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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 DS0.
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 Σ(n) action
DX=∨n(X∧n/Σ(n)).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 Symk(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 Δ (later δ) is induced from the diagonal X→X∨X (towards the bottom of p. 2).
Interestingly now, while the ordinary symmetric algebra Symk(k) of the ground ring itself is trivial, the extended power spectrum of the sphere spectrum itself is nontrivial. This is because 𝕊∧n≃𝕊 but hence the homotopy quotient by Σn contributes a copy of Σ∞+BΣ(n) at each stage.
That makes me want to say that the DS0 in Strickland-Turner is usefully thought of as the “polynomial ring” Sym𝕊𝕊 being like functions on the “absolute spectral affine line”. Or something like this. Does this make sense?
Yes, DS0 is the free E∞-algebra on one generator (Lurie writes this as S{t}, where S denotes the sphere spectrum), aka the symmetric algebra SymS(S), and it is indeed the ring of functions on the absolute spectral affine line (which I think Lurie writes as 𝔸1sm). It should probably be distinguished from the polynomial ring S[t]=Σ∞+(N) though (which is the ring of functions on the “flat affine line”, which I think Lurie writes as 𝔸1; this guy base changes over Spec(Hℤ) to the usual affine line).
More generally one can take X=Spec(SymS(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).
the absolute spectral affine line (which I think Lurie writes as 𝔸1sm).
Where?
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.
Okay, thanks.
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