Simplicial Commutative Rings – What is a Simplicial Commutative Ring in Homotopy Theory?

Question

Let $k$ be a field. There are two natural categories to consider:

• The category of simplicial commutative $k$-algebras.
• The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of $k$-vector spaces in nonnegative dimensions with a coherently associative and commutative multiplication law).

These categories are not the same if $k$ does not have characteristic zero. Simplicial commutative $k$-algebras are rather special, and (for example) not every commutative dga over $k$ (which determines an $E_\infty$-algebra over $k$) comes from a simplicial commutative $k$-algebra. (The homotopy groups of a simplicial commutative ring have divided powers, by an explicit construction that I don't really understand.) The category of $E_\infty$-algebras over $k$ has a nice interpretation via homotopy theory: it is the category of commutative algebra objects (in an appropriate sense) in the category of connective $k$-module spectra. (In particular, it is monadic over connective $k$-module spectra, in the $\infty$-categorical sense.) I don't know how to think of simplicial commutative rings in this way; all I know for motivation is that they form a nice homotopy theory (e.g. presented by a fairly concrete model category) that allows you to extend the category of ordinary commutative rings (e.g., to resolve non-smooth objects by smooth ones).

Is there an analog of the above discussion for $E_\infty$-algebras that works for simplicial commutative $k$-algebras? In particular, can they be described as algebras over a nice monad for $k$-module spectra?

Answer 1

I don't know a really satisfying answer to this question, but here are a few observations.

1) The $\infty$-category of simplicial commutative $k$-algebras is monadic over the $\infty$-category of connective $k$-module spectra. The relevant monad is the nonabelian left derived functor of the "total symmetric power" on ordinary $k$-modules, which is different from the construction $M \mapsto \bigoplus_{n} (M^{\otimes n})_{h \Sigma_n}$ unless $k$ has characteristic zero.

2) The $\infty$-category of simplicial commmutative rings is freely generated under sifted colimits by the ordinary category of finitely generated polynomial algebras over $k$. In other words, it can be realized as the $\infty$-category of product-preserving functors from the ordinary category of $k$-schemes which are affine spaces to the $\infty$-category of spaces.

3) Let $X$ be the affine line over $k$ (in the sense of classical algebraic geometry). Then $X$ represents the forgetful functor {commutative $k$-algebras} -> {sets}. Consequently, $X$ has the structure of a commutative $k$-algebra in the category of schemes. Also, $X$ is flat over $k$.

Now, any ordinary scheme can be regarded as a spectral scheme over $k$: that is, it also represents a functor {connective E-infty algebras over k} -> {spaces}. In general, products in the category of ordinary $k$-schemes need not coincide with products in the $\infty$-category of spectral $k$-schemes. However, they do agree for flat $k$-schemes. Consequently, $X$ can be regarded as a commutative $k$-algebra in the $\infty$-category of derived $k$-schemes. In particular, $X$ represents a functor {connective E-infty algebras over k} -> {connective E_infty algebras over k}. This functor has the structure of a comonad whose comodules are the simplicial commutative $k$-algebras.

You can summarize the situation more informally by saying: derived algebraic geometry (based on simplicial commutative $k$-algebras) is what you get when you take spectral algebraic geometry (based on E-infty-algebras over $k$) by forcing the two different versions of the affine line to coincide.

4) The forgetful functor {simplicial commutative $k$-algebras} -> {E-infty algebras over $k$} is both monadic and comonadic. In particular, you can think of a simplicial commutative $k$-algebra $R$ as an E-infty algebra over $k$ with some additional structure. As Tyler mentioned, one way of thinking about that additional structure is that it gives you the ability to form symmetric powers of connective modules. Of course, if $M$ is any $R$-module spectrum, you can always form the construction $(M^{\otimes n})_{h \Sigma_n}$. However, this doesn't behave the way you might expect based on experience in ordinary commutative algebra: for example, if $M$ is free (i.e. a sum of copies of $R$) then $(M^{\otimes n})_{h \Sigma_n}$ need not be free (unless $R$ is of characteristic zero). However, when $R$ is a simplicial commutative ring, there is a related construction on connective $R$-module spectra, given by nonabelian left derived functors of the usual symmetric power. This will carry free $R$-module spectra to free $R$-module spectra (of the expected rank).

It is possible to describe the $\infty$-category of simplicial commutative $k$-algebras along the following lines: a simplicial commutative $k$-algebra is a connective E-infty algebra over $R$ together with a collection of symmetric power functors Sym^{n} from connective $R$-modules to itself, plus a bunch of axioms and coherence data. I don't remember the exact statement (my recollection is that spelling this out turned out to be more trouble than it was worth).