Sunday, 21 June 2009

Stacks and Elliptic Cohomology

This week I became interested in two different topics due to conversations that I overheard. The first is the topic of stacks and the second is elliptic cohomology.


Apparently there are numerous different kinds of stacks - Deligne-Mumford Stacks, Artin Stacks and for the die hard, apparently more general kinds of stacks.

The following is probably completely wrong, but is my understanding of what stacks are about.

Consider elliptic curves defined over the complex numbers K = C say. It is a classical result that up to isomorphism, these can be parameterised by points in the complex upper half plane, modulo the action of the modular group.

Now the upper half plane, modulo the modular group can be compactified by adding a point at the cusp, and made into a Riemann surface (of genus zero in this case). We can put coordinates on this Riemann surface (the complex j-function as it happens) and turn it into an algebraic curve of genus 0.

In fact, two elliptic curves are isomorphic iff their j-invariants are equal. In other words the Riemann sphere or j-line as it is often called, can be thought of as classifying all elliptic curves. In fact we call C the course moduli space for elliptic curves.

Now suppose we try to construct a universal space for families of elliptic curves over this base space. The problem is that an elliptic curve can have extra automorphisms and it is possible for a family of elliptic curves to contain isomorphic elliptic curves for this reason. That prevents us from having a univeral space for families of elliptic curves.

The way we get around this is using stacks. We define the stack of elliptic curves which is a category whose objects are families of elliptic curves over a base space (fixed for that family) and we define a morphism to be a map between families of elliptic curves along with a map between the corresponding base spaces such that the map between families is compatible with the map between base spaces.

Furthermore, for it to be a morphism (X'->B') -> (X->B), we require that if we pull the family of elliptic curves X back along the map B'->B of base spaces, we get a family of elliptic curves isomorphic to the family X'.

We can restrict to the subcategory of families of elliptic curves over a fixed base space B if we want. We call this subcategory the fibre over B.

Now note that the fibre over a base space B is a groupoid (a categories whose only morphisms are isomorphisms). We say that the stack of elliptic curves is a category fibred in groupoids.

Now it is clear that there is a universal family of elliptic curves with respect to this construction.

There's more to stacks than this (some of the critical components of the definition are omitted above).

A stack is of Deligne-Mumford type (formerly an algebraic stack) if it satisfies some additional conditions, in particular that there is an etale surjective morphism (called an atlas) from a scheme U to the stack F, amongst other things.

An Artin stack (nowadays what is referred to as an algebraic stack) simply replaces etale with smooth in the previous definition.

Anyhow, what was interesting to me is that nowadays stacks are replacing schemes as the ultimate objects of interest. A lot of work has been done to popularise them. Hey, if you want to know more theres only 1000 pages to read: The Stacks Project.

Elliptic cohomology

The second topic which piqued my interest this week was elliptic cohomology. I thought maybe this might be related to parabolic cohomology, which is defined in terms of parabolic cusps (fixed points of parabolic elements of SL_2(R)). But I don't know that this is the origin of the term.

Instead I found this enormous survey on the web, which is written helpfully:

That's a long document to be called a summary, so I'll give a summary of the summary.

To a topological space X we can associate the singular cohomology groups A^n(X) = H^n(X; Z) which can be characterised by the Eilenberg-Steenrod axioms. Any collection of functors and connecting maps satisfying these axioms necessarily gives you the usual integral cohomology functors (X \subseteq Y) -> H^n(X, Y; Z). More generally we can replace Z with any abelian group M.

Now if we drop the last of the E-S axioms:

* If X is a point then A^n(X) = { 0 if n \neq 0 and Z is n = 0 }

then we get something more general, called a cohomology theory.

Interestingly, complex K theory is an example of such a cohomology theory!

Complex K-theory is a so-called multiplicative cohomology theory, because A^n(*) is a graded commutative ring. Another nice feature is it is periodic:

* According to the Bott periodicity theorem for complex K-theory, there is an element \beta in K^2(*) such that multiplication by \beta induces an isomorphisms: \beta : K^n(X) -> K^{n+2}(X)

and even:

* K^i(*) = 0 if i is odd.

Ordinary cohomology H*(X; A) for a commutative ring A is even, but to make it periodic we need to take products over every second cohomology group A^n(X; A) = \prod_k H^{n+2k}(X; A)

Now when A is an even periodic cohomology, it turns out that A(CP^\infty) of the infinite dimensional complex projective space, is isomorphic to a formal power series ring A(*)[[t]] over the commutative ring A(*).

We can view the element t as the first Chern class of the universal line bundle O(1). The space CP^\infty is a classifying space for complex line bundles, i.e. for any complex line bundle L on a space X there is a (classifying) map \phi : X -> CP^\infty and an isomorphism L <-> \phi*(O(1)).

We then define c1(L) = \phi*(t) \in A(X), the first Chern class of the cohomology A.

In ordinary cohomology the first Chern class of a tensor product of line bundles is simply the sum of the first Chern classes of the line bundles.

In the case of complex K-theory line bundles L can be thought of as representatives of elements of K(X) itself. We write such an element [L]. Then c1(L) = [L] - 1.

Now c1(L1 \tensor L2) = c1(L1) + c1(L2) + c1(L1)c1(L2).

In the general case we have:

c1(L1 \tensor L2) = f(c1(L1), c1(L2)) for some f \in A(*)[[t1, t2]].

It turns out that the following properties hold:

* f(0, t) = f(t, 0) = t
* f(u, v) = f(v, u)
* f(a, f(b, c)) = f(f(a, b), c)

A power series with these properties is called a commutative 1-dimensional formal group law over the commutative ring A(*). It gives a group structure on the formal scheme Spf A*(X)[[t]] = Spf A(CP^\infty).

We call the first of the group laws above f(a, b) = a + b the additive formal group law denoted \hat{G_a} and the other f(a, b) = a + b + a*b the multiplicative formal group law, denoted \hat{G_m}.

Now one wonders if there are other possible formal group laws. It turns out that the Lazard ring is a ring classifying formal group laws. Quillen proved that it comes from a cohomology called periodic complex cobordism denoted MP. There is a canonical isomorphism MP(CP^\infty) <-> MP(*)[[t]] and the coefficient ring MP(*) is the Lazard ring.

One may construct the moduli stack of all formal group laws M_{FGL} so that for a commutative ring R, the set of homomorphisms Hom(Spec R, M_{FGL}) can be identified with the power series f(u, v) \in R[[u, v]] satisfying the three conditions above. Then M_{FGL} is an affine scheme, in fact M_{FGL} = Spec MP(*), as we'd expect for a moduli stack.

Well it gets a bit more complicated than that. One must mod out by the action on M_{FGL} by the group of automorphisms of the formal affine line Spf Z[[x]]. This yields the stack of formal groups M_{FG}.

Now what is an elliptic cohomology?

Well we need to relax one thing slightly. Instead of demanding that we have a periodic cohomology, we'll just require our multiplicative cohomology A to be weakly-periodic:

* The natural map A^2(*) \tensor_{A(*)} A^n(*) -> A^{n+2}(*) is an isomorphism for all n \in Z.

Now we can define an elliptic cohomology A:

* R is a commutative ring

* E is an elliptic curve over R

* A is a multiplicative cohomology which is even and weakly-perioidic

* There are isomorphisms A(*) <-> R and \hat{E} <-> Spf A(CP^\infty) of formal groups, over R which is isomorphic to A(*)

Here \hat{E} represents the formal completion of E along its identity section. It is a commutative 1-dimensional formal group over R. It is classified by a map \phi : Spec R -> M_{FG}.

For finite cell complexes X we may interpret the complex cobordism groups MP^n(X) as quasi-coherent sheaves on the moduli stack M_{FG} and in the case of a formal group over R we can define A^n(X) to be the pullback of these sheaves along \phi.

When \phi is flat then we call the formal group Landweber-exact. Landweber gave a criterion for determining when a formal group is Landweber-exact.

Anyhow it turns out that in the elliptic cohomology case, when the formal group is Landweber-exact, the elliptic curve and the isomorphism of the definition of elliptic cohomology are uniquely given. In this case, giving the elliptic cohomology theory is exactly the same thing as giving an elliptic curve.

Now in the case of the formal multiplicative group, there is a universal formal group. But in the case of elliptic cohomology, there is no such thing as a universal elliptic curve over a commutative ring. The moduli stack of elliptic curves M_{1,1} is not an affine variety, and not even a scheme. However, it is a Deligne-Mumford stack.

For each etale morphism \phi : R \to M_{1,1} there is an elliptic curve E_\phi and happily, when \phi is etale, the associated formal group \hat{E_\phi} is Landweber-exact.

Well, this allows us to create on M_{1,1} a presheaf taking on values in the category of cohomology theories. But this is a nasty construction and so we try to represent our cohomology theories by representing spaces.

Roughly speaking this is where the theory of E-\infty rings or E-\infty spectra comes in. Basically a cohomology theory A_\phi can be represented by a spectrum. That eventually allows one to develop a universal elliptic cohomology theory.

We define a presheaf O_{M}^{Der} of E_\infty rings representing elliptic cohomology theories on the category {\phi : Spec R -> M_{1,1}} mentioned above.

Now we extract our universal cohomology theory by taking a "homotopy limit" of the functor O_{M}^{Der}. This gives us the E-\infty ring of topological modular forms tmf[\Delta^{-1}].

(Note tmf is what you get when you replace M_{1,1} by its Deligne-Mumford compactification. After inverting 2 and 3 [to do with the extra automorphisms on elliptic curves] there is an isomorphism from tmf to the ring of integral holomorphic modular forms.)

OK that summarises the first 9 pages of the long summary above. But probably it isn't much of an improvement on the original. But it helps the memory to write it all down somewhere.

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