nLab Steenrod algebra

under construction

Idea

For $p$ a prime number, the Steenrod algebra $\mathcal{A}$ is the associative algebra over the prime field $\mathbb{F}_p$ of cohomology operations on ordinary cohomology with coefficients in $\mathbb{F}_p$. For $p = 2$ this is the $\mathbb{F}_2$-algebra generated by the Steenrod square operations. (Often the case $p = 2$ is understood by default.)

For any prime $p$, the mod-$p$ Steenrod algebra furthermore has the structure of a Hopf algebra over $\mathbb{F}_p$, non-commutative but co-commutative. Under forming linear duals this gives the dual Steenrod algebra, traditionally denoted $\mathcal{A}_\ast$ instead of $\mathcal{A}^\ast$. This is a Hopf algebra over $\mathbb{F}_p$ that is commutative, but non-co-commutative.

The dual $\mathbb{F}_p$ Steenrod algebra is a special case of a commutative Hopf algebroid structure canonically induced on the self generalized homology $E_\bullet(E)$ of any ring spectrum $E$ for which $E_\bullet \to E_\bullet(E)$ is a flat morphisms. Therefore in this general case one sometimes speaks of “dual $E$-Steenrod algebras”.

The Ext-groups between comodules for these commutative Hopf algebroids $E_\bullet(E)$ prominently appear on the second page of the $E$-Adams spectral sequence, see there for more.

Definition

For ordinary cohomology

(…) (e.g. Mosher-Tangora 68, section 6, Lurie 07) (…)

For generalized cohomology

The Steenrod algebra and its standard properties, such as the Adem relations, follow abstractly from the Cotor groups of comodules over any commutative Hopf algebroid.

This is due to (May 70, 11.8). A review is in (Ravenel, appendix 1, theorem A1.5.2).

In particular for $E$ a suitable E-infinity ring, its self-generalized homology $E_\bullet(E)$ forms a (graded-)commutative Hopf algebroid over $E_\bullet$.

See at Hopf algebroid structure – For generalized cohomology below. For more see at Adams spectral sequence – The first page.

Properties

Presentation by generators and relations

Definition

The Serre-Cartan basis is the subset of elements of the Steenrod algebra on those of the form

$$Sq^{i_1} \circ Sq^{i_2} \circ \cdots \circ Sq^{i_n}$$

where $i_k \in \mathbb{N}$ subject to the relation

$$i_k \geq 2 i_{k+1}.$$

This is indeed a linear basis for the $\mathbb{F}_2$-vector space underlying the Steenrod algebra.

Proposition

The Steenrod square operations satisfy the following relation, for all for all $0 \lt i \lt 2 j$:

$$Sq^i \circ Sq^j = \sum_{k = 0}^{[i/2]} \left( \array{ j - k - 1 \\ i - 2k } \right)_{mod 2} Sq^{i + j -k} \circ Sq^k$$

These are called the Ádem relations (Ádem 52).

Proposition

The Ádam relations precisely generate the ideal of relations among the Serre-Cartan basis elements, def. , in the Steenrod algebra.

More generally, for other prime numbers:

Definition

Let $p$ be a prime number. Write $\mathbb{F}_p$ for the corresponding prime field.

The mod $p$-Steenrod algebra $\mathcal{A}_{\mathbb{F}_p}$ is the graded co-commutative Hopf algebra over $\mathbb{F}_p$ which is

• for $p = 2$ generated by elements denoted $Sq^n$ for $n \in \mathbb{N}$, $n \geq 1$;

• for $p \gt 2$ generated by elements denoted $\beta$ and $P^n$ for $n \in \mathbb{N}$, $n \geq 1$

(called the Serre-Cartan basis elements) whose product is subject to the following relations (called the Ádem relations):

for $p = 2$:

for $0 \lt h \lt 2k$ the

$$Sq^h Sq^k \;=\; \underoverset{i = 0}{[h/2]}{\sum} \left( \array{ k -i - 1 \\ h - 2i } \right) Sq^{h + k -i} Sq^i \,,$$

for $p \gt 2$:

for $0 \lt h \lt p k$ then

$$P^h P^k \;=\; \underoverset{i = 0}{[h/p]}{\sum} (-1)^{h+i} \left( \array{ (p-1)(k-i) - 1 \\ h - p i } \right) P^{h +k - i}P^i$$

and if $0 \lt h \le p k$ then

$$\begin{aligned} P^h \beta P^k & =\; \underoverset{[h/p]}{i = 0}{\sum} (-1)^{h+i} \left( \array{ (p-1)(k-i) \\ h - p i } \right) \beta P^{h+k-i}P^i \\ & + \underoverset{[(h-1)/p]}{i = 0}{\sum} (-1)^{h+i-1} \left( \array{ (p-1)(k-i) - 1 \\ h - p i - 1 } \right) P^{h+k-i} \beta P^i \end{aligned}$$

and whose coproduct $\Psi$ is subject to the following relations:

for $p = 2$:

$$\Psi(Sq^n) \;=\; \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}$$

for $p \gt 2$:

$$\Psi(P^n) \;=\; \underoverset{n}{k = 0}{\sum} P^k \otimes P^{n-k}$$

and

$$\Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta \,.$$

e.g. (Kochmann 96, p. 52)

Dual Steenrod algebra

Definition

The $\mathbb{F}_p$-linear dual of the mod $p$-Steenrod algebra (def. ) is itself naturally a graded commutative Hopf algebra (with coproduct the linear dual of the original product, and vice versa), called the dual Steenrod algebra $\mathbb{A}_{\mathbb{F}_p}^\ast$.

Remark

There is an isomorphism

$$\mathcal{A}^\ast_{\mathbb{F}_p} \simeq H_\bullet( H \mathbb{F}_p, \mathbb{F}_p ) = \pi_\bullet( H \mathbb{F}_p \wedge H \mathbb{F}_p ) \,.$$

(e.g. Rognes 12, remark 7.24)

Milnor’s theorem

We now give the generators-and-relations description of the dual Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ from def. , in terms of linear duals of the generators for $\mathcal{A}_{\mathbb{F}_p}$ itself, according to def. .

In the following, we use for $p = 2$ the notation

$$P^n \coloneqq Sq^{2n}$$

$$\beta \coloneqq Sq^1 \,.$$

This serves to unify the expressions for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$

• $P^n$ has even degree $deg(P^n) = 2n(p-1)$;

• $\beta$ has odd degree $deg(\beta) = 1$.

Theorem

(Milnor's theorem)

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. ) is, as an associative algebra, the free graded commutative algebra

$$\mathcal{A}^\ast_{\mathbb{F}_p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)$$

on generators:

• $\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$, has degree $2p^i - 2$;

• $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$, has degree $2p^i - 1$.

Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by

$$\Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k$$

and

$$\Psi(\tau_n) = \tau_n \otimes 1 + \underoverset{k=0}{n}{\sum} \xi_{n-k}^{p^k}\otimes \tau_k \,,$$

where we set $\xi_0 \coloneqq 1$.

This is due to (Milnor 58). See for instance (Kochmann 96, theorem 2.5.1)

Hopf algebroid structure

For ordinary cohomology

The Steenrod algebra for mod $p$ coefficients is a Hopf algebra over $\mathbb{F}_p$ which is graded commutative and non-co-commutative.

This is due to (Milnor 58). A review is in Ravenel, ch. 3, section 1.

For generalized cohomology

More generally:

Lemma

Let $R$ be an E-∞ ring and let $A$ an E-∞ algebra over $R$. The self-generalized homology $A^R_\bullet(A)$ is naturally a module over the cohomology ring $A_\bullet$ via applying the homotopy groups $\infty$-functor $\pi_\bullet$ to the canonical inclusion

$$A \stackrel{\simeq}{\rightarrow} A \underset{R}{\wedge} R \stackrel{}{\rightarrow} A \underset{R}{\wedge} A \,.$$

Proposition

Let $R$ be an E-∞ ring and let $A$ an E-∞ algebra over $R$. If the the $A_\bullet$-module $A^R_\bullet(A)$ of lemma is a flat module, then

1. $(A_\bullet, A_\bullet(A))$ is a commutative Hopf algebroid over $R_\bullet$;

2. $A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-∞-module $X$.

This is due to (Baker-Lazarev 01), further discussed in (Baker-Jeanneret 02) (there expressed in terms of the presentation by highly structured ring spectra). A review is also in (Ravenel, chapter 2, prop. 2.2.8).

Proof sketch

Using the arguments of (Adams 74, Ravenel 86).

The flatness condition implies that there is an equivalence

$$\pi_\bullet\left( A \underset{R}{\wedge} A \underset{R}{\wedge} X \right) \stackrel{\simeq}{\longrightarrow} A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X) \,.$$

Combining this with the map in lemma yields the coaction

$$A_\bullet^R(X) \longrightarrow \pi_\bullet\left( A \underset{R}{\wedge} A \underset{R}{\wedge} X \right) \stackrel{\simeq}{\longrightarrow} A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X) \,.$$

These (dual) $E$-Steenrod algebra Hopf algebroids have also been called “brave new Hopf algebroids” (Baker, Baker-Jeanneret 02)

Example

For $E = H \mathbb{F}_2$ the Eilenberg-MacLane spectrum, this reproduces the Hopf algebra structure on the dual ordinary Steenrod algebra as above.

Example

For $E =$ MU then by Quillen's theorem on MU $\pi_\bullet(MU) \simeq L$ is the Lazard ring and the Hopf algebroid $(\pi_\bullet(MU), MU_\bullet(MU))$ is described by the Landweber-Novikov theorem.

Example

For $E =$ BP the analog is the content of the Adams-Quillen theorem. The Landweber exact functor theorem was proven using the $B P_\bullet(B P)$-Hopf algebroid.

Relation to Adams spectral sequence

For ordinary cohomology

By construction, the total cohomology $H^\bullet(X,\mathbb{Z}_2)$ of a topological space $X$, naturally is a module over the Steenrod algebra:

a cohomology element is represented by a cocycle which is a map $X \longrightarrow B^n \mathbb{Z}_2$ and the action of a Steenrod square on this is just by composition.

In this form the Steenrod algebra appears in the second page of the Adams spectral sequence which computes $[\Sigma^\infty X, \Sigma^\infty Y]$ for topological spaces $X$ and $Y$: that second page is given by the Ext-groups

$$E_2 = Ext_A(H^\bullet(X, \mathbb{Z}_2), H^\bullet(Y,\mathbb{Z}_2))$$

computed in the category of A-modules for $A$ the Steenrod algebra.

For generalized cohomology

More generally, For $R$ an E-infinity ring such that its dual $R$-Steenrod algebra in the form of the self-homology $R_\bullet(R)$ is a (graded-)commutative Hopf algebroid over $R_\bullet = \pi_\bullet(R)$ (see at Steenrod algebra – Hopf algebroid structure), then the $E^2$-term of the $E$-Adams spectral sequence is an Ext of $E_\bullet(E)$-comodules

$$E^2 \simeq Ext_{R_\bullet(R)}(R_\bullet, R_\bullet(X)) \,.$$

See the references below.

Related concepts

• cohomology operation

• power operation

• Pi-algebra

• Lambda-algebra

• Adams resolution, Adams spectral sequence, Adams-Novikov spectral sequence

• Brown-Gitler spectrum

• generic representation theory

References

General

Original articles include

• José Ádem, The iteration of the Steenrod squares in algebraic topology , Proceedings of the National Academy of Sciences of the United States of America 38: 720–726 (1952)

• John Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150–171.

• Norman Steenrod, David Epstein, Cohomology operations, Ann. of Math. Studies, no. 50, Princeton University Press, Princeton, N. J., 1962.

• John Frank Adams, part III, section 12 of Stable homotopy and generalised homology, University of Chicago Press (1974).

Comprehensive discussion of the ordinary Steenrod algebra, with proof is the Adem relations includes

• Robert Mosher, Martin Tangora, Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968) (pdf)

• Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007. (MIT OpenCourseWare: Massachusetts Institute of Technology), Lecture notes

  Lecture 2 Steenrod operations (pdf)

  Lecture 3 Basic properties of Steenrod operations (pdf)

  Lecture 4 The Adem relations (pdf)

  Lecture 5 The Adem relations (cont.) (pdf)

The general algebraic approach was laid out in

• Peter May, A general algebraic approach to Steenrod operations, in The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. (pdf)

• Robert Bruner, Peter May, J. E. McClure, M. Steinberger,

  $H_\infty$ ring spectra and their applications, Lecture Notes in Math., Springer-Verlag, Berlin,

reviewed in (Ravenel 86, A1.5).

Further textbook accounts include

• Harvey Margolis, Spectra and the Steenrod algebra, 1983 North-Holland

• Stanley Kochmann, section 2.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Stefan Schwede, chapter II section 10 of Symmetric spectra, 2012 (pdf)

Lecture notes include

• John Rognes, section 3 of The Adams spectral sequence, 2012 (pdf)

Review:

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres Academic Press 1986

  (appendix 1, section 5 reviews the abstract algebraic definition).

• Agnès Beaudry, Jonathan A. Campbell, §3 in: A Guide for Computing Stable Homotopy Groups, in Topology and Quantum Theory in Interaction, Contemporary Mathematics 718, Amer. Math. Soc. (2018) [arXiv:1801.07530, ams:conm/718]

  (with an eye towards functorial quantum field theory)

• Cary Malkievich, The Steenrod Algebra (pdf)

On the dual Steenrod algebra:

• J. Palmieri, Some quotient Hopf algebras of the dual Steenrod algebra (pdf)

For a commented list of furhter references see also

• Steenrod algebra learning seminar

See also

• online Adem relation calculator

$E_\bullet$-Hopf algebroid structure

The commutative Hopf algebroid structure on the dual $E$-Steenrod algebra $E_\bullet(E)$ and its relation to the $E^2$-term in the Adams spectral sequence is discussed in

• Andrew Baker, Brave new Hopf algebroids, pdf

• Andrew Baker, Andrey Lazarev, On the Adams Spectral Sequence for R-modules, Algebr. Geom. Topol. 1 (2001) 173-199 (arXiv:math/0105079)

• Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)

• Mark Hovey, Homotopy theory of comodules over a Hopf algebroid (arXiv:math/0301229)