spectral super-scheme


Super-Algebra and Super-Geometry

Higher geometry



The concept of spectral super-scheme is supposed to be the refinement of the concept of super-scheme as one passes to spectral geometry in the sense of derived algebraic geometry over E-infinity rings (E-infinity geometry).


The following is an argument for a good definition of spectral supergeometry. This was originally motivated from the observation in Kapranov 13 and uses results due to Rezk 09 and Sagave-Schlichtkrull 2011.

Observe that

  1. E-∞ geometry is already in itself a higher geometric version of \mathbb{Z}-graded supergeometry (in the sense discussed at geometry of physics – superalgebra).


    At the level of homotopy groups this is the following basic fact:

    For EE a homotopy commutative ring spectrum, its stable homotopy groups π (E)\pi_\bullet(E) inherit the structure of a \mathbb{Z}-graded super-commutative ring (according to this). See at Introduction to Stable homotopy theory in the section 1-2 Homotopy commutative ring spectra this proposition.


    But more is true: the E E_\infty-analog of the integers \mathbb{Z} is the sphere spectrum 𝕊Σ S 0\mathbb{S} \,\simeq\, \Sigma^\infty S^0, and every E-infinity ring (E,)(E, \cdot) is canonically 𝕊\mathbb{S}-graded, in that (Sagave-Schlichtkrull 11, theorem 1.7-1.8):

    on underlying E E_\infty -spaces E 0Ω (E)E_0 \,\coloneqq\, \Omega^\infty(E), at least, realized as Ω 𝒥(E)\Omega^{\mathcal{J}}(E) in SaSc11 (4.4), they are canonically equipped with an E E_\infty -monoid homomorphism

    (E 0,)(𝕊 0,+) (E_0, \cdot) \xrightarrow{\;} (\mathbb{S}_0, +)

    to the additive E E_\infty-space underlying the sphere spectrum (traditionally denoted QS 0Q S^0 , which is notation for a construction that yields Ω Σ S 0\Omega^\infty \Sigma^\infty S^0).

    The E E_\infty-monoid homomorphisms of this form are the evident homotopy-theoretic generalization of morphisms of commutative monoids to the additive group (,+)(\mathbb{Z}, +) of the integers, and these are evidently equivalent to \mathbb{Z}-gradings on the domain monoid.


    So E-∞ geometry in itself is already a categorified/homotopified version of supergeometry, but of \mathbb{Z}-graded supergeometry, not of the proper /2\mathbb{Z}/2-graded supergeometry.


    (That grading over the sphere spectrum is closely related to superalgebra had been highlighted in Kapranov 13, but the issue of the difference between homotopified \mathbb{Z}-grading compared to homotopified /2\mathbb{Z}/2-grading had been left open.)


  2. But ordinary /2\mathbb{Z}/2-graded supercommutative superalgebra is equivalently \mathbb{Z}-graded supercommutative superalgebra over the free even periodic \mathbb{Z}-graded supercommutative superalgebra (this prop.).


  3. In view of the first point, the second point has an evident analog in E-∞ geometry:

    The higher/derived analog of an even periodic \mathbb{Z}-graded commutative algebra is an E-infinity algebra over an even periodic ring spectrum.


    That E-infinity algebras over even periodic ring spectra are usefully regarded from the point of view of supercommutative superalgebra was highlighted in Rezk 09, section 2.

Hence it makes sense to say:

Definition. Spectral/E E_\infty super-geometry is simply the E-∞ geometry over even periodic ring spectra.


Spectral superpoint

The ordinary superpoint over some field kk is the spectrum of a commutative ring of the graded symmetric algebra on a single odd generator (“graded ring of dual numbers”)

𝔸 k 0|1Spec(Sym k(k[1])) \mathbb{A}_k^{0 \vert 1} \;\simeq\; Spec( \,Sym_k (k[1])\, )

Accordingly, for RR an even periodic ring spectrum, then the spectral superpoint R 0|1R^{0 \vert 1} should be the spectral scheme given by the spectral symmetric algebra on the suspension spectrum of RR:

R 0|1 Spec(Sym R(ΣR)) Spec(R(nBΣ(n) τ n) +) Spec(RSym 𝕊(Σ𝕊)). \begin{aligned} R^{0 \vert 1} &\coloneqq Spec \left( Sym_R (\Sigma R) \right) \\ & \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B \Sigma(n)^{\tau_n} \right)_+ \right) \\ & \simeq Spec\left( R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S}) \right) \end{aligned} \,.

where on the right we have the Thom space of the vector bundle τ n\tau_n associated to the Σ(n)\Sigma(n)-universal principal bundle via the canonical action of Σ(n)\Sigma(n) on n\mathbb{R}^n (see also at symmetric group – Classifying space and Thom space).


The proposal for spectral super-geometry above invokes observations from

The proposal above was originally motivated from the discussion of the sphere spectrum in relation to super algebra highlighted in

A closely related suggestion later appears in: