebra, with compatibility via the Nishida relations. A first ingredient is a functor defined on modules over the Steenrod algebra that arose in the work of Kuh…
Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sens…
The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^m-1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show …
This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is proved that for certain irreducible representations L(lambda) of the full matrix semigroup M_n(F_p), the first occurrence of L(lambda) as a composition factor in the polynomial algebra P=F_p[x_1,...,x_n] is linked by a Steenrod operation to the first occurrence of L(lambda) as a submodule in P. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra A_p …
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 powers of its augmentation ideal was studied by J. P. May in his Princeton thesis of 1964. We extend some of May's results to ${\mathsf A}_q$ and obtain a convenient set of defining relations for the graded algebra $E^0({\mathsf A}_q)$. In the case $q=p$, we recover the observation of S. B. Priddy that the subalgebra $E^0({\mathsf A}_p(n-2))$ of $E^0({\mathsf …
Let $p$ be an odd prime number. Denote by $GL_n = GL(n,\mathbb F_p)$ the general linear group over the prime field $\mathbb F_p$. Each subgroup of $GL_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$ in the usual manner. We grade $P_n$ by assigning $\dim x_i=1$ and $\dim y_i=2.$ This algebra is a module over the mod $p$ Steenrod algebra $\mathcal A_p$. The purpose of the paper is to compute the action of the Steenrod-Milnor operations on the generators of $P_2^{GL_2}$. More precisely, we explicitly determine the action of $St^{(i,j)}$ on the Dickson invariants …
Let $P_n$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_n]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field of two elements. The Peterson hit problem is to find a minimal generating set for $P_n$ regarded as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. Equivalently, we want to find a vector space basis for $\mathbb F_2 \otimes_{\mathcal A} P_n$ in each degree $d$. Such a basis may be represented by a list of monomials of degree $d$. In this paper, we present a construction for the $\mathcal A$-generators of $P_n$ and pro…
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 $\mathcal{A}^{\mathbb{R}}$-module structures. We also show that all of these $\mathcal{A}^{\mathbb{R}}$-modules can be realized as the cohomology of a $2$-local finite $\mathbb{R}$-motivic spectrum. The realization results are obtained using an $\mathbb{R}$ -motivic analogue of the Toda realization theorem. We notice that each realization of $\mathcal{A}^{\mathbb{R}…
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)=F_2[x_1,...,x_n], over the field F_2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F_2). The cohits Q^d(n)=P^d(n)/P^d(n)\cap A^+(P(n)) form a modular representation of GL(n,F_2) and the hit problem is to analyze this module. In certain generic degrees d we show how the semistandard Young tableaux can be used to index a set of monomials which span Q^d(n). The hook…
Let $P_{k}=H^{*}((\mathbb{R}P^{\infty})^{k})$ be the modulo-$2$ cohomology algebra of the direct product of $k$ copies of infinite dimensional real projective spaces $\mathbb{R}P^{\infty}$. Then, $P_{k}$ is isomorphic to the graded polynomial algebra $\mathbb{F}_{2}[x_{1},\ldots,x_{k}]$ of $k$ variables, in which each $x_{j}$ is of degree 1, and let $GL_k$ be the general linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$. Here the cohomology is taken with coefficients in the prime field $\mathbb F_2$ of two elements. We study the {\it hit problem}, set up by Frank P…
Let $\mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb F_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra over the prime field of two elements, $\mathbb F_2.$ We investigate the Peterson hit problem for the polynomial algebra $\mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. The purpose of this paper is to continue our study of the hit problem by developing a result in \cite{ph31} for $\mathcal P_n$ in the generic degree $r(2^s-1)…
In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $φ_3^{\mathbb{F}_p}$ and $φ_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$.
