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A Survey of Elliptic Cohomology - descent ss and coefficients

Abstract This entry discusses the descent spectral sequence and sheaves in homotopy theory. Using said spectral sequence we compute π *tmf (3)\pi_* tmf_{(3)}.

This is a sub-entry of

see there for background and context.

Here are the entries on the previous sessions:


The Descent Spectral Sequence

The spectral sequence

We would like to understand the following theorem.

Theorem. Let (X,O)( X, \mathbf{O}) be a derived Deligne-Mumford stack. Then there is a spectral sequence

H s(X;π tO)π tsΓ(X,O).H^s (X ; \pi_t \mathbf{O}) \Rightarrow \pi_{t-s} \Gamma (X , \mathbf{O}).

Recalling what is what

Let XX be an \infty-topos, heuristically XX is ”sheaves of spaces on an \infty-category CC.” Further O\mathbf{O} is a functor O:{E } opX\mathbf{O} : \{ E_\infty \}^{op} \to X, which for a cover UU of CC formally assigns

A(UHom(A,O(U))).A \mapsto (U \mapsto \mathrm{Hom} (A , \mathbf{O} (U))).

Via DAG V 2.2.1 we can make sense of global sections and Γ(X,O)\Gamma (X , \mathbf{O}) is an E E_\infty-ring.

Given an \infty-category CC we can form the subcategory of nn truncated objects τ nC\tau_{\le n} C which consists of all objects such that all mapping spaces have trivial homotopy groups above level nn. Further τ n:CC\tau_{\le n} : C \to C defines a functor which serves the role of the Postnikov decomposition.

Let XX be an \infty-topos, define DiscX:=τ 0X\mathrm{Disc} \; X : = \tau_{\le 0} X. Further define functors π n:X *N(DiscX)\pi_n : X_* \to N (\mathrm{Disc} \; X) by

Yτ 0Map(S n,Y)=π n(Y).Y \mapsto \tau_{\le 0} \mathrm{Map} (S^n , Y) = \pi_n (Y) .

Facts.

  1. For ADiscXA \in \mathrm{Disc} \; X an abelian group object there exists K(A,n)XK(A,n) \in X, such that

    H n(X,A):=π 0Map(1 X,K(A,n)) H^n (X, A) := \pi_0 \mathrm{Map} (1_X , K(A,n))

    corresponds to sections of K(A,n)K(A,n) along the identity of CC.

  2. If CC is an ordinary site, H n(X,A)H^n (X, A) corresponds to ordinary sheaf cohomology (HTT 7.2.2.17).

The non-derived descent ss

Let us define a mapping space TotX=hom Δ(Δ,X)\mathrm{Tot} \; X = \mathrm{hom}^\Delta (\Delta , X), this is the hom-set as simplicial objects. Now

TotX=lim(Tot nXTot n1XTot 0*),\mathrm{Tot} \; X = \lim ( \dots \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X \to \dots \to \mathrm{Tot}^0 \to * ) ,

where Tot nX=Tot(cosk nX)\mathrm{Tot}^n \; X = \mathrm{Tot} (\mathrm{cosk}_n X ). We have a homotopy cofiber sequence

F nTot nXTot n1XF_n \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X

and it is a fact that

F sΩ s( I=s+1O(U I)),F_s \simeq \Omega^s ( \prod_{|I|=s+1} \mathbf{O} (U_I )),

for the fibered product U IU_I corresponding to the cover {U iN}\{ U_i \to N\} of an object NN of the etale site of M 1,1M_{1,1}.

Applying π *\pi_* to the cofiber sequence we obtain an exact couple and hence a spectral sequence with

E t,s 1=π tsF slim nπ tsTot nX=π tsTotX=π tsO(N). E^1_{t,s} = \pi_{t-s} F_s \Rightarrow \lim_n \pi_{t-s} \mathrm{Tot}^n X = \pi_{t-s} \mathrm{Tot} X = \pi_{t-s} \mathbf{O} (N).

Note that π tsF s\pi_{t-s} F_s is the Čech complex of the cover, so the E 2E^2-page calculates Čech cohomology. If we choose an affine cover, hence acyclic and lim 1=0\lim^1 =0, then

E t,s 2H s(N,π tO).E^2_{t,s} \Rightarrow H^s (N, \pi_t \mathbf{O}) .

Stacks and Hopf Algebroids

Let XX be a (non-derived) Deligne-Mumford stack on Aff\mathrm{Aff} and let SpecAX\mathrm{Spec} \; A \to X be a faithfully flat cover, then

SpecA× XSpecA=SpecΓ, \mathrm{Spec} \; A \times_X \mathrm{Spec} \; A = \mathrm{Spec} \; \Gamma,

for some commutative ring Γ\Gamma. Via the projection maps (which are both flat) we have a groupoid in Aff\mathrm{Aff}, by definition it is a commutative Hopf algebroid (A,Γ)(A, \Gamma).

Now let (A,Γ)(A,\Gamma) be a commutative Hopf algebroid, then the collection of principal bundles form a stack M A,ΓM_{A,\Gamma}. Here a principal bundle is a map of schemes PXP \to X, a SpecΓ\mathrm{Spec} \; \Gamma equivariant map PSpecAP \to \mathrm{Spec} \; A, where the action is given by a map P× SpecASpecΓPP \times_{\mathrm{Spec} A} \mathrm{Spec} \; \Gamma \to P. In this we have an equivalence of 2-categories

{DMStacks}{HopfAlgebroids,bibundles}. \{DM \; Stacks\} \simeq \{Hopf \; Algebroids, \; bibundles\} .

and

{DMstacksequippedwithcover}{Hopfalgebroids,functorsofgroupoids}. \{DM \; stacks \; equipped \; with \; cover\} \simeq \{Hopf \; algebroids, \; functors \; of \; groupoids\}.

Let XX be a scheme then a sheaf of abelian groups is a functor

:Aff/X opAb.\mathfrak{I} : \mathrm{Aff}/X^{op} \to \mathrm{Ab} .

The structure sheaf O X\mathbf{O}_X is defined by

O X(SpecAX)=A. \mathbf{O}_X ( \mathrm{Spec} \; A \to X) = A .

Let \mathfrak{I} be a sheaf of O X\mathbf{O}_X modules. \mathfrak{I} is quasi-coherent if for any map SpecBSpecA\mathrm{Spec} \; B \to \mathrm{Spec} \; A and maps f:SpecAXf: \mathrm{Spec} \; A \to X, g:SpecBXg: \mathrm{Spec} \; B \to X we have

B A(f)(g). B \otimes_{A} \mathfrak{I} (f) \simeq \mathfrak{I} (g) .

We have an equivalence of categories QCSh/SpecAA\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A-mod via the assignment (1 A)\mathfrak{I} \mapsto \mathfrak{I} (1_A).

Now consider the stack M A,ΓM_{A,\Gamma} from above. One can show that quasi-coherent sheaves over M A,ΓM_{A, \Gamma} is nothing but a (A,Γ)(A,\Gamma) comodule, that is an AA-module, MM, and a coaction map of AA-modules

MΓ A d 1MM \to \Gamma \otimes^{d_1}_{A} M

where the right hand side is an AA-module via the map d 0d_0.

Cohomology of Sheaves

Recall that sheaf cohomology is obtained by deriving the global sections functor. If XX is a noetherian scheme/stack then we restrict to deriving

Γ():QCSh/XAb. \Gamma (-) : \mathrm{QCSh}/X \to \mathrm{Ab} .

Suppose further that X=SpecAX = \mathrm{Spec} \; A, so Γ\Gamma lands in AA-modules, however from above we know QCSh/SpecAA\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A-mod, hence Γ\Gamma is exact and all higher cohomology groups vanish.

Let N\mathfrak{I}_N be a quasi-coherent sheaf on a DM stack M A,ΓM_{A,\Gamma}. Then global sections of N\mathfrak{I}_N induce global sections nNn \in N such that the two pullbacks to Γ\Gamma correspond to each other

Γ A d 0NΓ A d 1N;1n1n. \Gamma \otimes_A^{d_0} N \to \Gamma \otimes_A^{d_1} N; \; 1 \otimes n \mapsto 1 \otimes n .

That is the coaction map n1nn \mapsto 1 \otimes n is well defined and n:AN;1nn: A \to N; \; 1 \mapsto n is a map of comodules. This allows us to interpret global sections as

Hom A,Γ(A,):Comod A,ΓAmod, \mathrm{Hom}_{A,\Gamma} (A, -) : \mathrm{Comod}_{A,\Gamma} \to A-\mathrm{mod} ,

so a section is a map from the trivial sheaf to the given sheaf. It follows that

H n(M A,Γ, N)=Ext A,Γ n(A,N). H^n ( M_{A,\Gamma} , \mathfrak{I}_N ) = \mathrm{Ext}^n_{A,\Gamma} (A,N) .

To simplify notation we write the above as H n(A,Γ;N)H^n (A, \Gamma ; N) and if the NN is suppressed it is assumed that N=AN=A. In general we compute these Ext groups via the cobar complex?.

Change of Rings

Let (A,Γ)(A,\Gamma) be a commutative Hopf algebroid and f:ABf: A \to B a ring homomorphism. Define

Γ B=B A d 0Γ A d 1B, \Gamma_B = B \otimes_A^{d_0} \Gamma \otimes_A^{d_1} B ,

so we have a map of Hopf algebroids f *:(A,Γ)(B,Γ B)f_* : (A, \Gamma) \to (B, \Gamma_B) and of stacks

f *:M B,Γ BM A,Γ. f^* : M_{B,\Gamma_B} \to M_{A,\Gamma} .

Theorem. If there exists a ring RR and a homomorphism Γ ABR\Gamma \otimes_A B \to R such that

AΓ ABRA \to \Gamma \otimes_A B \to R

is faithfully flat, then f *f^* is an equivalence of stacks.

The Weierstrass Stack

Given C/SC/S an elliptic curve, Riemann–Roch gives us (locally on SS) sections xΓ(C,O(2e)),yΓ(C,O(3e))x \in \Gamma (C, \mathbf{O} (2e)) , \; y \in \Gamma (C, \mathbf{O} (3e)) such that x 3y 2Γ(C,O(5e))x^3-y^2 \in \Gamma (C, \mathbf{O}(5e)) and CC a̲ 2C \simeq C_{\underline{a}} \subset \mathbb{P}^2 is given by

y 2+a 1xy+a 3y=x 3+a 2x 2+a 4X+a 6 y^2 + a_1 xy + a_3 y = x^3 +a_2 x^2 +a_4 X + a_6

for a iO Sa_i \in \mathbf{O}_S and e=[0:1:0]e = [0: 1:0]. Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.

Two Weierstrass curves (C a̲,e)(C_\underline{a} , e) and (C a̲,e)(C_{\underline{a}'},e) are isomorphic if and only if they are related by a coordinate change of the form

(X,Y)(λ 2X+r,λ 3y+sλ 2x+t). (X,Y) \mapsto (\lambda^{-2} X + r, \lambda^{-3} y + s \lambda^{-2} x +t ) .

For instance, this means that a 1=λ(a 1+2s)a_1'= \lambda (a_1 +2s). We then build a Hopf algebroid (A,Γ)(A, \Gamma) by defining

A=[a 1,,a 4,a 6],Γ=A[r,s,t,λ ±]. A = \mathbb{Z} [a_1 , \dots , a_4 , a_6 ] , \; \Gamma = A [r,s,t, \lambda^\pm ] .

Further, define the stacks M Weir=M A,ΓM_{Weir} = M_{A, \Gamma} and M ell=M A[Δ 1],Γ[Δ 1]M_{ell} = M_{A[\Delta^{-1}] , \Gamma[\Delta^{-1}]}. Note that

M ellM ell¯M Weir. M_{ell} \subset \overline{M_{ell}} \subset M_{Weir} .

Let ω C/S=π *Ω C/S 1\omega_{C/S} = \pi_* \Omega^1_{C/S} (which is locally free) and π 2nO=ω n\pi_{2n} \mathbf{O} = \omega^n. If CC is a Weierstrass curve, then ω\omega is free with generator of degree 2

η=dx2y+a 1x+a 3.\eta = \frac{dx}{2y + a_1 x +a_3} .

Let ω *\omega^* correspond to the graded comodule

A *=A[η ±]Γ[η ±]=Γ *;ηλ 1η. A_* = A [ \eta^\pm ] \to \Gamma [\eta^\pm ] = \Gamma_* ; \; \eta \mapsto \lambda^{-1} \eta .

It is classical that

H 0,*(A,Γ;A *)=[c 4,c 6,Δ]/(12 3Δc 4 3+c 6 2) H^{0,*} (A, \Gamma ; A_*) = \mathbb{Z} [c_4 , c_6 , \Delta]/ (12^3 \Delta -c_4^3 +c_6^2 )

that is the ring of modular forms. So we get a map

π *tmf{modularforms} \pi_* tmf \to \{modular \; forms \}

as the edge homomorphism of our spectral sequence

H s,t(A,Γ;A *)H s,t(A *,Γ *)π tstmf. H^{s,t} (A,\Gamma ; A_*) \simeq H^{s,t} (A_* , \Gamma_*) \Rightarrow \pi_{t-s} tmf .

It should be noted that we have a comparison map with the Adams-Novikov spectral sequence for MUMU.

pp-local Coefficients

With 6 inverted

Note that if 2 is invertible than we can complete the square in the Weierstrass equation to obtain

y¯ 2=x 3+1/4b 2x 2+1/2b 4x+1/4b 6 \overline{y}^2 = x^3 + 1/4 b_2 x^2 + 1/2 b_4 x + 1/4 b_6

and the only automorphisms of the curve are xx+rx \mapsto x+r. Now if 3 is invertible we complete the cube and have

y¯ 2=x¯ 31/48c 4x¯1/864c 6 \overline{y}^2 = \overline{x}^3 -1/48 c_4 \overline{x} -1/864 c_6

and this curve is rigid. Define C=[c 4,c 6]C= \mathbb{Z} [c_4 , c_6 ] and Γ C=C\Gamma_C = C, then

H s,*(A *,Γ *)[1/6]H s,*(C,C)[1/6]=[1/6,c 4,c 6] H^{s,*} (A_* , \Gamma_* ) [1/6] \simeq H^{s,*} (C,C) [1/6] = \mathbb{Z} [1/6 , c_4 , c_6 ]

if s=0s=0 and 0 otherwise.

Localized at 3

It is true that H s,t(A *,Γ *)=H s,t(B,Γ B)H^{s,t} (A_* , \Gamma_* ) = H^{s,t} (B, \Gamma_B) where

B= (3)[b 2,b 4,b 6]Γ B=B[r];b 2b 2+12r B = \mathbb{Z}_{(3)} [ b_2 , b_4, b_6 ] \to \Gamma_B = B[r] ; \; b_2 \mapsto b_2 + 12r

and the degree of rr is 4. We have the class α=[r]H 1,4\alpha = [r] \in H^{1,4} and β=[1/2(r 2r+rr 2)]H 2,12\beta = [ -1/2 (r^2 \otimes r + r \otimes r^2)] \in H^{2,12}.

Let I=(3,b 2,b 4)I = (3 , b_2 , b_4 ) and consider the Hopf algebroid (B/I,Γ B/I)(B/I , \Gamma_B /I) which by change of rings theorem is equivalent to (F 3,F 3[r]/(r 3))(\mathbf{F}_3 , \mathbf{F}_3 [r]/(r^3)). A spectral sequence obtained by filtering by powers of II gives:

Theorem. H *,*(B,Γ B)=[c 4,c 6,Δ,α,β]H^{*,*} (B, \Gamma_B) = \mathbb{Z} [c_4 , c_6 , \Delta , \alpha, \beta ] subject to the following relations

  1. 12 3Δc 4 3+c 6 2=α 2=3α=3β=0;12^3 \Delta -c_4^3 +c_6^2 = \alpha^2 = 3\alpha = 3 \beta =0;

  2. c 4α=c 6α=c 4β=c 6β=0.c_4 \alpha = c_6 \alpha = c_4 \beta = c_6 \beta = 0.

By using the comparison map with the Adams-Novikov spectral sequence one can prove the following theorem.

Theorem. The edge homomorphism π *tmf (3){ModularForms} (3)\pi_* tmf_{(3)} \to \{Modular Forms\}_{(3)} has

  1. Cokernel given by /3[Δ n]\mathbb{Z}/3\mathbb{Z} [\Delta^n] for n0n \ge 0 and not divisble by 3;

  2. Kernel consisting a copy of /3\mathbb{Z}/3\mathbb{Z} in degrees 3,10,13,20,27,30,37,40 modulo 72. This is the 3-torsion in π *tmf\pi_* tmf.

For more see

Tilman Bauer, Computation of the homotopy of the spectrum tmf. In Geom. Topol. Monogr., 13, 2008.

Revised on January 13, 2014 15:04:25 by Urs Schreiber (89.204.155.62)