Why derived algebraic geometry?

I’ll give a topologist’s motivation for why one should care about derived algebraic geometry. Recall the Landweber exact functor theorem:

Theorem: Let be a formal group law over a torsion-free ring , classified by a map . If is Landweber-exact, i.e., if, when we write , the sequence is a regular sequence in , the functor on spaces defined by

is a homology theory, and in particular, defines a spectrum with .

In general, there is no reason for this to be an -ring, but in some cases, it is; for instance, complex K-theory, , satisfies , where the formal group law on is the multiplicative formal group law .

Algebraic geometry already begins showing its usefulness in this story. Indeed, the condition that the sequence (the elements are typically denoted ) forms a regular sequence is equivalent to asking that the map , classifying the underlying formal group of is flat, where is the moduli stack of formal groups (this can be constructed from , the moduli of formal group laws, by quotienting out by the group scheme of all ‘‘coordinate transformations’’, i.e., the group scheme that sends ).

Hello, world!

If we want to try to construct new cohomology theories, then this rephrasing of Landweber exactness tells us that it’s a good idea to try to construct flat maps to . Let us consider a baby case – we’ll return to a much harder and more interesting case later. Let denote the classifying stack of the cyclic group on two elements. This stack classifies forms of the multiplicative group over , since . There is a representable map of stacks , and this map is flat.

This is good, because for every flat map , we get a (Landweber exact) homotopy commutative ring spectrum . (We might as well throw on the adjective ‘‘even periodic’’.) This means that we get a presheaf of homotopy commutative ring spectra on the ‘‘flat site’’ of . In derived algebraic geometry, though, it is important to consider phenomena on the etale site of a stack, mainly because etale maps behave very well for -rings, as evidenced by the following theorem of Lurie’s:

Theorem (Lurie): The affine etale site of an -ring is equivalent to the affine etale site of .

In our case, we have the following theorem:

Theorem: There is a sheaf of -rings on such that , and the underlying formal group of is given by the composite .

The idea behind the proof of this theorem is that is an -ring, along with a (coherent) -action coming from complex conjugation. We might try to understand the global sections . One can show, however, that (as is expected)

$$\Mod(KU)^{\Z/2} \simeq \QCoh(B\Z/2)$$

In particular, we have , which is, by a theorem of Atiyah’s, exactly . This is really intriguing: from this purely chromatic/algebro-geometric point of view, we have constructed a cohomology theory – as an -ring – which is not even complex oriented!

The example of that we constructed above is -local, and so we might ask if there is such a construction at height , that runs along similar lines. The theory of topological modular forms provides an answer to this question, and we will return to this in the next post.