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Steenrod algebra

Contents

under construction

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebra

Contents

Idea

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

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

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

The Ext-groups between comodules for these commutative Hopf algebroids E (E)E_\bullet(E) prominently appear on the second page of the EE-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 EE a suitable E-infinity ring, its self-generalized homology E (E)E_\bullet(E) form a (graded-)commutative Hopf algebroid over E E_\bullet.

See at Hopf algebroid structure – For generalized cohomology below.

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 1Sq i 2Sq i n Sq^{i_1} \circ Sq^{i_2} \circ \cdots \circ Sq^{i_n}

where i ki_k \in \mathbb{N} subject to the relation

i k2i k+1. i_k \geq 2 i_{k+1}.

This is indeed a linear basis for the 𝔽 2\mathbb{F}_2-vector space underlying the Steenrod algebra.

Proposition

The Steenrod square operations satisfy the following relation, for all for all 0<i<2j0 \lt i \lt 2 j:

Sq iSq j= k=0 [i/2](jk1 i2k) mod2Sq i+jkSq k 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 pp be a prime number. Write 𝔽 p\mathbb{F}_p for the corresponding prime field.

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

  • for p=2p = 2 generated by elements denoted Sq nSq^n for nn \in \mathbb{N}, n1n \geq 1;

  • for p>2p \gt 2 generated by elements denoted β\beta and P nP^n for nn \in \mathbb{N}, n1n \geq 1

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

for p=2p = 2:

for 0<h<2k0 \lt h \lt 2k the

Sq hSq k=i=0[h/2](ki1 h2i)Sq h+kiSq i, 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>2p \gt 2:

for 0<h<pk0 \lt h \lt p k then

P hP k=i=0[h/p](1) h+i((p1)(ki)1 hpi)P h+kiP i 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<h<pk0 \lt h \lt p k then

P hβP k =[h/p]i=0(1) h+i((p1)(ki) hpi)βP h+kiP i +[(h1)/p]i=0(1) h+i1((p1)(ki)1 hpi1)P h+kiβP i \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 Ψ\Psi is subject to the following relations:

for p=2p = 2:

Ψ(Sq n)=k=0nSq kSq nk \Psi(Sq^n) \;=\; \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}

for p>2p \gt 2:

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

and

Ψ(β)=β1+1β. \Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta \,.

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

Dual Steenrod algebra

Definition

The 𝔽 p\mathbb{F}_p-linear dual of the mod pp-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 𝔸 𝔽 p *\mathbb{A}_{\mathbb{F}_p}^\ast.

Remark

There is an isomorphism

𝒜 𝔽 p *H (H𝔽 p,𝔽 p)=π (H𝔽 pH𝔽 p). \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 𝒜 𝔽 p *\mathcal{A}^\ast_{\mathbb{F}_p} from def. , in terms of linear duals of the generators for 𝒜 𝔽 p\mathcal{A}_{\mathbb{F}_p} itself, according to def. .

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

P nSq 2n P^n \coloneqq Sq^{2n}
βSq 1. \beta \coloneqq Sq^1 \,.

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

  • P nP^n has even degree deg(P n)=2n(p1)deg(P^n) = 2n(p-1);

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

Theorem

(Milnor's theorem)

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

𝒜 𝔽 p *Sym 𝔽 p(ξ 1,ξ 2,,τ 0,τ 1,) \mathcal{A}^\ast_{\mathbb{F}_p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)

on generators:

  • ξ n\xi_n being the linear dual to P p n1P p n2P pP 1P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1;

  • τ n\tau_n being linear dual to P p n1P p n2P pP 1βP^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta.

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

Ψ(ξ n)=k=0nξ nk p kξ k \Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k

and

Ψ(τ n)=τ n1+k=0nξ nk p kξ nk p kτ k, \Psi(\tau_n) = \tau_n \otimes 1 + \underoverset{k=0}{n}{\sum} \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k \,,

where we set ξ 01\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 pp coefficients is a Hopf algebra over 𝔽 p\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 RR be an E-∞ ring and let AA an E-∞ algebra over RR. The self-generalized homology A R(A)A^R_\bullet(A) is naturally a module over the cohomology ring A A_\bullet via applying the homotopy groups \infty-functor π \pi_\bullet to the canonical inclusion

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

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

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

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

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

π (ARARX)A R(A)A A R(X). \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 R(X)π (ARARX)A R(A)A A R(X). 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) EE-Steenrod algebra Hopf algebroids have also been called “brave new Hopf algebroids” (Baker, Baker-Jeanneret 02)

Example

For E=H𝔽 2E = 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=E = MU then by Quillen's theorem on MU π (MU)L\pi_\bullet(MU) \simeq L is the Lazard ring and the Hopf algebroid (π (MU),MU (MU))(\pi_\bullet(MU), MU_\bullet(MU)) is described by the Landweber-Novikov theorem.

Example

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

Relation to Adams spectral sequence

For ordinary cohomology

By construction, the total cohomology H (X, 2)H^\bullet(X,\mathbb{Z}_2) of a topological space XX, naturally is a module over the Steenrod algebra:

a cohomology element is represented by a cocycle which is a map XB n 2X \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 [Σ X,Σ Y][\Sigma^\infty X, \Sigma^\infty Y] for topological spaces XX and YY: that second page is given by the Ext-groups

E 2=Ext A(H (X, 2),H (Y, 2)) E_2 = Ext_A(H^\bullet(X, \mathbb{Z}_2), H^\bullet(Y,\mathbb{Z}_2))

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

For generalized cohomology

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

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

See the references below.

References

General

Original articles include

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

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

reviewed in (Ravenel 86, A1.5).

Further textbook accounts include

Lecture notes include

Reviews include

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

  • 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

See also

E E_\bullet-Hopf algebroid structure

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

Last revised on April 21, 2019 at 10:27:18. See the history of this page for a list of all contributions to it.