# solid commutative ring spectra

Let $R$ be a discrete (i.e. an ordinary) commutative ring and let $HR\rightarrow T$ be a map of $E_{\infty}$-ring spectra where $HR$ is the associated Eilenberg-Mac Lane ring spectrum. We say that $T$ is $R$-solid (in the derived sense) if the induced map of $E_{\infty}$-ring spectra $$T\wedge_{HR}^{\mathbb{L}}T\rightarrow T$$ is a weak equivalence. Are there some nontrivial examples of such solid ring spectra? By "nontrivial" I mean that $T$ is not discrete. The discrete case (over $\mathbf{Z}$) is classified here .

• This doesn't answer your question but, over the sphere spectrum, you can have a look at arxiv.org/pdf/1303.5265.pdf. Also that paper mostly works over symmetric monoidal categories so, although it doesn't explicitly consider the specific case you're asking about, you still can see how the general results apply. Commented Aug 12, 2015 at 6:30
• So the underlying map of chain complexes is a quasi-iso. This seems unlikely to be a common thing, but I don't know the motivation or the classical situation you are generalizing. Commented Aug 12, 2015 at 12:01
• @FernandoMuro thanks for the reference, I read the paper quickly, I did not find a way (at least for now) to see a possible way to unswear my initial question. Maybe you have an idea ? Commented Aug 12, 2015 at 17:35
• I have undeleted this question which you have deleted last year. -- Please do not self-delete your useful questions! Commented Apr 30, 2021 at 21:06

## 1 Answer

Here is an example of such pair $T$ and $R$.

1. $k$ is a finite field.
2. $G$ is the Lie group $SO(2)$ and $G^{\delta}$ the same group but with dicrete topology.
3. $R$ is the group algebra $k[G^{\delta}]$.
4. $T$ is $C_{\ast}(G,k)$ the singular chain complex associated to $G$.

Then the natural map $k[G^{\delta}]\rightarrow C_{\ast}(G,k)$ of $E_{\infty}$-algebras induces a quasi-isomorphism $$C_{\ast}(G,k)\otimes_{k[G^{\delta}]}^{\mathbf{L}} C_{\ast}(G,k)\rightarrow C_{\ast}(G,k)$$ of $E_{\infty}$-algebras.