symmetric monoidal (∞,1)-category of spectra
Abstract This entry discusses the descent spectral sequence and sheaves in homotopy theory. Using said spectral sequence we compute $\pi_* tmf_{(3)}$.
Here are the entries on the previous sessions:
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 - derived group schemes and (pre-)orientations
A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves
A Survey of Elliptic Cohomology - compactifying the derived moduli stack
We would like to understand the following theorem.
Theorem. Let $( X, \mathbf{O})$ be a derived Deligne-Mumford stack. Then there is a spectral sequence
Let $X$ be an $\infty$-topos, heuristically $X$ is ”sheaves of spaces on an $\infty$-category $C$.” Further $\mathbf{O}$ is a functor $\mathbf{O} : \{ E_\infty \}^{op} \to X$, which for a cover $U$ of $C$ formally assigns
Via DAG V 2.2.1 we can make sense of global sections and $\Gamma (X , \mathbf{O})$ 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 $\mathrm{Disc} \; X : = \tau_{\le 0} X$. Further define functors $\pi_n : X_* \to N (\mathrm{Disc} \; X)$ by
Facts.
For $A \in \mathrm{Disc} \; X$ an abelian group object there exists $K(A,n) \in X$, such that
corresponds to sections of $K(A,n)$ along the identity of $C$.
If $C$ is an ordinary site, $H^n (X, A)$ corresponds to ordinary sheaf cohomology (HTT 7.2.2.17).
Let us define a mapping space $\mathrm{Tot} \; X = \mathrm{hom}^\Delta (\Delta , X)$, this is the hom-set as simplicial objects. Now
where $\mathrm{Tot}^n \; X = \mathrm{Tot} (\mathrm{cosk}_n X )$. We have a homotopy cofiber sequence
and it is a fact that
for the fibered product $U_I$ corresponding to the cover $\{ U_i \to N\}$ 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
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
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
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 $(A, \Gamma)$.
Now let $(A,\Gamma)$ 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 $\mathrm{Spec} \; \Gamma$ 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
and
Let $X$ be a scheme then a sheaf of abelian groups is a functor
The structure sheaf $\mathbf{O}_X$ is defined by
Let $\mathfrak{I}$ be a sheaf of $\mathbf{O}_X$ modules. $\mathfrak{I}$ 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
We have an equivalence of categories $\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A$-mod via the assignment $\mathfrak{I} \mapsto \mathfrak{I} (1_A)$.
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
where the right hand side is an $A$-module via the map $d_0$.
Recall that sheaf cohomology is obtained by deriving the global sections functor. If $X$ is a noetherian scheme/stack then we restrict to deriving
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 $\mathfrak{I}_N$ be a quasi-coherent sheaf on a DM stack $M_{A,\Gamma}$. Then global sections of $\mathfrak{I}_N$ induce global sections $n \in N$ such that the two pullbacks to $\Gamma$ correspond to each other
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
so a section is a map from the trivial sheaf to the given sheaf. It follows that
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?.
Let $(A,\Gamma)$ be a commutative Hopf algebroid and $f: A \to B$ a ring homomorphism. Define
so we have a map of Hopf algebroids $f_* : (A, \Gamma) \to (B, \Gamma_B)$ and of stacks
Theorem. If there exists a ring $R$ and a homomorphism $\Gamma \otimes_A B \to R$ such that
is faithfully flat, then $f^*$ is an equivalence of stacks.
Given $C/S$ an elliptic curve, Riemann–Roch gives us (locally on $S$) sections $x \in \Gamma (C, \mathbf{O} (2e)) , \; y \in \Gamma (C, \mathbf{O} (3e))$ such that $x^3-y^2 \in \Gamma (C, \mathbf{O}(5e))$ and $C \simeq C_{\underline{a}} \subset \mathbb{P}^2$ is given by
for $a_i \in \mathbf{O}_S$ 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 $(C_{\underline{a}'},e)$ are isomorphic if and only if they are related by a coordinate change of the form
For instance, this means that $a_1'= \lambda (a_1 +2s)$. We then build a Hopf algebroid $(A, \Gamma)$ by defining
Further, define the stacks $M_{Weir} = M_{A, \Gamma}$ and $M_{ell} = M_{A[\Delta^{-1}] , \Gamma[\Delta^{-1}]}$. Note that
Let $\omega_{C/S} = \pi_* \Omega^1_{C/S}$ (which is locally free) and $\pi_{2n} \mathbf{O} = \omega^n$. If $C$ is a Weierstrass curve, then $\omega$ is free with generator of degree 2
Let $\omega^*$ correspond to the graded comodule
It is classical that
that is the ring of modular forms. So we get a map
as the edge homomorphism of our spectral sequence
It should be noted that we have a comparison map with the Adams-Novikov spectral sequence for $MU$.
Note that if 2 is invertible than we can complete the square in the Weierstrass equation to obtain
and the only automorphisms of the curve are $x \mapsto x+r$. Now if 3 is invertible we complete the cube and have
and this curve is rigid. Define $C= \mathbb{Z} [c_4 , c_6 ]$ and $\Gamma_C = C$, then
if $s=0$ and 0 otherwise.
It is true that $H^{s,t} (A_* , \Gamma_* ) = H^{s,t} (B, \Gamma_B)$ where
and the degree of $r$ is 4. We have the class $\alpha = [r] \in H^{1,4}$ and $\beta = [ -1/2 (r^2 \otimes r + r \otimes r^2)] \in H^{2,12}$.
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 $(\mathbf{F}_3 , \mathbf{F}_3 [r]/(r^3))$. 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
$12^3 \Delta -c_4^3 +c_6^2 = \alpha^2 = 3\alpha = 3 \beta =0;$
$c_4 \alpha = c_6 \alpha = c_4 \beta = c_6 \beta = 0.$
By using the comparison map with the Adams-Novikov spectral sequence one can prove the following theorem.
Theorem. The edge homomorphism $\pi_* tmf_{(3)} \to \{Modular Forms\}_{(3)}$ has
Cokernel given by $\mathbb{Z}/3\mathbb{Z} [\Delta^n]$ for $n \ge 0$ and not divisble by 3;
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 $\pi_* tmf$.
For more see
Tilman Bauer, Computation of the homotopy of the spectrum tmf. In Geom. Topol. Monogr., 13, 2008.