Cosimplicial commutative rings in stable homotopical algebra

Question

When doing "derived commutative algebra" over a discrete commutative ring $R$ that doesn't contain $\mathbb{Q}$, it's fairly well known that you generally have two flavors of "totally commutative object," given by simplicial commutative $R$-algebras on the one hand, and $E_\infty$-algebras in connective complexes over $R$ (equivalently connective $E_\infty$-algebra spectra over $HR$) on the other. The former category maps to the latter, and there are various ways to describe its image, as detailed in the answers to the question "What is a simplicial commutative ring from the point of view of homotopy theory? "

I'm interested in the dual question involving cosimplicial commutative $R$-algebras and coconnective $E_\infty$-algebra spectra over $HR$. There seems again to be a natural functor given by co-Dold-Kan, but I have no feeling for the objects it picks out. Are there similar intrinsic characterizations of the image of this functor as in the connective case? I'd be happy to know anything about this, even/especially when $R$ is a finite field.

Answer 1

Take $R = \mathbb{F}_2$. This isn't anywhere near a characterization, but here is a necessary condition that the $E_\infty$-ring associated to a cosimplicial $\mathbb{F}_2$-algebra must satisfy (that is not always satisfied for coconnective $E_\infty$-algebras under $\mathbb{F}_2$).

Namely, if $R$ arises as the totalization $R = \mathrm{Tot} (R^\bullet)$ of a cosimplicial diagram $R^\bullet$ of discrete commutative $\mathbb{F}_2$-algebras, then $R$ is the homotopy inverse limit of the tower $\{ \mathrm{Tot}_n(R^\bullet)\}$. Observe that each $\mathrm{Tot}_n(R^\bullet)$ has homotopy groups concentrated in degrees $[-n, 0]$. Moreover, the map $R \to \mathrm{Tot}_n(R^\bullet)$ induces an isomorphism on homotopy groups in the range $(-n, 0]$.

A consequence is that any $x \in \pi_{-n-1}(R)$ is mapped to zero in $\pi_*(\mathrm{Tot}_n(R^\bullet))$. Let ${Q^2}$ be the second Dyer-Lashof operation, which raises the degree by two. If $Q^2$ carries $x$ to a nonzero element in $\pi_{-n + 1} (R)$, we obtain a contradiction by naturality of the Dyer-Lashof operations and the map $R \to \mathrm{Tot}_n(R)$.

In the most common class of examples of coconnective $E_\infty$ $\mathbb{F}_2$-algebras (namely, cochain algebras on spaces), $Q^2$ acts by zero, but this is not necessary. Consider the free $E_\infty$-ring on a generator in degree $-3$; this is $\bigoplus_{n \geq 0} (\mathbb{F}_2^{\otimes n})_{h \Sigma_n}$. Now truncate this by throwing away everything with $n \geq 3$ and making the multiplication "cube zero." Finally, replace the top term, i.e., $(\mathbb{F}_2^{\otimes 2})_{h \Sigma_2}$, with its $\tau_{\leq 0}$ (which it maps to). The result is an $E_\infty$-ring under $\mathbb{F}_2$, but $Q^2$ of the class in degree $-3$ is not zero. Therefore, this $E_\infty$-ring does not arise as a cosimplicial $\mathbb{F}_2$-algebra.