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 .

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    $\begingroup$ 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. $\endgroup$ Commented Aug 12, 2015 at 6:30
  • $\begingroup$ 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. $\endgroup$ Commented Aug 12, 2015 at 12:01
  • $\begingroup$ @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 ? $\endgroup$
    – Ilias A.
    Commented Aug 12, 2015 at 17:35
  • $\begingroup$ I have undeleted this question which you have deleted last year. -- Please do not self-delete your useful questions! $\endgroup$
    – Stefan Kohl
    Commented Apr 30, 2021 at 21:06

1 Answer 1


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.


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