Motivation and potential applications of spectral algebraic geometry

Question

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry.

Now I'm curious what future is there for spectral algebraic geometry. That is, what problems could be possibly attacked using this formalism, rather than arithmetic algebraic geometry or derived algebraic geometry.

I know historically you have used it to study topological modular forms.

Since it's also ''higher arithmetic geometry'' I'm curious if it can be useful in number theoretic problems. J.Lurie has a joint paper with Gaitsgory about Weil conjecture for function field. Did he use spectral algebraic geometry there? Or ''simple'' derived algebraic geometry?

I know that spectral algebraic geometry is also a type of ''higher''/''derived'' geometry, so I'm interested what advantages it has over derived algebraic geometry over simplicial commutative rings.

I wonder if spectral algebraic geometry could be potentially a useful tool for number theory or Langlands program (geometric). And other applications are also would be appreciated compared to DAG.

Answer 1

This is not really an answer to your question, just an attempt to address your question from the comments.

There are various flavours of homotopical or higher algebraic geometry that are commonly considered, which have different levels of connectivity, linearity, or strictness of commutativity. These include:

1) simplicial commutative rings

2) connective $E_\infty$-algebras over $H\mathbf{Z}$

3) connective $E_\infty$-ring spectra

4) (nonconnective) $E_\infty$-algebras over $H\mathbf{Z}$

5) (nonconnective) $E_\infty$-ring spectra

In characteristic zero, one also considers (connective) commutative dg-algebras.

The flavour most suited for algebraic geometry purposes is (1): this is the minimal extension of algebraic geometry where derived tensor products and cotangent complexes live. This was the flavour originally studied by Lurie in his thesis, and Toen-Vezzosi in HAG II.

Any of the other theories might be called "spectral algebraic geometry". (2) is similar to (1), but is less suited for algebraic geometry purposes, because deformation theory in the $E_\infty$-world is different than in the setting of simplicial commutative rings. In fact, the affine line is not even smooth in the $E_\infty$-world.

The difference between (2) and (3), as between (4) and (5), is linearity: in (3) and (5), you only consider objects which are linear over the sphere spectrum, so these settings are well suited to purposes of stable homotopy theory.

The main difference between the connective and nonconnective settings is the lack of converging Postnikov towers. That is, every connective $E_\infty$-ring spectrum $R$ can be written as a homotopy limit of square zero extensions of $\pi_0(R)$. This allows one to establish analogues of many results from classical algebraic geometry, by using induction along square zero extensions. The nonconnective world, on the other hand, behaves much differently, and geometric intuition very often fails.

I don't know much stable homotopy theory, but I believe the main point of spectral algebraic geometry is to be able to consider $E_\infty$-ring spectra as affine schemes, and to apply algebro-geometric techniques to study them. For example, the main application so far is Lurie's construction of tmf, the spectrum of topological modular forms, as the global sections of a sheaf of $E_\infty$-ring spectra on the moduli stack of spectral elliptic curves.