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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 ${\pi }_{*}{\mathrm{tmf}}_{(3)}$.

This is a sub-entry of

• A Survey of Elliptic Cohomology

see there for background and context.

Here are the entries on the previous sessions:

• A Survey of Elliptic Cohomology - cohomology theories

• A Survey of Elliptic Cohomology - formal groups and cohomology

• A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

• A Survey of Elliptic Cohomology - elliptic curves

• A Survey of Elliptic Cohomology - equivariant cohomology

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

• A Survey of Elliptic Cohomology - A-equivariant cohomology

• A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

• A Survey of Elliptic Cohomology - towards a proof

• A Survey of Elliptic Cohomology - compactifying the derived moduli stack

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The spectral sequence

We would like to understand the following theorem.

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

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

Recalling what is what

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

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

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

Given an $\mathrm{\infty }$-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 $\mathrm{\infty }$-topos, define $\mathrm{Disc}X:={\tau }_{\le 0}X$. Further define functors ${\pi }_n:X_{*}\to N(\mathrm{Disc}X)$ by

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

Facts.

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

   $$H^n(X,A):={\pi }_0\mathrm{Map}(1_X,K(A,n))$$H^n (X, A) := \pi_0 \mathrm{Map} (1_X , K(A,n))

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

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

The non-derived descent ss

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

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

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

$$F_n\to {\mathrm{Tot}}^{n}X\to {\mathrm{Tot}}^{n-1}X$$F_n \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X

and it is a fact that

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

for the fibered product $U_I$ corresponding to the cover $\{U_i\to N\}$ of an object $N$ of the etale site of $M_{\mathrm{1,1}}$.

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

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

$$E_{t,s}^2\Rightarrow H^s(N,{\pi }_{t}O).$$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 $\mathrm{Spec}A\to X$ be a faithfully flat cover, then

$$\mathrm{Spec}A{\times }_X\mathrm{Spec}A=\mathrm{Spec}\Gamma ,$$\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 Hopf algebroid $(A,\Gamma )$.

Now let $(A,\Gamma )$ be a 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 $\mathrm{Spec}\Gamma$ equivariant map $P\to \mathrm{Spec}A$, where the action is given by a map $P{\times }_{\mathrm{Spec}A}\mathrm{Spec}\Gamma \to P$. In this we have an equivalence of 2-categories

$$\{\mathrm{DM}\mathrm{Stacks}\}\simeq \{\mathrm{Hopf}\mathrm{Algebroids},\mathrm{bibundles}\}.$$\{DM \; Stacks\} \simeq \{Hopf \; Algebroids, \; bibundles\} .

and

$$\{\mathrm{DM}\mathrm{stacks}\mathrm{equipped}\mathrm{with}\mathrm{cover}\}\simeq \{\mathrm{Hopf}\mathrm{algebroids},\mathrm{functors}\mathrm{of}\mathrm{groupoids}\}.$$\{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

$$O_X(\mathrm{Spec}A\to X)=A.$$\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 $\mathrm{Spec}B\to \mathrm{Spec}A$ and maps $f:\mathrm{Spec}A\to X$, $g:\mathrm{Spec}B\to X$ we have

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

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

Now consider the stack $M_{A,\Gamma }$ from above. One can show that quasi-coherent sheaves over $M_{A,\Gamma }$ is nothing but a $(A,\Gamma )$ 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

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

Suppose further that $X=\mathrm{Spec}A$, so $\Gamma$ lands in $A$-modules, however from above we know $\mathrm{QCSh}/\mathrm{Spec}A\simeq A$-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

$$\Gamma {\otimes }_A^{d_0}N\to \Gamma {\otimes }_A^{d_1}N;1\otimes n\mapsto 1\otimes n.$$\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 $n\mapsto 1\otimes n$ is well defined and $n:A\to N;1\mapsto n$ is a map of comodules. This allows us to interpret global sections as

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

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

To simplify notation we write the above as $H^n(A,\Gamma ;N)$ 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 $(A,\Gamma )$ be a 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 $f_{*}:(A,\Gamma )\to (B,{\Gamma }_B)$ 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 $x\in \Gamma (C,O(2e)),y\in \Gamma (C,O(3e))$ such that $x^3-y^2\in \Gamma (C,O(5e))$ and $C\simeq C_{\underline{a}}\subset {\mathbb{P}}^2$ 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 $e=[0:1:0]$. Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.

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

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

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

$$A=\mathbb{Z}[a_1,\dots ,a_4,a_6],\Gamma =A[r,s,t,{\lambda }^{\pm }].$$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 $M_{\mathrm{ell}}=M_{A[{\Delta }^{-1}],\Gamma [{\Delta }^{-1}]}$. 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

$$A_{*}=A[{\eta }^{\pm }]\to \Gamma [{\eta }^{\pm }]={\Gamma }_{*};\eta \mapsto {\lambda }^{-1}\eta .$$A_* = A [ \eta^\pm ] \to \Gamma [\eta^\pm ] = \Gamma_* ; \; \eta \mapsto \lambda^{-1} \eta .

It is classical that

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

$${\pi }_{*}\mathrm{tmf}\to \{\mathrm{modular}\mathrm{forms}\}$$\pi_* tmf \to \{modular \; forms \}

as the edge homomorphism of our spectral sequence

$$H^{s,t}(A,\Gamma ;A_{*})\simeq H^{s,t}(A_{*},{\Gamma }_{*})\Rightarrow {\pi }_{t-s}\mathrm{tmf}.$$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

$${\overline{y}}^2=x^3+1/4b_2{x}^2+1/2b_{4}x+1/4b_6$$\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 $x\mapsto x+r$. Now if 3 is invertible we complete the cube and have

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

and this curve is rigid. Define $C=\mathbb{Z}[c_4,c_6]$ and ${\Gamma }_C=C$, then

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

if $s=0$ and 0 otherwise.

Localized at 3

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

$$B={\mathbb{Z}}_{(3)}[b_2,b_4,b_6]\to {\Gamma }_B=B[r];b_2\mapsto b_2+12r$$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 $\alpha =[r]\in H^{\mathrm{1,4}}$ and $\beta =[-1/2(r^2\otimes r+r\otimes r^2)]\in H^{\mathrm{2,12}}$.

Let $I=(3,b_2,b_4)$ and consider the Hopf algebroid $(B/I,{\Gamma }_B/I)$ which by change of rings theorem is equivalent to $(F_3,F_3[r]/(r^3))$. A spectral sequence obtained by filtering by powers of $I$ gives:

Theorem. $H^{*,*}(B,{\Gamma }_B)=\mathbb{Z}[c_4,c_6,\Delta ,\alpha ,\beta ]$ 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 ${\pi }_{*}{\mathrm{tmf}}_{(3)}\to \{\mathrm{Modular}\mathrm{Forms}{\}}_{(3)}$ has

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

2. Kernel consisting a copy of $\mathbb{Z}/3\mathbb{Z}$ 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.