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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2017
   • (edited Mar 14th 2017)
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   Author: Urs
   Format: MarkdownItexI was looking around a little for natural examples of [[Hopf ring spectra]] discussed in the literature. In [Strickland-Turner 97](https://ncatlab.org/nlab/show/Hopf+E-%E2%88%9E+algebra#StricklandTurner97) is discussed Hopf (semi-)(co-)ring structure on the extended power spectrum of the sphere spectrum, which they write $D S^0$. Now the extended power spectrum of any spectrum $X$, that's the direct sum over $n$ of the homotopy quotients of the $n$-fold smash powers of $X$ by the canonical symmetric group $\Sigma(n)$ action $$ D X = \underset{n}{\vee} \left( X^{\wedge^n}/\Sigma(n) \right) \,. $$ I suppose this may be thought of as the spectral analog of the [[symmetric algebra]] construction where for $V$ a $k$-vector space we form $Sym_k(V)$. (If $k$ is of char 0 then we may form this from the tensor algebra by quotienting out the symmetric group action.) The analog of the ground field $k$ is now the sphere spectrum. This should be the intuitive explanation of why there may be Hopf-like structure on these extended power spectra: They are like rings of functions on affine lines, and hence the additive group structure of the affine line induces a coproduct on its ring of functions. If I understand well, this matches with what Strickland-Turner have, where the coproduct $\Delta$ (later $\delta$) is induced from the diagonal $X \to X \vee X$ (towards the bottom of p. 2). Interestingly now, while the ordinary symmetric algebra $Sym_k(k)$ of the ground ring itself is trivial, the extended power spectrum of the sphere spectrum itself is nontrivial. This is because $\mathbb{S}^{\wedge^n} \simeq \mathbb{S}$ but hence the homotopy quotient by $\Sigma_n$ contributes a copy of $\Sigma^\infty_+ B \Sigma(n)$ at each stage. That makes me want to say that the $D S^0$ in Strickland-Turner is usefully thought of as the "polynomial ring" $Sym_{\mathbb{S}} \mathbb{S}$ being like functions on the "absolute spectral affine line". Or something like this. Does this make sense?

   I was looking around a little for natural examples of Hopf ring spectra discussed in the literature. In Strickland-Turner 97 is discussed Hopf (semi-)(co-)ring structure on the extended power spectrum of the sphere spectrum, which they write $D S^0$.

   Now the extended power spectrum of any spectrum $X$, that’s the direct sum over $n$ of the homotopy quotients of the $n$-fold smash powers of $X$ by the canonical symmetric group $\Sigma(n)$ action

   $$D X = \underset{n}{\vee} \left( X^{\wedge^n}/\Sigma(n) \right) \,.$$

   I suppose this may be thought of as the spectral analog of the symmetric algebra construction where for $V$ a $k$-vector space we form $Sym_k(V)$. (If $k$ is of char 0 then we may form this from the tensor algebra by quotienting out the symmetric group action.) The analog of the ground field $k$ is now the sphere spectrum.

   This should be the intuitive explanation of why there may be Hopf-like structure on these extended power spectra: They are like rings of functions on affine lines, and hence the additive group structure of the affine line induces a coproduct on its ring of functions. If I understand well, this matches with what Strickland-Turner have, where the coproduct $\Delta$ (later $\delta$) is induced from the diagonal $X \to X \vee X$ (towards the bottom of p. 2).

   Interestingly now, while the ordinary symmetric algebra $Sym_k(k)$ of the ground ring itself is trivial, the extended power spectrum of the sphere spectrum itself is nontrivial. This is because $\mathbb{S}^{\wedge^n} \simeq \mathbb{S}$ but hence the homotopy quotient by $\Sigma_n$ contributes a copy of $\Sigma^\infty_+ B \Sigma(n)$ at each stage.

   That makes me want to say that the $D S^0$ in Strickland-Turner is usefully thought of as the “polynomial ring” $Sym_{\mathbb{S}} \mathbb{S}$ being like functions on the “absolute spectral affine line”. Or something like this. Does this make sense?

2. • CommentRowNumber2.
   • CommentAuthoradeelkh
   • CommentTimeMar 18th 2017
   • (edited Mar 18th 2017)
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   Author: adeelkh
   Format: MarkdownItexYes, $DS^0$ is the free $E_\infty$-algebra on one generator (Lurie writes this as $S\{t\}$, where $S$ denotes the sphere spectrum), aka the symmetric algebra $Sym_{S}(S)$, and it is indeed the ring of functions on the absolute spectral affine line (which I think Lurie writes as $\mathbb{A}^1_{sm}$). It should probably be distinguished from the polynomial ring $S[t] = \Sigma^\infty_+(\mathbf{N})$ though (which is the ring of functions on the ``flat affine line'', which I think Lurie writes as $\mathbb{A}^1$; this guy base changes over $Spec(H\mathbb{Z})$ to the usual affine line). More generally one can take $X = Spec(Sym_S(M))$ for any connective $S$-module $M$. Even when $M$ is non-connective there is a non-schematic spectral stack associated to $M$, which is a spectral Artin stack when $M$ is perfect. Its infinitesimal theory is as "simple" as (the tor-amplitude of) $M$ is (for example if $M$ is of tor-amplitude $[0,0]$, aka locally free (without shifts) then $X$ is just a vector bundle).

   Yes, $DS^0$ is the free $E_\infty$-algebra on one generator (Lurie writes this as $S\{t\}$, where $S$ denotes the sphere spectrum), aka the symmetric algebra $Sym_{S}(S)$, and it is indeed the ring of functions on the absolute spectral affine line (which I think Lurie writes as $\mathbb{A}^1_{sm}$). It should probably be distinguished from the polynomial ring $S[t] = \Sigma^\infty_+(\mathbf{N})$ though (which is the ring of functions on the “flat affine line”, which I think Lurie writes as $\mathbb{A}^1$; this guy base changes over $Spec(H\mathbb{Z})$ to the usual affine line).

   More generally one can take $X = Spec(Sym_S(M))$ for any connective $S$-module $M$. Even when $M$ is non-connective there is a non-schematic spectral stack associated to $M$, which is a spectral Artin stack when $M$ is perfect. Its infinitesimal theory is as “simple” as (the tor-amplitude of) $M$ is (for example if $M$ is of tor-amplitude $[0,0]$, aka locally free (without shifts) then $X$ is just a vector bundle).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2017
   • (edited Mar 19th 2017)
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   Author: Urs
   Format: MarkdownItex> the absolute spectral affine line (which I think Lurie writes as $\mathbb{A}^1_{sm}$). Where?

   the absolute spectral affine line (which I think Lurie writes as $\mathbb{A}^1_{sm}$).

   Where?

4. • CommentRowNumber4.
   • CommentAuthoradeelkh
   • CommentTimeMar 22nd 2017
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   Author: adeelkh
   Format: MarkdownItexIt looks like Lurie only discusses the projective versions so far (5.4 and 19.2.6 in SAG). Presumably discussion of the affine versions will also be added at some point, but who knows. Have you seen his new paper [Elliptic Cohomology I](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf), though? That might have what you're looking for, see especially Example 3.5.4.

   It looks like Lurie only discusses the projective versions so far (5.4 and 19.2.6 in SAG). Presumably discussion of the affine versions will also be added at some point, but who knows. Have you seen his new paper Elliptic Cohomology I, though? That might have what you’re looking for, see especially Example 3.5.4.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 30th 2017
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   Author: Urs
   Format: MarkdownItexOkay, thanks.

   Okay, thanks.