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nLab > Latest Changes: spectral group scheme

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2016

    Made a start at spectral group scheme.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2016

    I see we have derived group scheme, which has this

    warning careful, this needs a bit more attention. The general idea is obvious, but the details require care. One problem is that in the Elliptic Survey “derived scheme” really refers to Spectral Schemes and not to the derived schemes discussed in Structured Spaces.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2016

    After a dizzying tour round our entries, would that be right to work on the ’derived equals ∞\infty’ principle here, and so merge the pages?

    A good reference for ’derived’ group scheme seems to be

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2016

    So Iwanari’s definition can be used at spectral group scheme?

    Definition A.3. A derived group scheme over RR is a functor

    G:CAlg R→Grp(S)G : CAlg_R \to Grp(S)

    such that the composite CAlg R→Grp(S)→SCAlg_R \to Grp(S) \to S is representable by a derived scheme

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2016
    • (edited Dec 22nd 2016)

    Yes, so there is no real mystery in the definition. We talk about group objects in functor categories (freely throwing in “∞\infty” everywhere), and by Yoneda these are equivalently functors to the category of groups (we have some discussion of this at group object – In terms of presheaves of groups).

    So the structure of a group object on a derived (or not) and affine (or not) scheme is a lift of its functor of points to groups (∞\infty-groups), just as you guessed it is.

    Explicitly for affine spectral group schemes this is def. 3.9.7 in Lurie’s note (pdf), but the achievement of that note is not to come up with this definition. This is the ovious definition.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 22nd 2016

    Should we merge with derived group scheme?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2016
    • (edited Dec 22nd 2016)

    It would make sense to merge, yes.

    I believe the reason for switching terminology from “derived scheme” to “spectral schemes” was that most people these days when hearing “derived scheme” think of something way less general than was subsumed under this term in “Structured Spaces”, namely they think of dg-schemes of sorts and in particular not of E ∞E_\infty-schemes. However there is a considerable qualitative shift in passing from dg- to spectral.

    • CommentRowNumber8.
    • CommentAuthoradeelkh
    • CommentTimeDec 25th 2016
    • (edited Dec 25th 2016)

    Note that there is some subtlety regarding group schemes in spectral AG, in that there are typically smooth and flat versions. For example, the GL n,RGL_{n,R} example on the page spectral group scheme is smooth, and is not a group scheme in the sense of the definition at derived group scheme. In particular when RR is an Eilenberg-MacLane spectrum, this is not the classical group scheme GL n,RGL_{n,R} (it is not even discrete). In his paper on elliptic cohomology, Lurie only works with flat group schemes.

    In derived AG (AG based on simplicial commutative rings) this distinction does not arise; for SCR’s, smoothness is equivalent to being smooth on π 0\pi_0 and flat. Hence derived group schemes are really a very conservative extension of classical group schemes, while the theory of group schemes in SAG is less straightforward.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 27th 2016

    Thanks, Adeel. So what would you recommend? Keep the two pages separate, and then include some explanation as above?

    • CommentRowNumber10.
    • CommentAuthoradeelkh
    • CommentTimeJan 3rd 2017
    • (edited Jan 3rd 2017)

    That’s what I would do, but I think the nLab tends to view derived AG as a subset of spectral AG (and just uses the term “derived AG” for the latter). I don’t agree with this personally, but still one might prefer to merge both pages into “derived group scheme” to agree with the conventions already established on the nLab.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2017

    the nLab tends to view derived AG as a subset of spectral AG

    Not sure what this is referring to. My impression is the opposite:

    The little material we have mostly followed Lurie’s Structured Spaces in saying “derived geometry” for the very general concept of derived geometry, of which ordinary geometry and dg-geometry and spectral geometry are special cases.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 3rd 2017

    It would be useful to standardize terminology, unless it’s still so much in flux out there that this would be premature.

    We have derived algebraic geometry (where ’brave new algebraic geometry’ is called for), homotopical algebraic geometry, spectral algebraic geometry.

    We also have E-∞ geometry

    What may be called E ∞E_{\infty}-geometry, or spectral algebraic geometry is the “full” version of derived algebraic geometry

    Then we sometimes use ’higher’ as in higher differential geometry.

    I see that our group scheme is quite long and makes the distinctions that Adeel mentioned, smooth and flat (étale).

    • CommentRowNumber13.
    • CommentAuthoradeelkh
    • CommentTimeJan 4th 2017
    • (edited Jan 4th 2017)

    the nLab tends to view derived AG as a subset of spectral AG

    Not sure what this is referring to.

    What I meant is that the nLab tends to view AG based on simplicial rings as a sub-theory of AG based on E_oo-rings, i.e. that the latter theory is a generalization or even more “correct” version of the former (see e.g. the quote cited in #12). My point was that there are basic AG concepts which extend to both settings but which are genuinely different in the E_oo-setting; an example is smoothness as I mentioned above, and another is vector bundles: if you take a classical scheme and view it as a spectral scheme, then the category of (spectral) vector bundles over it is not even discrete (except in characteristic zero).

    The little material we have mostly followed Lurie’s Structured Spaces in saying “derived geometry” for the very general concept of derived geometry, of which ordinary geometry and dg-geometry and spectral geometry are special cases.

    Actually, this is not the terminology used by Lurie either. In Structured Spaces, Lurie uses simply “geometry” for the general theory, and “derived algebraic geometry” specifically for the specialization to simplicial rings. See page 5, the last paragraph before “Overview”, for example. I believe Lurie has followed this terminology consistently in everything he’s written, including the new book SAG. Similarly Toen-Vezzosi use the term homotopical algebraic geometry for the general theory, and derived algebraic geometry for the specialization to simplicial rings.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 4th 2017

    So ’higher’ and ’homotopical’ are synonymous when applied to geometry and its branches, and if you believe it is the future, may be omitted all together.

    Then there are specializations, as set out at homotopical algebraic geometry.

    If so, then derived algebraic geometry sets out OK, but then

    a version of derived algebraic geometry which is locally modelled on E-∞ rings, called spectral algebraic geometry.

    and

    The adjective “derived” means pretty much the same as the “∞\infty-”

    suggest “derived” is the general form.

    How does the ’derived’ of derived algebraic geometry (simplicial commutative rings) relate to that of derived differential geometry?

    Derived differential geometry is higher differential geometry in an ambient (∞,1)-topos which is not 1-localic.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 4th 2017

    In support of Adeel in #13, Toen writes in Derived Algebraic Geometry

    Homotopical algebraic geometry is the general form of derived algebraic geometry, E∞-algebraic geometry and spectral geometry.

    But that raises the issue that we have treated the latter two as synonymous at E-∞ geometry, while Toen sees E∞-algebraic geometry as over HℤH \mathbb{Z}-algebras.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2017
    • (edited Jan 4th 2017)

    Okay, thanks.

    Regarding terminology, I suppose there was once a series called “Derived Algebraic Geometry” which included installments titled “Spectral Schemes”. But anyway, terminology is not the issue. The point about the difference between regarding ordinary schemes as dg-schemes versus spectral schemes should be added to the nnLab, with any statement suggesting something contrary removed.

    • CommentRowNumber17.
    • CommentAuthoradeelkh
    • CommentTimeJan 4th 2017

    How does the ’derived’ of derived algebraic geometry (simplicial commutative rings) relate to that of derived differential geometry?

    It is the same “derived”. Let T algT_{alg} be the category of polynomial rings k[x 1,…,x n]k[x_1,\ldots,x_n] (n≥0)(n \ge 0). The (∞,1)(\infty,1)-category of simplicial commutative rings is the free completion of T algT_{alg} by sifted (∞,1)(\infty,1)-colimits; in other words, it is the non-abelian derived category in the sense of HTT, hence the natural home for derived functors on the non-abelian category of commutative rings.

    In derived differential geometry, the rings of functions are simplicial C ∞C^\infty-rings. The (∞,1)(\infty,1)-category of simplicial C ∞C^\infty-rings admits the same description as above where instead of T algT_{alg} we take T diffT_{diff}, the category of Euclidean spaces R n\mathbf{R}^n (and smooth maps between them).

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 4th 2017

    Re #17, so were we to adopt the ’derived = simplicial ring’ policy, then derived differential geometry would be a special case of the general form. At the moment this general form is denoted ’higher’, so perhaps odd that it differs from the general epithet ’homotopical’ of algebraic geometry.

    From Spectral Algebraic Geometry

    In Part VII, we study several variants of spectral algebraic geometry:

    • derived differential topology, whose basic objects (derived manifolds) are analogous to smooth manifolds in the same way that spectral schemes are analogous to smooth algebraic varieties.

    • derived complex analytic geometry, whose basic objects (derived complex-analytic spaces) are analogous to complex-analytic manifolds in the same way that spectral schemes are analogous to smooth algebraic varieties.

    • derived algebraic geometry, a variant of spectral algebraic geometry which uses simplicial commutative rings in place of E ∞E_{\infty}-rings. The resulting theory is equivalent to spectral algebraic geometry in characteristic zero, but is quite different (and more closely connected to classical algebraic geometry) in positive and mixed characteristic

    Is it that there’s no need/possibility to separate off a ’spectral’ from a ’derived’ form of differential topology, etc. ?

    But, yes, re #16, enough about terminology. Anyone competent with a little spare time?

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