One of the greatest recent achievements in algebraic topology is the construction of the spectrum as the global section of a certain (infinity,1)-sheaf of commutative ring spectra over the moduli stack of elliptic curves.
From this sheaf, one can recover the Adams-type spectral sequence associated to . According to SEC, this sheaf is actually the structure sheaf of the moduli stack classifying “oriented elliptic curves” over commutative ring spectra, or, to be in the correct variance, over derived affine schemes.
There is a canonical homomorphism of E-∞ rings
in the -generalized cohomology of its base space .
|chromatic level||complex oriented cohomology theory||E-∞ ring/A-∞ ring|
|0||ordinary cohomology||Eilenberg-MacLane spectrum|
|0th Morava K-theory|
|1||complex K-theory||complex K-theory spectrum|
|first Morava K-theory|
|first Morava E-theory|
|second Morava K-theory|
|second Morava E-theory|
|algebraic K-theory of KU|
|th Morava K-theory|
|th Morava E-theory|
|algebraic K-theory applied to chrom. level||(red-shift conjecture)|
There is a way to “construct” the tmf-spectrum as the E-∞ ring of global sections of a structured (∞,1)-topos whose underlying space is essentially the moduli stack of elliptic curves. We sketch some main ideas of this construction.
The discussion happens in the context of derived geometry in the (∞,1)-topos over a small version of the (∞,1)-site of formal duals of E-∞ rings (ring spectra). This is equipped with some subcanonical coverage. For we write for its image under the (∞,1)-Yoneda embedding .
Because the sphere spectrum is the initial object in .
The crucial input for the entire construction is the following statement.
This means that plays the role of a cover of the point. This allows to do some computations with ring spectra locally on the cover . Since is the Lazard ring, this explains why formal group laws show up all over the place.
To see this, first notice that the problem of realizing or any other ring spectrum as the ring of global sections on something has a tautological solution : almost by definition (see generalized scheme) there is an -ring valued structure sheaf on and its global sections is . So we have in particular
In order to get a less tautological and more insightful characterization, the strategy is now to pass on the right to the -cover by forming the (∞,1)-pullback
which by formal duality is
As a groupoid object this is still equivalent to just .
To simplify this we take a drastic step and apply a lot of decategorification: by applying the homotopy group (∞,1)-functor to all the -rings involved these are sent to graded ordinary rings , etc. The result is an ordinary simplicial scheme
This general Ansatz is discussed in (Hopkins).
This simplicial scheme, which is degreewise the formal dual of a graded ring of generalized homology-groups one can show is in fact a groupoid, hence a stack: effectively the moduli stack of elliptic curves. . See (Henriques).
In fact if in this construction one replaced by the point, one obtains the simplicial scheme
which one finds is the moduli stack of formal group laws.
Now, a priori these underived stacks remember little about the original derived schemes etc. They may not even carry any -ring valued structure sheaf anymore (though some of them do).
If they do carry an -ring valued structure sheaf , one can compute the homotopy groups of its global sections by a spectral sequence
But it turns out that even if the derived structure sheaf does not exist, this spectral sequence may still converge and may still compute the homotopy groups of the ring spectrum that one started with. This gives one way to compute the homotopy groups of .
For the case of one finds that the homotopy sheaves are simple: they vanish in odd degree and are tensor powers of the canonical line bundle in even degree , where the fiber of over an elliptic curve is the tangent space of that curve at its identity element. A section of is a modular form of weight . So the whole problem of computing the homotopy groups of boils down to computing the abelian sheaf cohomology of the moduli stack of elliptic curves with coefficients in these abelian groups of modular forms — and then examining the resulting spectral sequence.
This can be done quite explicitly in terms of a long but fairly elementary computation in ordinary algebra. A detailed discussion of this computation is in (Henriques)
A course that goes through the construction of tmf and the computation of its homotopy groups is
Talbot workshop on TMF (web)
More details are at
and in the series of lecture notes linked to there.
see also remark 1.4 of
and for more on the sigma-orientation see
The self-Anderson duality of is discussed in