# nLab A Survey of Elliptic Cohomology - descent ss and coefficients

higher algebra

universal algebra

## Theorems

Abstract This entry discusses the descent spectral sequence and sheaves in homotopy theory. Using said spectral sequence we compute ${\pi }_{*}{\mathrm{tmf}}_{\left(3\right)}$.

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 $\left(X,O\right)$ be a derived Deligne-Mumford stack. Then there is a spectral sequence

${H}^{s}\left(X;{\pi }_{t}O\right)⇒{\pi }_{t-s}\Gamma \left(X,O\right).$H^s (X ; \pi_t \mathbf{O}) \Rightarrow \pi_{t-s} \Gamma (X , \mathbf{O}).

### Recalling what is what

Let $X$ be an $\infty$-topos, heuristically $X$ is ”sheaves of spaces on an $\infty$-category $C$.” Further $O$ is a functor $O:\left\{{E}_{\infty }{\right\}}^{\mathrm{op}}\to X$, which for a cover $U$ of $C$ formally assigns

$A↦\left(U↦\mathrm{Hom}\left(A,O\left(U\right)\right)\right).$A \mapsto (U \mapsto \mathrm{Hom} (A , \mathbf{O} (U))).

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

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

Let $X$ be an $\infty$-topos, define $\mathrm{Disc}\phantom{\rule{thickmathspace}{0ex}}X:={\tau }_{\le 0}X$. Further define functors ${\pi }_{n}:{X}_{*}\to N\left(\mathrm{Disc}\phantom{\rule{thickmathspace}{0ex}}X\right)$ by

$Y↦{\tau }_{\le 0}\mathrm{Map}\left({S}^{n},Y\right)={\pi }_{n}\left(Y\right).$Y \mapsto \tau_{\le 0} \mathrm{Map} (S^n , Y) = \pi_n (Y) .

Facts.

1. For $A\in \mathrm{Disc}\phantom{\rule{thickmathspace}{0ex}}X$ an abelian group object there exists $K\left(A,n\right)\in X$, such that

${H}^{n}\left(X,A\right):={\pi }_{0}\mathrm{Map}\left({1}_{X},K\left(A,n\right)\right)$H^n (X, A) := \pi_0 \mathrm{Map} (1_X , K(A,n))

corresponds to sections of $K\left(A,n\right)$ along the identity of $C$.

2. If $C$ is an ordinary site, ${H}^{n}\left(X,A\right)$ corresponds to ordinary sheaf cohomology (HTT 7.2.2.17).

### The non-derived descent ss

Let us define a mapping space $\mathrm{Tot}\phantom{\rule{thickmathspace}{0ex}}X={\mathrm{hom}}^{\Delta }\left(\Delta ,X\right)$, this is the hom-set as simplicial objects. Now

$\mathrm{Tot}\phantom{\rule{thickmathspace}{0ex}}X=\mathrm{lim}\left(\dots \to {\mathrm{Tot}}^{n}\phantom{\rule{thickmathspace}{0ex}}X\to {\mathrm{Tot}}^{n-1}\phantom{\rule{thickmathspace}{0ex}}X\to \dots \to {\mathrm{Tot}}^{0}\to *\right),$\mathrm{Tot} \; X = \lim ( \dots \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X \to \dots \to \mathrm{Tot}^0 \to * ) ,

where ${\mathrm{Tot}}^{n}\phantom{\rule{thickmathspace}{0ex}}X=\mathrm{Tot}\left({\mathrm{cosk}}_{n}X\right)$. We have a homotopy cofiber sequence

${F}_{n}\to {\mathrm{Tot}}^{n}\phantom{\rule{thickmathspace}{0ex}}X\to {\mathrm{Tot}}^{n-1}\phantom{\rule{thickmathspace}{0ex}}X$F_n \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X

and it is a fact that

${F}_{s}\simeq {\Omega }^{s}\left(\prod _{\mid I\mid =s+1}O\left({U}_{I}\right)\right),$F_s \simeq \Omega^s ( \prod_{|I|=s+1} \mathbf{O} (U_I )),

for the fibered product ${U}_{I}$ corresponding to the cover $\left\{{U}_{i}\to N\right\}$ of an object $N$ of the etale site of ${M}_{1,1}$.

Applying ${\pi }_{*}$ to the cofiber sequence we obtain an exact couple and hence a spectral sequence with

${E}_{t,s}^{1}={\pi }_{t-s}{F}_{s}⇒\underset{n}{\mathrm{lim}}{\pi }_{t-s}{\mathrm{Tot}}^{n}X={\pi }_{t-s}\mathrm{Tot}X={\pi }_{t-s}O\left(N\right).$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 ${\pi }_{t-s}{F}_{s}$ is the Čech complex of the cover, so the ${E}^{2}$-page calculates Čech cohomology. If we choose an affine cover, hence acyclic and ${\mathrm{lim}}^{1}=0$, then

${E}_{t,s}^{2}⇒{H}^{s}\left(N,{\pi }_{t}O\right).$E^2_{t,s} \Rightarrow H^s (N, \pi_t \mathbf{O}) .

## Stacks and Hopf Algebroids

Let $X$ be a (non-derived) Deligne-Mumford stack on $\mathrm{Aff}$ and let $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A\to X$ be a faithfully flat cover, then

$\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A{×}_{X}\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A=\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}\Gamma ,$\mathrm{Spec} \; A \times_X \mathrm{Spec} \; A = \mathrm{Spec} \; \Gamma,

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

Now let $\left(A,\Gamma \right)$ be a commutative Hopf algebroid, then the collection of principal bundles form a stack ${M}_{A,\Gamma }$. Here a principal bundle is a map of schemes $P\to X$, a $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}\Gamma$ equivariant map $P\to \mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A$, where the action is given by a map $P{×}_{\mathrm{Spec}A}\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}\Gamma \to P$. In this we have an equivalence of 2-categories

$\left\{\mathrm{DM}\phantom{\rule{thickmathspace}{0ex}}\mathrm{Stacks}\right\}\simeq \left\{\mathrm{Hopf}\phantom{\rule{thickmathspace}{0ex}}\mathrm{Algebroids},\phantom{\rule{thickmathspace}{0ex}}\mathrm{bibundles}\right\}.$\{DM \; Stacks\} \simeq \{Hopf \; Algebroids, \; bibundles\} .

and

$\left\{\mathrm{DM}\phantom{\rule{thickmathspace}{0ex}}\mathrm{stacks}\phantom{\rule{thickmathspace}{0ex}}\mathrm{equipped}\phantom{\rule{thickmathspace}{0ex}}\mathrm{with}\phantom{\rule{thickmathspace}{0ex}}\mathrm{cover}\right\}\simeq \left\{\mathrm{Hopf}\phantom{\rule{thickmathspace}{0ex}}\mathrm{algebroids},\phantom{\rule{thickmathspace}{0ex}}\mathrm{functors}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}\mathrm{groupoids}\right\}.$\{DM \; stacks \; equipped \; with \; cover\} \simeq \{Hopf \; algebroids, \; functors \; of \; groupoids\}.

Let $X$ be a scheme then a sheaf of abelian groups is a functor

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

The structure sheaf ${O}_{X}$ is defined by

${O}_{X}\left(\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A\to X\right)=A.$\mathbf{O}_X ( \mathrm{Spec} \; A \to X) = A .

Let $\Im$ be a sheaf of ${O}_{X}$ modules. $\Im$ is quasi-coherent if for any map $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}B\to \mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A$ and maps $f:\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A\to X$, $g:\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}B\to X$ we have

$B{\otimes }_{A}\Im \left(f\right)\simeq \Im \left(g\right).$B \otimes_{A} \mathfrak{I} (f) \simeq \mathfrak{I} (g) .

We have an equivalence of categories $\mathrm{QCSh}/\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A\simeq A$-mod via the assignment $\Im ↦\Im \left({1}_{A}\right)$.

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

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

where the right hand side is an $A$-module via the map ${d}_{0}$.

## Cohomology of Sheaves

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

$\Gamma \left(-\right):\mathrm{QCSh}/X\to \mathrm{Ab}.$\Gamma (-) : \mathrm{QCSh}/X \to \mathrm{Ab} .

Suppose further that $X=\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A$, so $\Gamma$ lands in $A$-modules, however from above we know $\mathrm{QCSh}/\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}A\simeq A$-mod, hence $\Gamma$ is exact and all higher cohomology groups vanish.

Let ${\Im }_{N}$ be a quasi-coherent sheaf on a DM stack ${M}_{A,\Gamma }$. Then global sections of ${\Im }_{N}$ induce global sections $n\in N$ such that the two pullbacks to $\Gamma$ correspond to each other

$\Gamma {\otimes }_{A}^{{d}_{0}}N\to \Gamma {\otimes }_{A}^{{d}_{1}}N;\phantom{\rule{thickmathspace}{0ex}}1\otimes n↦1\otimes n.$\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 $n↦1\otimes n$ is well defined and $n:A\to N;\phantom{\rule{thickmathspace}{0ex}}1↦n$ is a map of comodules. This allows us to interpret global sections as

${\mathrm{Hom}}_{A,\Gamma }\left(A,-\right):{\mathrm{Comod}}_{A,\Gamma }\to A-\mathrm{mod},$\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

${H}^{n}\left({M}_{A,\Gamma },{\Im }_{N}\right)={\mathrm{Ext}}_{A,\Gamma }^{n}\left(A,N\right).$H^n ( M_{A,\Gamma} , \mathfrak{I}_N ) = \mathrm{Ext}^n_{A,\Gamma} (A,N) .

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

### Change of Rings

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

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

so we have a map of Hopf algebroids ${f}_{*}:\left(A,\Gamma \right)\to \left(B,{\Gamma }_{B}\right)$ and of stacks

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

Theorem. If there exists a ring $R$ and a homomorphism $\Gamma {\otimes }_{A}B\to R$ such that

$A\to \Gamma {\otimes }_{A}B\to R$A \to \Gamma \otimes_A B \to R

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

### The Weierstrass Stack

Given $C/S$ an elliptic curve, Riemann–Roch gives us (locally on $S$) sections $x\in \Gamma \left(C,O\left(2e\right)\right),\phantom{\rule{thickmathspace}{0ex}}y\in \Gamma \left(C,O\left(3e\right)\right)$ such that ${x}^{3}-{y}^{2}\in \Gamma \left(C,O\left(5e\right)\right)$ and $C\simeq {C}_{\underline{a}}\subset {ℙ}^{2}$ is given by

${y}^{2}+{a}_{1}\mathrm{xy}+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}X+{a}_{6}$y^2 + a_1 xy + a_3 y = x^3 +a_2 x^2 +a_4 X + a_6

for ${a}_{i}\in {O}_{S}$ and $e=\left[0:1:0\right]$. Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.

Two Weierstrass curves $\left({C}_{\underline{a}},e\right)$ and $\left({C}_{\underline{a}\prime },e\right)$ are isomorphic if and only if they are related by a coordinate change of the form

$\left(X,Y\right)↦\left({\lambda }^{-2}X+r,{\lambda }^{-3}y+s{\lambda }^{-2}x+t\right).$(X,Y) \mapsto (\lambda^{-2} X + r, \lambda^{-3} y + s \lambda^{-2} x +t ) .

For instance, this means that ${a}_{1}\prime =\lambda \left({a}_{1}+2s\right)$. We then build a Hopf algebroid $\left(A,\Gamma \right)$ by defining

$A=ℤ\left[{a}_{1},\dots ,{a}_{4},{a}_{6}\right],\phantom{\rule{thickmathspace}{0ex}}\Gamma =A\left[r,s,t,{\lambda }^{±}\right].$A = \mathbb{Z} [a_1 , \dots , a_4 , a_6 ] , \; \Gamma = A [r,s,t, \lambda^\pm ] .

Further, define the stacks ${M}_{\mathrm{Weir}}={M}_{A,\Gamma }$ and ${M}_{\mathrm{ell}}={M}_{A\left[{\Delta }^{-1}\right],\Gamma \left[{\Delta }^{-1}\right]}$. Note that

${M}_{\mathrm{ell}}\subset \overline{{M}_{\mathrm{ell}}}\subset {M}_{\mathrm{Weir}}.$M_{ell} \subset \overline{M_{ell}} \subset M_{Weir} .

Let ${\omega }_{C/S}={\pi }_{*}{\Omega }_{C/S}^{1}$ (which is locally free) and ${\pi }_{2n}O={\omega }^{n}$. If $C$ is a Weierstrass curve, then $\omega$ is free with generator of degree 2

$\eta =\frac{\mathrm{dx}}{2y+{a}_{1}x+{a}_{3}}.$\eta = \frac{dx}{2y + a_1 x +a_3} .

Let ${\omega }^{*}$ correspond to the graded comodule

${A}_{*}=A\left[{\eta }^{±}\right]\to \Gamma \left[{\eta }^{±}\right]={\Gamma }_{*};\phantom{\rule{thickmathspace}{0ex}}\eta ↦{\lambda }^{-1}\eta .$A_* = A [ \eta^\pm ] \to \Gamma [\eta^\pm ] = \Gamma_* ; \; \eta \mapsto \lambda^{-1} \eta .

It is classical that

${H}^{0,*}\left(A,\Gamma ;{A}_{*}\right)=ℤ\left[{c}_{4},{c}_{6},\Delta \right]/\left({12}^{3}\Delta -{c}_{4}^{3}+{c}_{6}^{2}\right)$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

${\pi }_{*}\mathrm{tmf}\to \left\{\mathrm{modular}\phantom{\rule{thickmathspace}{0ex}}\mathrm{forms}\right\}$\pi_* tmf \to \{modular \; forms \}

as the edge homomorphism of our spectral sequence

${H}^{s,t}\left(A,\Gamma ;{A}_{*}\right)\simeq {H}^{s,t}\left({A}_{*},{\Gamma }_{*}\right)⇒{\pi }_{t-s}\mathrm{tmf}.$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 $\mathrm{MU}$.

## $p$-local Coefficients

### With 6 inverted

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

${\overline{y}}^{2}={x}^{3}+1/4{b}_{2}{x}^{2}+1/2{b}_{4}x+1/4{b}_{6}$\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 $x↦x+r$. Now if 3 is invertible we complete the cube and have

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

and this curve is rigid. Define $C=ℤ\left[{c}_{4},{c}_{6}\right]$ and ${\Gamma }_{C}=C$, then

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

if $s=0$ and 0 otherwise.

### Localized at 3

It is true that ${H}^{s,t}\left({A}_{*},{\Gamma }_{*}\right)={H}^{s,t}\left(B,{\Gamma }_{B}\right)$ where

$B={ℤ}_{\left(3\right)}\left[{b}_{2},{b}_{4},{b}_{6}\right]\to {\Gamma }_{B}=B\left[r\right];\phantom{\rule{thickmathspace}{0ex}}{b}_{2}↦{b}_{2}+12r$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 $r$ is 4. We have the class $\alpha =\left[r\right]\in {H}^{1,4}$ and $\beta =\left[-1/2\left({r}^{2}\otimes r+r\otimes {r}^{2}\right)\right]\in {H}^{2,12}$.

Let $I=\left(3,{b}_{2},{b}_{4}\right)$ and consider the Hopf algebroid $\left(B/I,{\Gamma }_{B}/I\right)$ which by change of rings theorem is equivalent to $\left({F}_{3},{F}_{3}\left[r\right]/\left({r}^{3}\right)\right)$. A spectral sequence obtained by filtering by powers of $I$ gives:

Theorem. ${H}^{*,*}\left(B,{\Gamma }_{B}\right)=ℤ\left[{c}_{4},{c}_{6},\Delta ,\alpha ,\beta \right]$ subject to the following relations

1. ${12}^{3}\Delta -{c}_{4}^{3}+{c}_{6}^{2}={\alpha }^{2}=3\alpha =3\beta =0;$

2. ${c}_{4}\alpha ={c}_{6}\alpha ={c}_{4}\beta ={c}_{6}\beta =0.$

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

Theorem. The edge homomorphism ${\pi }_{*}{\mathrm{tmf}}_{\left(3\right)}\to \left\{\mathrm{Modular}\mathrm{Forms}{\right\}}_{\left(3\right)}$ has

1. Cokernel given by $ℤ/3ℤ\left[{\Delta }^{n}\right]$ for $n\ge 0$ and not divisble by 3;

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

For more see

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

Revised on November 25, 2013 06:13:31 by Zoran Škoda (161.53.130.104)