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A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations

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This is a sub-entry of

see there for background and context.

This entry contains a basic introduction to derived group schemes and their orientations.

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the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish


Contents

Introduction

Recall from last time that given GG an algebraic group such that the formal spectrum SpfA(P )Spf A(\mathbb{C}P^\infty) is the completion G^\hat G, we could define A S 1(*):= GA_{S^1}({*}) := \mathcal{o}_{G} then passing to germs gave a completion map

A S 1(*)A(P )=A S 1 Bor(*). A_{S^1}({*}) \to A(\mathbb{C}P^\infty) = A^{Bor}_{S^1}({*}).

The problem we (begin) to address here is how to extend this equivariant cohomology to other spaces besides the point. This requires derived algebraic geometry.

Derived group schemes

Recall that a commutative group scheme over a scheme XX is a functor

G:Sch/X opAb G: \mathrm{Sch} /X^{op} \to \mathrm{Ab}

such that composition with the forgetful functor F:AbSetF: \mathrm{Ab} \to \mathrm{Set} is representable.

We would like extend this definition to the world of derived schemes. There are two problems

  1. Because of the higher categorical nature of derived schemes Hom sets are spaces.

  2. Everything should in the \infty-setting, that is defined only up to homotopy.

We will not worry about the second concern and address the first by replacing the category Ab\mathrm{Ab} with TopAb\mathrm{TopAb} and Set\mathrm{Set} with Top\mathrm{Top}.

The following definition is somewhat restrictive and really should incorporate more of the \infty-structure.

Definition A commutative derived group scheme over a derived scheme XX is a topological functor

G:DSch/X opTopAb G: \mathrm{DSch} / X^{op} \to \mathrm{TopAb}

such that composition with the forgetful functor F:TopAbTopF: \mathrm{TopAb} \to \mathrm{Top} is representable (up to weak equivalence) by an object which is flat? over XX.

Examples

  1. Let XX be a scheme, then we have an associated derived scheme X¯\overline{X}. The structure sheaf of X¯\overline{X} is obtained by viewing the structure sheaf of XX as a presheaf of E E_\infty-rings and then sheafifying. We then have an equivalence between commutative derived group schemes over X¯\overline{X} and commutative group schemes which are flat over XX.

  2. For XX a derived scheme we have a map from commutative derived group schemes over XX to commutative group schemes which are flat over π 0X\pi_0 X.

Preorientations

Throughout AA will be an E E_\infty-ring, XX the affine derived scheme SpecA\mathrm{Spec} A, GG a commutative derived group scheme over XX, A S 1A_{S^1} the E E_\infty-ring given by Γ(G)\Gamma (G), and A P A^{\mathbb{C}P^\infty} the E E_\infty-ring given by

A P =Hom E (P ,A). A^{\mathbb{C}P^\infty} = \mathrm{Hom}_{E_\infty} (\mathbb{C}P^\infty , A).

Definition(Preliminary) A preorientation of GG is a morphism of commutative derived group schemes over XX

σ:SpfA P G, \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G,

where SpfA P \mathrm{Spf} A^{\mathbb{C}P^\infty} is the completion wrt the ideal ker(A P A pt=A)\mathrm{ker} (A^{\mathbb{C}P^\infty} \to A^{pt} = A). A preorientation is an orientation if the induced map

σ^:SpfA P G^ \hat \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to \hat G

is an isomorphism.

Suppose that GG is affine, then a map

σ:SpfA P G \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G

corresponds to a map A S 1A P A_{S^1} \to A^{\mathbb{C}P^\infty} which is the same as a map

P Hom(X,G)=G(X). \mathbb{C}P^\infty \to \mathrm{Hom} (X, G) = G(X).

Hence we are led to the following definition.

Definition Let XX be a derived scheme and GG a commutative derived group scheme over XX. A preorientation of GG is a morphism of topological commutative monoids

([α])=P G(X). \mathbb{P} (\mathbb{C} [\alpha]) = \mathbb{C}P^\infty \to G(X).

Notice that P \mathbb{C}P^\infty is nearly freely generated. Indeed it follows from the fundamental theorem of algebra that as a topological monoid P \mathbb{C}P^\infty is generated by P 1\mathbb{C}P^1 subject to the single relation that pt=P 0P 1\mathrm{pt} = \mathbb{C}P^0 \subset \mathbb{C}P^1 is the monoidal unit.

Proposition A preorientation up to homotopy is a map

S 2P 1G(X)S^2 \simeq \mathbb{C}P^1 \to G(X)

that is an element of π 2(G(X))\pi_2 (G(X)).

Hence, we always have at least one preorientation: the trivial one which corresponds to 0π 2(G(X))0 \in \pi_2 (G(X)).

Orientations

As motivation recall that a map s:ABs: A \to B of 1-dim formal groups is an isomorphism if and only if ss' is invertible. We would like to encode this in our derived language (without defining ss').

Definition Let AA be an E E_\infty-ring, GG a commutative derived group scheme over SpecA\mathrm{Spec} A and σ:S 2G(A)\sigma : S^2 \to G(A) a preorientation. Then σ\sigma is an orientation if

  1. π 0GSpecπ 0A\pi_0 G \to \mathrm{Spec} \pi_0 A is smooth of relative dimension 1, and

  2. The map induced by β:ωπ 2A\beta : \omega \to \pi_2 A

    π nA π 0Aωπ n+2A \pi_n A \otimes_{\pi_0 A} \omega \to \pi_{n+2} A

    is an isomorphism for each nn.

Note that (2) implies that AA is weakly periodic. Conversely, if AA is weakly periodic then (2) is equivalent to β\beta being an isomorphism.

Before defining β\beta and ω\omega we extend the above definition to derived group schemes over an arbitrary derived scheme.

Definition Let XX be a derived scheme, GG a commutative derived group scheme over XX and σ:S 2G(X)\sigma: S^2 \to G(X) a preorientation. Then σ\sigma is an orientation if (G,σ) U(G, \sigma) |_U is an orientation for all UXU \subset X affine.

We now define the module ω\omega and the map β\beta. Let AA be an E E_\infty-ring, GG a commutative derived group scheme over SpecA\mathrm{Spec} A and let G 0=π 0GG_0 = \pi_0 G a scheme over Specπ 0A\mathrm{Spec} \pi_0 A. Let Ω\Omega denote the sheaf of differentials on G 0/π 0AG_0 / \pi_0 A. Then define

ω:=i *Ω,i:Specπ 0AG\omega := i^* \Omega, \; i: \mathrm{Spec} \pi_0 A \to G

is the identity section.

Now let UGU \hookrightarrow G be an open affine subscheme containing the identity section, so U=SpecBU = \mathrm{Spec} B for some E E_\infty-ring BB. Then σ\sigma induces a map BA S 2B \to A^{S^2} which is the same as a map

π 0BA(S 2)=π 0Aπ 2A. \pi_0 B \to A(S^2)= \pi_0 A \oplus \pi_2 A .

It is a fact that the map π 0Bπ 2A\pi_0 B \to \pi_2 A is a derivation over π 0A\pi_0 A and hence has a classifying map which yields a map

β:ωπ 2A.\beta : \omega \to \pi_2 A .

The Multiplicative Derived Group Scheme

The naive guess for G mG_m is GL 1GL_1, where GL 1(A)=A x=(π 0A) xGL_1 (A) = A^x = (\pi_0 A)^x. It is true that GL 1GL_1 is a derived scheme over SpecS\mathrm{Spec} \mathbf{S}, however it is not flat, nor is GL 1(A)GL_1 (A) an Abelian group as AA is an E E_\infty-ring and not an honestly commutative ring.

If AA is rational, that is there is a map HAH \mathbb{Q} \to A, then GL 1(A)GL_1 (A) can be given an Abelian group structure. Hence, GL 1GL_1 is a perfectly good group scheme defined on the category of rational E E_\infty-rings, however this category is too small; there are too few rational E E_\infty-rings.

Recall that for a ring RR, R x=Hom R(R[t,t 1],R)R^x = \mathrm{Hom}_R (R [ t, t^{-1} ] , R). Further, recall that for a group MM we can form the group algebra R[M]R[M] which is really a Hopf algebra. Then SpecR[M]\mathrm{Spec} R[M] is a group scheme over SpecA\mathrm{Spec} A. Further, R[m]R[m] is characterized by

Ring(R[M],B)=Ring(R,B)×Mon(M,B) \mathrm{Ring} ( R[M] , B) = \mathrm{Ring} (R, B) \times \mathrm{Mon} (M,B)

where Mon\mathrm{Mon} is the category of monoids and the ring BB is thought of as a monoid wrt to multiplication. Motivated by these observations we make the following definitions.

Definition Let AA be an E E_\infty-ring and MM a topological Abelian monoid, then we can define A[M]A[M] which is characterized by

Alg A(A[M],B)Alg A(A,B)×TopMon(M,B ×). \mathrm{Alg}_A (A[M], B) \simeq \mathrm{Alg}_A (A,B) \times \mathrm{TopMon} (M, B^\times ).

Recall that because of the higher categorical nature of things, the hom-sets above are spaces and the symbol \simeq indicates weak equivalence of spaces.

Definition Let AA be an E E_\infty-ring. We define the multiplicative group corresponding to AA as

G m=SpecA[]. G_m = \mathrm{Spec} A[ \mathbb{Z}].

G mG_m is a derived commutative group scheme over SpecA\mathrm{Spec} A.

Note that π *(A[])=(π *A)[]\pi_* ( A[ \mathbb{Z}]) = (\pi_* A) [ \mathbb{Z}]. Also, the map π 0G mπ 0SpecA\pi_0 G_m \to \pi_0 \mathrm{Spec} A is smooth of relative dimension 1.

Preorientations of G mG_m

Proposition For any E E_\infty-ring AA, we have a bijection (of sets) between preorientations of G mG_m and maps S[P ]A\mathbf{S} [ \mathbb{C} P^\infty ] \to A.

The proof follows from the fact that S\mathbf{S} is initial in the category of E E_\infty-rings and the mapping property of A[]A [ \mathbb{Z}].

Corollary SpecS[P ]\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ] is the moduli space of preorientations of G mG_m. That is, if G mG_m is defined over SpecA\mathrm{Spec} A, then a preorientation of G mG_m is the same as a map SpecASpecS[P ]\mathrm{Spec} A \to \mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ].

Orientations of G mG_m

We consider the map β:ωπ 2A\beta : \omega \to \pi_2 A where

ω=i *Ω π 0G/π 0SpecA 1 \omega = i^* \Omega^1_{\pi_0 G / \pi_0 \mathrm{Spec} A}

and ii is the identity section. Note that π 0G m=(π 0A)[t,t 1]\pi_0 G_m = (\pi_0 A ) [t, t^{-1}], hence it follows that ω\omega is canonically trivial, so an orientation is just an element β σπ 2A\beta_\sigma \in \pi_2 A such that β σ\beta_\sigma is invertible in π *A\pi_* A.

Let β\beta denote the (universal) orientation of S[P ]\mathbf{S} [ \mathbb{C} P^\infty]. Then we have the following.

Theorem SpecS[P ][β 1]\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty] [ \beta^{-1}] is the moduli space of orientations of G mG_m.

It is a theorem of Snaith, that this moduli space has the homotopy type of KUKU the spectrum of complex K-theory. Note that by considering the homtopy fixed points of a certain action there is a way to recover KOKO as well.

Connection to complex orientation

Let AA be an E E_\infty-ring, so in particular AA defines a cohomology theory. An orientation of G mG_m over SpecA\mathrm{Spec} A is a map KUAKU \to A. A complex orientation of AA is a map MUAMU \to A. Recalling that KUKU is complex oriented, we see that an orientation of G mG_m gives a complex orientation by precomposing with the map MUKUMU \to KU.

The Additive Derived Group Scheme

The naive definition of G aG_a is A 1\mathbf{A}^1, where A 1(A)\mathbf{A}^1 (A) is the additive group of AA. It is true that A 1\mathbf{A}^1 is a derived scheme over SpecS\mathrm{Spec} \mathbf{S}, however it is not flat as for an E E_\infty-ring AA

π kA A 1= n0A k(BΣ n) \pi_k \mathbf{A}^1_A = \oplus_{n \ge 0} A^{-k} (B \Sigma_n )

where as if it were flat we would have

π kA A 1=π kA[x]. \pi_k \mathbf{A}^1_A = \pi_k A [x] .

Also, A 1\mathbf{A}^1 is not commutative. A 1(A)\mathbf{A}^1 (A) is an infinite loop space, but not an Abelian monoid. Again A 1\mathbf{A}^1 is a derived group scheme when restricted to rational E E_\infty-rings.

We no restrict to the category of integral E E_\infty-rings, i.e. those equipped with a map HAH \mathbb{Z} \to A. Note that in this category HH \mathbb{Z} is initial.

Definition For AA an integral E E_\infty-ring define

G a A=Spec(A [x]). G^A_a = \mathrm{Spec} ( A \otimes_\mathbb{Z} \mathbb{Z} [x] ).

It can be shown that G a AG^A_a is flat and has the correct amount of commutativity.

Why can’t we just use SpecA[]\mathrm{Spec} A [ \mathbb{N}]?

Proposition For all integral E E_\infty-rings, preorientations of G a AG_a^A are in bijective correspondence with maps H[P ]AH \mathbb{Z} [ \mathbb{C} P^\infty] \to A. Consequently, SpecH[P ]\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty] is the moduli space of preorientations of G aG_a.

Now, π *H[P ]=H *(P ,)\pi_* H \mathbb{Z} [ \mathbb{C} P^\infty] = H_* (\mathbb{C} P^\infty , \mathbb{Z} ). The right side is a free divided power series on a generator β\beta where βπ 2H[P ]\beta \in \pi_2 H \mathbb{Z} [ \mathbb{C} P^\infty].

Proposition SpecH[P ][β 1]\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}] is the moduli space of orientations of G aG_a.

Proposition SpecH[P ][β 1]=KU\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}] = KU \otimes \mathbb{Q}. Hence the Chern character yields an isomorphism with rational periodic cohomology.

Revised on December 16, 2009 21:40:29 by Toby Bartels (173.60.119.197)