# What is derived algebraic geometry?

# 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.