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Abstract This entry discusses the descent spectral sequence and sheaves in homotopy theory. Using said spectral sequence we compute .
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 be a derived Deligne-Mumford stack. Then there is a spectral sequence
Recalling what is what
Let be an -topos, heuristically is ”sheaves of spaces on an -category .” Further is a functor , which for a cover of formally assigns
Via DAG V 2.2.1 we can make sense of global sections and is an -ring.
Given an -category we can form the subcategory of truncated objects which consists of all objects such that all mapping spaces have trivial homotopy groups above level . Further defines a functor which serves the role of the Postnikov decomposition.
Let be an -topos, define . Further define functors by
For an abelian group object there exists , such that
corresponds to sections of along the identity of .
If is an ordinary site, corresponds to ordinary sheaf cohomology (HTT 22.214.171.124).
The non-derived descent ss
Let us define a mapping space , this is the hom-set as simplicial objects. Now
where . We have a homotopy cofiber sequence
and it is a fact that
for the fibered product corresponding to the cover of an object of the etale site of .
Applying to the cofiber sequence we obtain an exact couple and hence a spectral sequence with
Note that is the Čech complex of the cover, so the -page calculates Čech cohomology. If we choose an affine cover, hence acyclic and , then
Stacks and Hopf Algebroids
Let be a (non-derived) Deligne-Mumford stack on and let be a faithfully flat cover, then
for some commutative ring . Via the projection maps (which are both flat) we have a groupoid in , by definition it is a commutative Hopf algebroid .
Now let be a commutative Hopf algebroid, then the collection of principal bundles form a stack . Here a principal bundle is a map of schemes , a equivariant map , where the action is given by a map . In this we have an equivalence of 2-categories
Let be a scheme then a sheaf of abelian groups is a functor
The structure sheaf is defined by
Let be a sheaf of modules. is quasi-coherent if for any map and maps , we have
We have an equivalence of categories -mod via the assignment .
Now consider the stack from above. One can show that quasi-coherent sheaves over is nothing but a comodule, that is an -module, , and a coaction map of -modules
where the right hand side is an -module via the map .
Cohomology of Sheaves
Recall that sheaf cohomology is obtained by deriving the global sections functor. If is a noetherian scheme/stack then we restrict to deriving
Suppose further that , so lands in -modules, however from above we know -mod, hence is exact and all higher cohomology groups vanish.
Let be a quasi-coherent sheaf on a DM stack . Then global sections of induce global sections such that the two pullbacks to correspond to each other
That is the coaction map is well defined and 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 and if the is suppressed it is assumed that . In general we compute these Ext groups via the cobar complex?.
Change of Rings
Let be a commutative Hopf algebroid and a ring homomorphism. Define
so we have a map of Hopf algebroids and of stacks
Theorem. If there exists a ring and a homomorphism such that
is faithfully flat, then is an equivalence of stacks.
The Weierstrass Stack
Given an elliptic curve, Riemann–Roch gives us (locally on ) sections such that and is given by
for and . Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.
Two Weierstrass curves and are isomorphic if and only if they are related by a coordinate change of the form
For instance, this means that . We then build a Hopf algebroid by defining
Further, define the stacks and . Note that
Let (which is locally free) and . If is a Weierstrass curve, then is free with generator of degree 2
Let 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 .
With 6 inverted
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 . Now if 3 is invertible we complete the cube and have
and this curve is rigid. Define and , then
if and 0 otherwise.
Localized at 3
It is true that where
and the degree of is 4. We have the class and .
Let and consider the Hopf algebroid which by change of rings theorem is equivalent to . A spectral sequence obtained by filtering by powers of gives:
Theorem. subject to the following relations
By using the comparison map with the Adams-Novikov spectral sequence one can prove the following theorem.
Theorem. The edge homomorphism has
Cokernel given by for and not divisble by 3;
Kernel consisting a copy of in degrees 3,10,13,20,27,30,37,40 modulo 72. This is the 3-torsion in .
For more see
Tilman Bauer, Computation of the homotopy of the spectrum tmf. In Geom. Topol. Monogr., 13, 2008.