Cooperations and representations

1. The general setup

Let A be a ring. We get a functor Spec(A) : Rings → Sets where given a ring R we set

Spec(A)(R) = Ring(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) : Ring(A, R) = Spec(A)(R) → R

given by a(R)(p) = p(a) for a map p : A → R of rings.

More generally, if Σ is an A-algebra, then there is a map of rings

A → Σ

which we may use to interpret Spec(Σ) as a scheme over Spec(A). Let p_{Σ} : Spec(Σ) → Spec(A) be the induced map. Note that A-algebra structures are in one-to-one correspondence with R-points of Spec(A), and if we fix one of these A-algebra structures on R (corresponding to x ∈ Spec(A)(R)), then

A − alg(Σ, R) = {a ∈ Spec(Σ)(R) : p_{Σ}(a) = x}.

We shall refer to the above set as Spec(Σ)(R)_{x}, the R-points of Spec(Σ) over x. If Σ and Λ are A-algebras, then

Spec(Σ ⊗_{A} Λ) = Spec(Σ) ×_{Spec(A)} Spec(Λ)

Therefore, if Σ is a hopf algebroid (commutative but not necessarily cocommutative) over A, then the coproduct leads to a composition pairing

Spec(Σ) ×_{Spec(A)} Spec(Σ) → Spec(Σ)

hence a groupoid scheme structure on the pair (Spec(A), Spec(Σ)). Given a ring R, we have the R-groupoid, a groupoid with objects Spec(A)(R) and morphisms Spec(Σ)(R) where an R-morphism f is a map from η_{L}^{∗}(f) → η_{R}^{∗}(f). η_{L} gives Σ its left A-algebra structure and η_{R} gives Σ its right A-module structure.

Fix a (commutative) hopf algebroid Σ and denote C = Spec(A), and Mor = Spec(Σ). Let M be a left Σ-comodule. Let Sym_{A}(M) be the symmetric A-algebra generated by M. Then the coaction map

M → Σ ⊗_{A} M

extends to a map of algebras

Sym_{A}(M) → Σ ⊗_{A} Sym_{A}(M).

Define an affine scheme E_{M} = Spec(Sym_{A}(M)). Since M is an A-module, it gives rise to a sheaf O_{M} over C. (This is actually the sheaf M̃^{∧}). The global sections of this sheaf may be identified with M^{∗} = hom_{A}(M, A). Alternatively, they may be thought of as sections of the ”bundle”

p_{E_{M}} : E_{M} → C

The correspondence is given by

Γ(E_{M}/C) = A − alg(Sym_{A}M, A) = hom_{A}(M, A) = M^{∗} = Γ(O_{M})

Given a global section s ∈ M^{∗}, we may evaluate it on an R-point x of C as follows. The fiber E_{M}(R)_{x} may be identified with A − alg(Sym_{A}(M), R) = hom_{A}(M, R) where R is given the A-algebra structure induced by x. The element of E_{M}(R)_{x} which represents s(R)(x) is the homomorphism of A-modules

M ⟶^{s} A ⟶^{x} R

Observe that the Σ-comodule sturcture on M gives rise to a groupoid representation of (C, Mor) on O_{M}. Specifically, the coaction gives rise to an action

Mor ×_{C} E_{M} → E_{M}

In terms of R-points, suppose f : x → y is an R-point of Mor(R), where x = η_{L}^{∗}(f) and y = η_{R}^{∗}(f) are the corresponding source and target R-points of C. Then f gives rise to a map of the R-points of the fibers

f^{∗} : E(R)_{y} → E(R)_{x}

This is the sense in which M is a groupoid representation. Observe that we may view E_{M}(R) as a contravariant functor from (C(R), Mor(R)) to sets.

Suppose that Σ is A-free with respect to its left A-module structure. (This assumption is relevent to our applications since we will have Σ = E_{∗}E for a spectrum E where E_{∗}E is E_{∗}-free.) Suppose furthermore that M is A-free (that is, E_{M} is a trivial vector bundle over C. Let Σ^{∗} = hom_{A}(Σ, A) be the dual hopf algebroid. and let M^{∗} be the dual of M. Then M^{∗} is a left Σ^{∗} module. We wish to establish the connection between this module structure and the representation O_{M}. Choose an A-basis (a collection of trivializing sections) {s_{i}} of M^{∗}. This corresponds to choosing projections π_{i} : E_{M} → 𝔸_{A}^{1}. For every R-point x of C, the map A → R maps {s_{i}} to an R-basis of E_{M}(R)_{x}. Given v ∈ E_{M}(R)_{x}, we may decompose v with respect to the basis, yielding

v = ∑_{i} π_{i}(v)s_{i}(x).

So if f ∈ Mor(R)(x, y), then

f^{∗}s_{i}(y) = ∑_{j} a_{i,j}(f)s_{j}(x)

where a_{i,j}(f) = π_{j}(f^{∗}s_{i}(y)). Therefore a_{i,j} are functions on Mor and may be viewed as elements of Σ. Given α ∈ Σ^{∗}, we get the formula

αs_{i} = ∑_{j} α(a_{i,j})s_{j}

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

If M^{∗} is a Σ^{∗}-algebra, then the induced (fiberwise) multiplication

E_{M} ×_{C} E_{M} → E_{M}

is Mor-equivariant, hence the Mor(R)-action on E_{M}(R) may be deduced from knowing the Mor(R)(x, y) action on a set of generators of E_{M}(R)_{x} as an R-algebra. This observation will come into play when we consider the cup product structure on the cohomology of topological spaces.

2. Complex cobordism, and the Landweber-Novikov algebra

In this section we explain how the algebra of operations on complex cobordism plays out in this framework. Let Σ = MU_{∗}MU be the complex cobordism hopf algebroid over A = MU_{∗} Recall that

MU_{∗}MU = MU_{∗}[b_{1}, b_{2}, …]

where |b_{i}| = 2i. Let (Mor, C) be the corresponding groupoid-scheme. Quillen says C = F GL, the scheme whose R-points are formal group laws over R. An R-point f of Mor is a strict isomorphism of formal groups. The generators b_{i} are chosen such that if ˜(f) : Σ → R is the map which represents f, then the strict isomorphism corresponding to f is given by

f(x) = x + f̃(b_{1})x^{2} + f̃(b_{2})x^{3} + ⋯

Thus on R-points, if F_{1}, F_{2}, and F_{3} are in F GL(R), then the map on R-points induced by the coproduct on Σ

Mor(R)(F_{1}, F_{2}) × Mor(R)(F_{2}, F_{3}) → Mor(R)(F_{1}, F_{3})

is given by composition of power series.

(f, g) → g ∘ f

The Landweber-Novikov algebra MU^{∗}MU = σ^{∗}. An A-basis of σ^{∗} can be obtained by taking the basis consisting of elements s_{α} which are A-dual to the monomials

b_{1}^{α_{1}}b_{2}^{α_{2}}b_{3}^{α_{3}} ⋯

of Σ. Here the indices α = (α_{1}, α_{2}, ⋯) are sequences of non-negative integers, of which all but finitely many are zero.

We shall now describe the comodule sturcture of MU_{∗}(ℂP^{∞}). Recall that MU_{∗}(ℂP^{∞}) is free over MU_{∗} on generators e_{2i} in dimension 2i for all i ≥ 0. If we write

MU^{∗}(ℂP^{∞}) = MU^{∗}[[x]]

where x is the cannonical complex orientation of MU, then the e_{2}i are dual to x^{i}. According to [1, p. 91], the MU_{∗} coproduct structure is given by

ψe_{2k} = ∑_{i+j=k} (∑_{l} b_{l})_{2i}^{j} ⊗ e_{2j}.

Here (∑_{l} b_{l})_{2i}^{j} means the summand in dimention 2i of the expression (∑_{l} b_{l})^{j}. This coaction has a very natural description when viewed as a representation of Σ. Let E_{ℂP^{∞}} be the vector bundle over F GL given by the Σ comodule M = MU_{∗}(ℂP^{∞}). Since M is A-free, this is a trivial vector bundle, with trivializing sections x^{i} ∈ M^{∗}. Given a ring R, and an R-point F in F GL(R) the R-points of the fiber over F may be described via

E_{ℂP^{∞}}(R)_{F} ∑_{i≥0} a_{i}x(F)^{i}

where a_{i} ∈ R. Here, if h is the A-module map M → R (with respect to the Amodule structure induced on R by the map which classifies F, then a_{i}h(e_{2i}) The action,

Mor(R)(F_{1}, F_{2}) × E_{M}(R)_{F_{2}} → E_{M}(R)_{F_{1}}

(for F_{1}, F_{2} ∈ F GL(R)) may be described. In light of the formula for the Mcomodule stucture given above, if f : F_{1} → F_{2}, then

f^{∗} ∑_{i} a_{i}x(F_{2})^{i} = ∑_{i} a_{i}(f(x(F_{1})))^{i}.

We can use the techniques of section 1 to evaluate the Novikov operation s_{α} on x ∈ M^{∗}. Specifically, if f = ∑_{j≥0} b_{j}x^{j+1}, then

f^{∗}x = ∑_{j≥0} b_{j}x^{j+1}.

We may conclude that

s_{α}(x) = ∑_{j>0}⟨s_{α}, b_{j}⟩x^{j+1}

which may be compared to [1, p. 12].

References

[1] J. F. Adams, Stable Homotopy and Generalised Homology, Chicago Lecture Notes in Math., Chicago, 1974.