Cooperations and group representations

1 The general setup

Let k be a field. If A is a ring , we get a functor spec(A) : k − algs → Sets where given a k-algebra R we set

spec(A)(R) = k − algs(A, R)

which we refer to as the R-points of spec(A). This terminology is used because an R-point of spec(A) may be thought of as a map spec(R) → spec(A). The ring A may be identified with the ring of functions spec(A) → A^{1}. Given a ∈ A, we get a function on R-points

a(R) : k − algs(A, R) = spec(A)(R) → R

given by a(R)(p) = p(a) for a map p : A → R of k-algebras.

If A and B are k-algebras, then

spec(A ⊗_{k} B) = spec(A) × spec(B)

Therefore, if A is a hopf algebra over k, then the comultiplication leads to a multiplication

spec(A) × spec(A) → spec(A)

hence a group structure on the R-points of spec(A).

Fix a hopf algebra A and denote G = spec(A). Let M be an A-comodule. Let Sym_{k}(M) be the symmetric k-algebra generated by M. Then the coaction map

M → A ⊗_{k} M

extends to a map of algebras

Sym_{k}(M) → A ⊗_{k} Sym_{k}(M).

Define an affine scheme V = spec(Sym_{k}(M)). Then as a scheme V is isomorphic to affine space of dimension equal to dim_{k}M. The coaction yields an action map

G × V → V

making V a linear representation of G on affine space.

Let A^{∗} be the dual hopf algebra A and let M^{∗} be the dual of M. Then M^{∗} is an A^{∗} module. We wish to establish the connection between this module structure and the representation V. First note that there are isomorphisms

M^{∗} = hom_{k}(M, k) = k − algs(Sym_{k}, k) = V (k)

which identifies M^{∗} with the k-points of V. Choose a k-basis {x_{i}} of M^{∗}. This corresponds to choosing projections π_{i} : V → A^{1}. For every k-algebra R, the unit k → R maps {x_{i}} to an R-basis of V (R). Given v ∈ V (R), we may decompose v with respect to the basis, yielding

v = ∑_{i} π_{i}(v)x_{i}.

So if g ∈ G(R), then

gx_{i} = ∑_{j} f_{i,j}(g)x_{j}

where f_{i,j}(g) = π_{j}(gx_{i}). Therefore f_{i,j} are functions on G and may be viewed as elements of A. Given α ∈ A^{∗}, we get the formula

αx_{i} = ∑_{j} α(f_{i,j})x_{j}

which gives the A^{∗} module structure of M^{∗}.

If M^{∗} is an A^{∗}-algebra, then the induced multiplication

V × V → V

is G - equivariant, hence the G(R)-action on V (R) may be deduced from knowing the G(R) action on a set of generators of V (R) as an R-algebra. This observation will come into play when we consider the cup product structure on the cohomology of topological spaces.

2 The mod 2 steenrod algebra

Let k = F_{2} and let A be the dual of the steenrod algebra.

A = P [ξ_{1}, ξ_{2}, …]

Then, of G = spec(A), G is the affine group scheme of 2-series. An R-point f ∈ G(R) may be identified with a power series

f(x) = x + ∑_{i>0} f_{i}x^{2^{i}}

where f_{i} = ξ_{i}(f) (viewing ξ_{i} as functions G(R) → R). We will agree that f_{0} is always 1 The multiplication

μ_{R} : G(R) × G(R) → G(R)

is given by reverse composition:

μ_{r} : (f(x), g(x)) → g(f(x)).

Explicitly

g(f(x)) = ∑_{i} g_{i}(∑_{j} f_{j}x^{2^{i}})^{2^{j}} = ∑_{i,j} f_{j}^{2^{i}}g_{i}x^{2^{i+j}}

where we are using the fact that R, being an F_{2}-algebra, has characteristic 2. This is indeed the group law given by the Milnor comultiplication formula

ψ(ξ_{k}) = ∑_{i+j=k} ξ_{j}^{2^{i}} ⊗ ξ_{i}.

For any spectrum X, H_{∗}(X; Z/2) is an A-comodule, hence determines a G representation which we shall denote V_{X}. We argue that sometimes it can be convenient to identify the representation V_{X} and then use the discussion above to extract how the milnor basis elements act on H^{∗}(X).

If X is a space, H^{∗}(X) is an A^{∗} -algebra, hence V_{X} is an algebra, with multiplication that is G-equivariant.

3 H^{∗}(RP^{∞})

Let X = RP^{∞}. Then M^{∗}(x) = P [[x]] as an algebra. The axioms for the steenrod operations imply that

Sq^{i}x^{k} = \binom{k}{i}x^{k+i}

Represent g ∈ V_{X}(R) as a power series

g(x) = g_{0} + g_{1}x + g_{2}x^{2} + g_{3}x^{3} + ⋯

where g_{i} are in R. The claim is that the G(R) action on V_{X}(R) is given by

μ(f(x), g(x)) = g(f(x)) ∈ V_{X}(R)

where f(x) is a 2-series in R. To verify the claim, we just must check that this representation induces the correct formula for the action of Sq^{i} on M^{∗}. But Sq^{i} ∈ A^{∗} is dual to ξ_{1}^{i} with respect to the monomial basis of A. Now, for f(x) ∈ G(R),

μ(f(x), x^{k}) = (x+f_{1}x^{2}+⋯)^{k} = \binom{k}{i}f_{1}^{i}x^{i+k}+terms with coefficients other then f_{1}^{i}

and so Sq^{i} is as it should be.

Now, we may give an ”instant” computation of the effect of the Milnor basis elements on x^{k}. Q_{i} ∈ A^{∗} is dual to ξ_{i+1}. If f is as before, then

μ(f(x), x^{k}) = (x + ⋯ + f_{i+1}x^{2^{i+1}} + ⋯)^{k} = kf_{i+1}x^{k+2^{i+1}−1} + ⋯

so Q_{i}x^{k} = kx^{k+2^{i+1}−1}, which is proven in Milnor’s paper.

4 H^{∗}(BO(n))

Let X = BO(n). We will identify the representation V_{X} of G. It is well known that

H^{∗}(BO(n)) = P [w_{1}, …, w_{n}]

and the inclusion of the diagonal O(1)×⋯ O(1) ↪ O(n) induces an inclusion

P [w_{1}, …, w_{n}] ↪ P [x_{1}, …, x_{n}]

where w_{i} is sent to the i^{th} elementary symmetric polynomial in x_{1}, …, x_{n}. Now, BO(1) = RP^{∞} so our previous calculations tell us that we may identify V_{X}(R) with

R[[σ_{1}, …, σ_{n}]] ⊆ R[[x_{1}, …, x_{n}]]

so the G(R) action is given by

μ(f(x), g(σ_{1}(x_{1}, …, x_{n}), …, σ_{n}(x_{1}, …, x_{n})))

= g(σ_{1}(f(x_{1}), …, f(x_{n})), …, σ_{n}(f(x_{1}), …, f(x_{n})))

We shall use this to understand the representation V_{M}O(n), and thus understand the A^{∗} action on H̃^{∗}(X^{ξ}) in terms of H^{∗}(X) by studying this universal example. The euler class of ξ_{n}, the cannonical n-plane bundle over BO(n) is w_{n}. Now the Thom isomorphism theorem implies that R-points of V_{MO(1)} may be thought of as R-points of V_{BO(1)} times a thom class. Therefore, we have the following formula in V_{MO(n)}(R)

μ(f(x), τ) = τ · \frac{f(x_{1}) ⋯ f(x_{n})}{x_{1} ⋯ x_{n}}

where τ is the image of the Thom class under the unit of R. Therefore, for any space X with a vector bundle ξ, if we identify R points of V_{X^{ξ}} with R-points of V_{X} times a Thom class, we get the formula

μ(f(x), τ · α) = τ · h_{f}(w_{1}, ⋯, w_{n})μ(f(x), α)

where τ is the thom class, w_{1}, …, w_{n} are the images of the Stiefel-Whitney classes of ξ under the unit of R, and h_{f} is the power series with coefficients in R such that

\frac{f(x_{1}) ⋯ f(x_{n})}{x_{1} ⋯ x_{n}} = h_{f}(σ_{1}, …, σ_{n})

where σ_{i}(x_{1}, …, x_{n}) is the i^{th} elementary symmetric polynomial.

As an application, we prove a formula for Q_{k−1} of the thom class of MO(1) that is central to the proof that HZ/2 is a thom spectrum. We will prove

Q_{k−1}τ = τ · (w_{2^{k}−1} + decomposibles).

For we have for an R-point f(x) in G,

μ(f(x), τ) = τ · \frac{(x_{1} + f_{k}x_{1}^{2^{k}} + ⋯) ⋯ (x + f_{k}x_{n}^{2^{k}} + ⋯)}{x_{1} ⋯ x_{n}}
= τ · (1 + f_{k}x_{1}^{2^{k}−1} + ⋯)(1 + f_{k}x_{n}^{2^{k}−1} + ⋯)
= f_{k}(x_{1}^{2^{k}−1} + ⋯ + x_{n}^{2^{k}−1}) + ⋯

so we just neet to write the image of x_{1}^{2^{k}−1}+⋯+x_{n}^{2^{k}−1} in the indecomposibles of the symmetric polynomials in n variables. Now

σ_{1}^{2^{k}−1} = x_{1}^{2^{k}−1} + ⋯ x_{n}^{2^{k}−1} + σ_{2^{k}−1} + decomposible symmetric polynomials

therefore

x_{1}^{2^{k}−1} + ⋯ + x_{n}^{2^{k}−1} = σ_{2^{k}−1} + decomposibles

and the result follows.