# 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:

$\newcommand{\Z}{\mathbf{Z}}$ Theorem: Let $f(x,y)$ be a formal group law over a torsion-free ring $R$, classified by a map $\Z[x_1,\cdots,x_n]\simeq L \simeq MU_\ast \to R_{}$. If $f(x,y)$ is Landweber-exact, i.e., if, when we write $[p]_f(x) = \sum a_i x^i$, the sequence $p,a_p,a_{p^2}, \cdots_{}$ is a regular sequence in $R$, the functor on spaces defined by

is a homology theory, and in particular, defines a spectrum $E$ with $\pi_\ast E \simeq R$.

$\newcommand{\Eoo}{\mathbf{E}_\infty}$ In general, there is no reason for this to be an $\Eoo$-ring, but in some cases, it is; for instance, complex K-theory, $KU$, satisfies $KU_\ast (X)\simeq \Z[\beta^{\pm 1}]\otimes_{MU_\ast} MU_\ast(X)_{}$, where the formal group law on $\pi_\ast KU\simeq \Z[\beta^{\pm 1}]$ is the multiplicative formal group law $x+y+\beta xy$.

$\newcommand{\Mfg}{\mathcal{M}_\mathbf{fg}}$ Algebraic geometry already begins showing its usefulness in this story. $\DeclareMathOperator{\spec}{Spec}$ Indeed, the condition that the sequence $p,a_p,a_{p^2},\cdots$ (the elements $a_{p^i}$ are typically denoted $v_i$) forms a regular sequence is equivalent to asking that the map $\spec R \to \Mfg$, classifying the underlying formal group of $f(x,y)$ is flat, where $\Mfg$ is the moduli stack of formal groups (this can be constructed from $\spec L$, the moduli of formal group laws, by quotienting out by the group scheme of all ‘‘coordinate transformations’’, i.e., the group scheme $G$ that sends $R\mapsto \{f(x)\in R[[x]]|f(x) = x + b_1 x^2 + b_2 x^3 + \cdots\}$).

# 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 $\Mfg$. Let us consider a baby case – we’ll return to a much harder and more interesting case later. Let $B\Z/2$ denote the classifying stack of the cyclic group on two elements. This stack classifies forms of the multiplicative group over $\Z$, since $\mathrm{Aut}(\mathbf{G}_m) = \Z^\times \simeq \Z/2$. There is a representable map of stacks $B\Z/2\to \Mfg$, and this map is flat. $\newcommand{\FF}{\mathbf{F}}$

This is good, because for every flat map $\spec R \to B\Z/2$, we get a (Landweber exact) homotopy commutative ring spectrum $E_R$. (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 $B\Z/2$. 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 $\Eoo$-rings, as evidenced by the following theorem of Lurie’s:

$\newcommand{\Aff}{\mathrm{Aff}^\text{et}}$ Theorem (Lurie): The affine etale site of an $\Eoo$-ring $R$ is equivalent to the affine etale site of $\pi_0 R$.

In our case, we have the following theorem:

$\newcommand{\co}{\mathcal{O}^\mathrm{top}}$ Theorem: There is a sheaf $\co$ of $\Eoo$-rings on $\Aff_{B\Z/2}$ such that $\co(\spec R \to B\Z/2) = E_R$, and the underlying formal group of $\co(\spec R \to B\Z/2)$ is given by the composite $\spec R \to B\Z/2\to \Mfg$.

The idea behind the proof of this theorem is that $KU$ is an $\Eoo$-ring, along with a (coherent) $\Z/2$-action coming from complex conjugation. We might try to understand the global sections $\Gamma(B\Z/2,\co)$. One can show, however, that (as is expected) $\newcommand{\Mod}{\mathrm{Mod}}$ $\newcommand{\QCoh}{\mathrm{QCoh}}$

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

In particular, we have $\Gamma(B\Z/2,\co) \simeq KU^{h\Z/2}$, which is, by a theorem of Atiyah’s, exactly $KO$. This is really intriguing: from this purely chromatic/algebro-geometric point of view, we have constructed a cohomology theory – as an $\Eoo$-ring – which is not even complex oriented!

The example of $KO$ that we constructed above is $K(1)$-local, and so we might ask if there is such a construction at height $2$, 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.