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\begin{document}

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\section*{spectral super-scheme}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{superalgebra_and_supergeometry}{}\paragraph*{{Super-Algebra and Super-Geometry}}\label{superalgebra_and_supergeometry}

[[!include supergeometry - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SpectralSuperpoint}{Spectral superpoint}\dotfill \pageref*{SpectralSuperpoint} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{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]]).

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

The following is an argument for a good definition of spectral supergeometry. This was originally motivated from the observation in \hyperlink{Kapranov13}{Kapranov 13} and uses results due to \hyperlink{Rezk09}{Rezk 09} and \hyperlink{SagaveSchlichtkrull11}{Sagave-Schlichtkrull 11}.

Observe that

\begin{enumerate}%
\item [[E-∞ geometry]] is \emph{already in itself} a [[higher geometry|higher geometric]] version of $\mathbb{Z}$-graded supergeometry (in the sense discussed at \emph{[[geometry of physics -- superalgebra]]}).

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

For $E$ a [[homotopy commutative ring spectrum]], its [[stable homotopy groups]] $\pi_\bullet(E)$ inherit the structure of a $\mathbb{Z}$-graded [[super-commutative ring]] (according to \href{geometry+of+physics+--+superalgebra#SupercommutativeSuperalgebraZGraded}{this}). See at \emph{[[Introduction to Stable homotopy theory]]} in the section \href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra}{1-2 Homotopy commutative ring spectra} \href{Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum}{this proposition}.

But more is true: the $E_\infty$-analog of the [[integers]] $\mathbb{Z}$ is the [[sphere spectrum]] $\mathbb{S}$, and every [[E-infinity ring]] $E$ is canonically $\mathbb{S}$-graded (\hyperlink{SagaveSchlichtkrull11}{Sagave-Schlichtkrull 11, theorem 1.7-18}).

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

(That grading over the [[sphere spectrum]] is closely related to [[superalgebra]] had been highlighted in \hyperlink{Kapranov13}{Kapranov 13}, but the issue of the difference between homotopified $\mathbb{Z}$-grading compared to homotopified $\mathbb{Z}/2$-grading had been left open.)


\item But ordinary $\mathbb{Z}/2$-graded [[supercommutative superalgebra]] is equivalently $\mathbb{Z}$-graded [[supercommutative superalgebra]] over the free even periodic $\mathbb{Z}$-graded supercommutative superalgebra (\href{geometry+of+physics+--+superalgebra#ModulesOverRbeta}{this prop.}).


\item 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 cohomology theory|even]] [[periodic ring spectrum]].

That [[E-infinity algebras]] over [[even cohomology theory|even]] [[periodic ring spectra]] are usefully regarded from the point of view of [[supercommutative superalgebra]] was highlighted in \hyperlink{Rezk09}{Rezk 09, section 2}.



\end{enumerate}
Hence it makes sense to say:

\textbf{Definition.} \emph{Spectral/$E_\infty$ super-geometry} is simply the [[E-∞ geometry]] over [[even cohomology theory|even]] [[periodic ring spectra]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{SpectralSuperpoint}{}\subsubsection*{{Spectral superpoint}}\label{SpectralSuperpoint}

The ordinary [[superpoint]] over some [[field]] $k$ is the [[spectrum of a commutative ring]] of the graded [[symmetric algebra]] on a single odd generator (``graded [[ring of dual numbers]]'')

\begin{displaymath}
\mathbb{A}_k^{0 \vert 1}
  \;\simeq\;
  Spec( \,Sym_k (k[1])\, )
\end{displaymath}
Accordingly, for $R$ an [[even cohomology theory|even]] [[periodic ring spectrum]], then the \emph{spectral superpoint} $R^{0 \vert 1}$ should be the [[spectral scheme]] given by the [[spectral symmetric algebra]] on the [[suspension spectrum]] of $R$:

\begin{displaymath}
\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}
  \,.
\end{displaymath}
where on the right we have the [[Thom space]] of the [[vector bundle]] $\tau_n$ [[associated bundle|associated]] to the $\Sigma(n)$-[[universal principal bundle]] via the canonical [[action]] of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at \href{permutation#ClassifyingSpaceAndThomSpace}{symmetric group -- Classifying space and Thom space}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

The proposal for spectral super-geometry \hyperlink{Definition}{above} invokes observations from

\begin{itemize}%
\item [[Charles Rezk]], section 2 of \emph{The congruence criterion for power operations in Morava E-theory}, Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.327-379 (\href{https://arxiv.org/abs/0902.2499}{arXiv:0902.2499})


\item [[Steffen Sagave]], [[Christian Schlichtkrull]], \emph{Diagram spaces and symmetric spectra}, Advances in Mathematics, Volume 231, Issues 3--4, October--November 2012, Pages 2116--2193 (\href{https://arxiv.org/abs/1103.2764}{arXiv:1103.2764})



\end{itemize}
The proposal \hyperlink{Definition}{above} was originally motivated from the discussion of the [[sphere spectrum]] in relation to [[super algebra]] highlighted in

\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Categorification of supersymmetry and stable homotopy groups of spheres}, talk at \emph{\href{http://mathserver.neu.edu/~bwebster/ACRT/}{Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)}} April 2013 (\href{http://mathserver.neu.edu/~bwebster/ACRT/calendar-with-abstracts.pdf}{abstract pdf}, \href{https://youtu.be/StbRti1fV7A}{video})


\item [[Mikhail Kapranov]], \emph{Supergeometry in mathematics and physics}, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) \emph{[[New Spaces for Mathematics and Physics]]} (\href{http://arxiv.org/abs/1512.07042}{arXiv:1512.07042})


\item [[Mikhail Kapranov]], \emph{Super-geometry}, talk at \emph{\href{http://ercpqg-espace.sciencesconf.org/program}{New Spaces for Mathematics \& Physics}}, IHP Paris, Oct-Sept 2015 (\href{https://www.youtube.com/watch?v=bjsNwKYT8JE}{video recording})



\end{itemize}
[[!redirects spectral super-schemes]]

[[!redirects spectral supergeometry]] [[!redirects E-∞ supergeometry]] [[!redirects E-infinity supergeometry]]

[[!redirects spectral super scheme]] [[!redirects spectral super schemes]]

[[!redirects spectral superpoint]] [[!redirects spectral superpoints]] [[!redirects spectral super point]] [[!redirects spectral super points]] [[!redirects spectral super-point]] [[!redirects spectral super-points]]

[[!redirects spectral superscheme]] [[!redirects spectral superschemes]]



\end{document}