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\begin{document}

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\section*{super Cartesian space}

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

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - 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{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{de_rham_complex_of_differential_forms}{De Rham complex of differential forms}\dotfill \pageref*{de_rham_complex_of_differential_forms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The analog of a [[Cartesian space]] in [[supergeometry]], a [[supermanifold]] with a single canonical [[coordinate chart]].

For suitable even|odd dimension this is naturally equipped with the structure of a [[supergroup]] that makes it a [[super-translation group]]. For Minkowski signature this is [[super-Minkowski spacetime]].

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

For $p,q \in \mathbb{N}$, let $\mathbb{R}^{p|q}$ the \emph{super Cartesian space} of [[dimension]]\_ $(p|q)$. This is the [[supermanifold]] defined by the fact that its [[algebra of functions]] is freely generated, as a [[smooth superalgebra]] by even-graded [[coordinate]]-functions $\langle \{x^a\}_{1}^p$ and odd-graded coordinate function $\{ \theta^\alpha\}_{\alpha= 1}^q$. Equivalently this is the [[tensor product]] of the [[smooth functions]] on $\mathbb{R}^p$ with the real [[Grassmann algebra]] on $q$ generators:

\begin{displaymath}
\begin{aligned}
      C^\infty(\mathbb{R}^{p|q}) 
      & \coloneqq 
      \langle \{x^a\}_{1}^p, \{ \theta^\alpha\}_{\alpha= 1}^q \rangle
      \\
      & = C^\infty(\mathbb{R}^p)\otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^q
  \end{aligned}
  \,.
\end{displaymath}
\hypertarget{details}{}\subsection*{{Details}}\label{details}

The following is taken from \emph{[[geometry of physics -- supergeometry]]}, see there for more:

\begin{defn}
\label{SuperCartesianSpace}\hypertarget{SuperCartesianSpace}{}
For $q\in \mathbb{N}$, the real [[Grassmann algebra]]

\begin{displaymath}
C^\infty(\mathbb{R}^{0|q})
  \coloneqq
  \wedge^\bullet \langle \theta^1, \cdots, \theta^q\rangle
\end{displaymath}
is the $\mathbb{R}$-algebra [[free functor|freely]] generated from $q$ generators $\{\theta^i\}_{i = 1}^q$ subject to the relations

\begin{displaymath}
\theta^i \theta^j = - \theta^j \theta^i
\end{displaymath}
for all $i,j \in \{1,\cdots, q\}$.

For $p,q \in \mathbb{N}$, the [[super-Cartesian space]] $\mathbb{R}^{p|q}$ is the [[formal dual]] of the [[supercommutative superalgebra]] written $\mathcal{O}(\mathbb{R}^{p\vert q})$ or $C^\infty(\mathbb{R}^{p|q})$ whose underlying $\mathbb{Z}/2\mathbb{Z}$-[[graded vector space]] is

\begin{displaymath}
C^\infty(\mathbb{R}^{p|q})
  \coloneqq
  C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet\langle\theta^1, \cdots, \theta^q\rangle
\end{displaymath}
with the product given by the relations

\begin{displaymath}
\begin{aligned}
  \left(
    f \theta^{i_1}\cdots \theta^{i_k}
  \right)
  \left(
    g \theta^{j_1}\cdots \theta^{j_l}
  \right)
  & =
  f \cdot g \; \theta^{i_1}\cdots \theta^{i_k} \theta^{j_1}\cdots \theta^{j_l}
  \\
  & =
  (-1)^{k l} g\cdot f \; \theta^{j_1}\cdots \theta^{j_l} \theta^{i_1}\cdots \theta^{i_k}
  \end{aligned}
\end{displaymath}
where $f \cdot g$ is the ordinary pointwise product of [[smooth functions]].

Write

\begin{displaymath}
SuperCartSp \hookrightarrow sCAlg_{\mathbb{R}}^{op}
\end{displaymath}
for the [[full subcategory]] of the [[opposite category]] of [[commutative superalgebras]] on those of this form. We write $\mathbb{R}^{p|q} \in SuperCartSp$ for the [[formal dual]] of $C^\infty(\mathbb{R}^{p|q})$.

\end{defn}
We write

\begin{displaymath}
CartSp \hookrightarrow SuperCartSp
\end{displaymath}
for the [[full subcategory]] on ordinary [[Cartesian spaces]] with [[smooth functions]] between them. These are the ``abstract coordinate charts'' from the discussion at \emph{[[geometry of physics -- smooth sets]]}, and so we are evidently entitled to think of the objects in $SuperCartSp$ as \textbf{abstract super coordinate systems} and to develop a geometry induced from these.

Recall the two magic algebraic properties of [[smooth functions]] that make the above algebraic description of [[differential geometry]] work:

\begin{enumerate}%
\item (\textbf{[[embedding of smooth manifolds into formal duals of R-algebras]]}) The functor that assigns [[algebra of functions|algebras of]] [[smooth function]] of [[smooth manifolds]]

\begin{displaymath}
C^\infty(-) \;\colon\; SmthMfd \hookrightarrow CAlg_{\mathbb{R}}^{op}
\end{displaymath}
is [[fully faithful functor|fully faithful]].


\item (\textbf{[[smooth Serre-Swan theorem]]}) The functor that assigns smooth [[sections]] of smooth [[vector bundles]] of [[finite number|finite]] [[rank]]

\begin{displaymath}
\Gamma_X(-) \;\colon\; SmthVectBund_{/X} \hookrightarrow C^\infty(X) Mod
\end{displaymath}
is [[fully faithful functor|fully faithful]] (its [[essential image]] being the [[finitely generated object|finitely generated]] [[projective modules]] over the $\mathbb{R}$-[[algebra of functions|algebra of]] [[smooth function]]).



\end{enumerate}
There is a third such magic algebraic property of smooth functions, which plays a role now:

\begin{defn}
\label{DerivationsOfSmoothFunctions}\hypertarget{DerivationsOfSmoothFunctions}{}
\textbf{([[derivations of smooth functions are vector fields]])}

Let $X$ be a [[smooth manifold]]. Write

\begin{displaymath}
der_X \;\colon\; \Gamma(T X) \longrightarrow Der(C^\infty(X))
\end{displaymath}
for the function that sends a smooth [[vector field]] $v \in \Gamma(T X)$ to the [[derivation]] of the [[algebra of functions|algebra of]] [[smooth functions]] on $X$ given by forming [[derivatives]]: $der_X(v)(f) \coloneqq v(f)$. This is a [[derivation]] by the [[chain rule]].

Then this function is a [[bijection]], hence every [[derivation]] of $C^\infty(X) \in CAlg_{\mathbb{R}}$ comes from [[differentiation]] along some smooth vector field, which is uniquely defined thereby.

\end{defn}
\begin{proof}
By the existence of [[partitions of unity]] we may restrict to the situation where $X = \mathbb{R}^n$ is a [[Cartesian space]]. By the [[Hadamard lemma]] every [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ may be written as

\begin{displaymath}
f(x) = f(0) + \sum_i x_i g_i(x)
\end{displaymath}
for [[smooth functions]] $\{g_i \in C^\infty(X)\}$ with $g_i(0) = \frac{\partial f}{\partial x_i}(0)$. Since any derivation $\delta : C^\infty(X) \to C^\infty(X)$ by definition satisfies the Leibniz rule, it follows that

\begin{displaymath}
\delta(f)(0) = \sum_i \delta(x_i) \frac{\partial f}{\partial x_i}(0)
  \,.
\end{displaymath}
Similarly, by translation, at all other points. Therefore $\delta$ is already fixed by its action of the coordinate functions $\{x_i \in C^\infty(X)\}$. Let $v_\delta \in T \mathbb{R}^n$ be the [[vector field]]

\begin{displaymath}
v_\delta \coloneqq \sum_i \delta(x_i) \frac{\partial}{\partial x_i}
\end{displaymath}
then it follows that $\delta$ is the derivation coming from $v_\delta$ under $\Gamma_X(T X) \to Der(C^\infty(X))$.

\end{proof}
Recall further from \emph{[[geometry of physics -- superalgebra]]} that the category of [[supercommutative superalgebras]] is related to that of ordinary [[commutative algebras]] over $\mathbb{R}$ by an [[adjoint cylinder]] (\href{geometry+of+physics+--+superalgebra#InclusionOfCAlgIntosCAlg}{this prop.}):

\begin{prop}
\label{AdjointCylinderOnSuperAffines}\hypertarget{AdjointCylinderOnSuperAffines}{}
The canonical inclusion of [[commutative algebras]] into [[supercommutative superalgebra]] is part of an [[adjoint triple]] of the form

\begin{displaymath}
CAlg_k
    \underoverset
      {\underset{(-)_{even}}{\longleftarrow}}
      {\overset{(-)/(-)_{odd}}{\longleftarrow}}
      {\hookrightarrow}
  sCAlg_k
  \,.
\end{displaymath}
The [[formal dual]] of this statement is that [[affine superschemes]] are related to ordinary [[affine schemes]] over $\mathbb{R}$ by an [[adjoint cylinder]] of this form

\begin{displaymath}
Aff(Vect_k)
    \underoverset
      {\underset
        {
          \phantom{AA}
          \overset
            {\phantom{A}}
            {\overset{\rightsquigarrow}{(-)}}
          \phantom{AA}
         }
         {\longleftarrow}
      }
      {\overset
         {
          \phantom{AA}
           \underset
            {\phantom{A}}
            { \overset{\rightrightarrows}{(-)} }
          \phantom{AA}
        }
        {\longleftarrow}
      }
      {\hookrightarrow}
  Aff(sVect_k)
  \,.
\end{displaymath}
(Beware that $\overset{\rightsquigarrow}{(-)}$ is the formal dual of $(-)/(-)_{odd}$ while $\overset{\rightrightarrows}{(-)}$ is the formal dual of $(-)_{even}$. That they change position in the diagrams is because we always draw [[left adjoints]] on top of [[right adjoints]] and the handedness of [[adjoints]] changes as we pass to [[opposite categories]].)

\end{prop}
The notation in prop. \ref{AdjointCylinderOnSuperAffines} is to serve as a convenient mnemonic for the nature of these functors:

In a [[Feynman diagram]]

\begin{enumerate}%
\item a single [[fermion]] is denoted by a solid arrow ``$\to$'' and the functor $\overset{\rightrightarrows}{(-)}$ produces a space whose [[algebra of functions]] is generated over smooth functions by the product of two fermionic functions


\item a single [[boson]] is denoted by a wiggly arrow $\rightsquigarrow$ and the functor denoted by this symbol produces a spaces whose [[algebra of functions]] contains only the bosonic smooth functions, no odd-graded functions.



\end{enumerate}
This highlights an important point: while the image of a [[super Cartesian space]] under $\rightsquigarrow$ is an ordinary [[Cartesian space]]

\begin{displaymath}
\overset{\rightsquigarrow}{\mathbb{R}^{p\vert q}}
   \simeq 
   \mathbb{R}^p
\end{displaymath}
its image under $\overset{\rightrightarrows}{(-)}$ is a bosonic space, but not an ordinary manifold. For instance

\begin{displaymath}
\begin{aligned}
   \mathcal{O}\left( \overset{\rightrightarrows}{\mathbb{R}^{0\vert 2}}\right)
    & = 
   \mathcal{O}(\mathbb{R}^{0\vert 2})_{even}
   \\
  & = 
  (\wedge^\bullet(\mathbb{R}^2)^\ast)_{even}
  \\
  &=
  \left\{a_0 + a_1 \, \theta_1 + a_2 \, \theta_2 + a_{12} \theta_1 \theta_2 \,\vert a_\bullet \in \mathbb{R}\,\right\}_{even}
  \\
  &=
  \left\{a_0  + a_{12} \theta_1 \theta_2 \,\vert a_{12} \in \mathbb{R}\,\right\}
  \\
  &=
  \mathbb{R}[\epsilon]/(\epsilon^2 = 0)
  \end{aligned}
\end{displaymath}
where in the last line we renamed $\theta_1 \theta_2$ to $\epsilon$.

This algebra $\mathbb{R}[\epsilon]/(\epsilon^2)$ is known as the \textbf{[[algebra of dual numbers]]} over $\mathbb{R}$. It is to be thouhgt of as the [[algebra of functions]] on a bosonic but [[infinitesimally thickened point]], a 1-dimensional neighbourhood of a point which is ``so very small'' that the canonical [[coordinate]] function $\epsilon$ on it takes values ``so tiny'' that its square, which is bound to be even tinier, is actually indistinguishable from zero.

In generalization of this we make the following definitions

\begin{defn}
\label{FormalCartSp}\hypertarget{FormalCartSp}{}
Write

\begin{displaymath}
InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
\end{displaymath}
for the [[full subcategory]] of the [[opposite category]] of [[commutative algebras]] over $\mathbb{R}$ on [[formal duals]] of [[commutative algebras]] over the [[real numbers]] of the form $\mathbb{R}\oplus V$ with $V$ a [[nilpotent ideal]] of [[finite number|finite]]-[[dimension]] over $\mathbb{R}$. We call this the category of \emph{[[infinitesimally thickened points]]}.

In [[synthetic differential geometry]] these algebras ar called \textbf{[[Weil algebra]]}, while in [[algebraic geometry]] they are known as \textbf{local [[Artin algebras]]} over $\mathbb{R}$.

Write moreover

\begin{displaymath}
FormalCartSp \coloneqq CartSp \rtimes InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
\end{displaymath}
for the [[full subcategory]] on [[formal duals]] of those algebras which are [[tensor products]] of commutative $\mathbb{R}$-algebras of the form

\begin{displaymath}
C^\infty(\mathbb{R}^n) \otimes C^\infty(D)
\end{displaymath}
of algebras $C^\infty(\mathbb{R}^p)$ of [[smooth functions]] $\mathbb{R}^n$ with algebras corresponding to infinitesimally thickened points $D$ as above.

\end{defn}
This kind of construction is traditionally more familiar from the theory of [[formal schemes]], but the same kind of general abstract theory goes through in the context of [[differential geometry]], a point of view known as \emph{[[synthetic differential geometry]]}.

The crucial property of [[infinitesimally thickened points]] (def. \ref{FormalCartSp}) is that they co-represent [[tangent vectors]] and [[jets]]:

\begin{example}
\label{}\hypertarget{}{}
Write $\mathbb{D}^1 = Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$ for the [[formal dual]] of the [[algebra of dual numbers]]. Then morphisms

\begin{displaymath}
\mathbb{R}^n \times \mathbb{D}\longrightarrow \mathbb{R}^n
\end{displaymath}
which are the identity after restriction along $\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}$, are equivalently algebra homomorphisms of the form

\begin{displaymath}
(C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
  \longleftarrow
  C^\infty(\mathbb{R}^n)
\end{displaymath}
which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to

\begin{displaymath}
f + (\partial f) \epsilon
\end{displaymath}
for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$

\begin{displaymath}
(f_1 f_2 + (\partial (f_1 f_2))\epsilon )
  =
  (f_1 + (\partial f_1) \epsilon)
  (f_2 + (\partial f_2) \epsilon)
  \,.
\end{displaymath}
Multiplying this out and using that $\epsilon^2 = 0$ this in turn is equivalent to

\begin{displaymath}
\partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
  \,.
\end{displaymath}
This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a [[derivation]]. But [[derivations of smooth functions are vector fields]] (prop. \ref{DerivationsOfSmoothFunctions}).

In particular one finds that maps

\begin{displaymath}
\mathbb{D} \longrightarrow \mathbb{R}^n
\end{displaymath}
are equivalently single [[tangent vectors]], hence for every $\mathbb{R}^n$ there is a [[natural bijection]]

\begin{displaymath}
Hom_{FormalCartSp}(\mathbb{D}^1, \mathbb{R}^n)
   \simeq
  \underset{\text{underlying} \atop \text{set}}{\underbrace{T \mathbb{R}^n}}
\end{displaymath}
between the [[hom-set]] from the formal dual of the [[ring of dual numbers]] and the set of [[tangent vectors]].

\end{example}
\begin{prop}
\label{CartSpCoreflectiveInclusion}\hypertarget{CartSpCoreflectiveInclusion}{}
The canonical inclusion $i$ of the category of ordinary [[Cartesian spaces]] into that of [[formal manifold|formal]] [[Cartesian spaces]] has a [[left adjoint]] $\Re$

\begin{displaymath}
CartSp 
    \underoverset
      {\underset{\Re}{\longleftarrow}}
      {\overset{i}{\hookrightarrow}}
      {\phantom{AAAA}}
  FormalCartSp
\end{displaymath}
given by

\begin{displaymath}
\Re(\mathbb{R}^n \times \mathbb{D}) \coloneqq \mathbb{R}^n
  \,.
\end{displaymath}
Hence exhibits $CartSp$ as a [[coreflective subcategory]] of that of formal cartesian spaces.

We say that $\mathbb{R}^n$ is the \textbf{[[reduced scheme]]} of $\mathbb{R}^n \times \mathbb{D}$.

\end{prop}
\begin{proof}
We check the [[natural isomorphism]] on [[hom-sets]] that characterizes a pair of [[adjoint functors]]:

By definition, a morphism of the form

\begin{displaymath}
f \;\colon\; i(\mathbb{R}^{n_1}) \longrightarrow \mathbb{R}^{n_2} \times \mathbb{D}
\end{displaymath}
is equivalently a [[homomorphism]] of [[commutative algebras]] of the form

\begin{displaymath}
f^\ast 
   \;\colon\;
  C^\infty(\mathbb{R}^{n_1}) 
    \longleftarrow
  C^\infty(\mathbb{R}^{n_2}) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V)
\end{displaymath}
where all elements $v \in V$ are nilpotent, in that there exists $n_v \in \mathbb{N}$ such that $(v)^{n_v} = 0$. Every algebra homomorphism needs to preserve this equation, and hence needs to send nilpotent elements to nilpotent elements. But the only nilpotent element in the ordinary function algebra $C^\infty(\mathbb{R}^n)$ is the zero-function, and so it follows that the above homomorphism has to vanish on all of $V$, hence has to factor (necessarily uniquely) through a homomorphism of the form

\begin{displaymath}
\tilde f^\ast 
   \;\colon\;
  C^\infty(\mathbb{R}^{n_1})
    \longleftarrow
  C^\infty(\mathbb{R}^{n_2})
  \,.
\end{displaymath}
This is dually a morphism of the form

\begin{displaymath}
\tilde f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
\end{displaymath}
in $CartSp$. This establishes a natural bijection $f \leftrightarrow \tilde f$.

\end{proof}
The above discussion following prop. \ref{AdjointCylinderOnSuperAffines} means that in passing to commutative superalgebras there are \emph{two} stages of generalizations of plain differential geometry involved:

\begin{enumerate}%
\item [[Cartesian spaces]] are generalized to [[formal manifold|formal]] [[Cartesian spaces]];


\item [[formal manifold|formal]] [[Cartesian spaces]] are further generalized to [[super formal Cartesian spaces]].



\end{enumerate}
In order to make this explicit, it is convenient to introduce the following slight generalization of [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}), which are simply [[Cartesian products]] of ordinary [[Cartesian spaces]] with an [[infinitesimally thickened point]] that may have both even and odd graded elements in its [[algebra of functions]].

\begin{defn}
\label{SuperFormalCartSp}\hypertarget{SuperFormalCartSp}{}
Write

\begin{displaymath}
SuperFormalCartSp
    \hookrightarrow
  sCAlg_{\mathbb{R}}^{op}
\end{displaymath}
for the [[full subcategory]] of that of the [[opposite category]] of [[supercommutative superalgebras]] on whose of the form

\begin{displaymath}
C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V)
\end{displaymath}
where $V$ is a [[nilpotent ideal]] of [[finite number|finite]] [[dimension]] over $\mathbb{R}$.

\end{defn}
One place where such super formal Cartesian spaces are made explicit is in \hyperlink{KonechnySchwarz97}{Konechny-Schwarz 97}

In conclusion we have the following situation:

\begin{prop}
\label{}\hypertarget{}{}
The [[coreflective subcategory]] inclusion of [[Cartesian spaces]] into [[formal manifold|formal]] [[Cartesian spaces]] from prop. \ref{CartSpCoreflectiveInclusion} and the coreflective as we all [[reflective subcategory]] inclusion\newline of affine schemes into [[affine superschemes]] from prop. \ref{AdjointCylinderOnSuperAffines} combine to give the following system of [[adjoint functors]] on our local model spaces

\begin{displaymath}
\itexarray{
   {\text{} \atop {\text{differential} \atop \text{geometry}}}
   &&
   {\text{formal} \atop {\text{differential} \atop {geometry}}}
   &&
   {\text{super} \atop {\text{differential} \atop \text{geometry}}}
   \\
   \\
  CartSp
   &
    \underoverset
      {\underset{\Re}{\longleftarrow}}
      {\overset{}{\hookrightarrow}}
      {}
   &
  FormalCartSp
  &
    \underoverset
      {\underset
        {
          \phantom{AA}
          \overset
            {\phantom{A}}
            {\overset{\rightsquigarrow}{(-)}}
          \phantom{AA}
         }
         {\longleftarrow}
      }
      {\overset
         {
          \phantom{AA}
           \underset
            {\phantom{A}}
            { \overset{\rightrightarrows}{(-)} }
          \phantom{AA}
        }
        {\longleftarrow}
      }
      {\hookrightarrow}
  &
  SuperFormalCartSp
  }
  \,.
\end{displaymath}
\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[superpoint]], [[odd line]]


\item [[super Minkowski spacetime]]



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{de_rham_complex_of_differential_forms}{}\subsubsection*{{De Rham complex of differential forms}}\label{de_rham_complex_of_differential_forms}

We discuss the [[de Rham complex]] of [[super differential forms]] on a super Cartesian space. (See also at \emph{[[signs in supergeometry]]}.)

Accordingly, by the discussion at \emph{[[Kähler forms]]}, the [[de Rham complex]] of [[super differential forms]] on $\mathbb{R}^{p|q}$ is freely generated as a super-[[module]] over the [[smooth superalgebra]] $C^\infty(\mathbb{R}^{p|q})$ by expressions of the form $\mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k} \wedge \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}$.

\begin{displaymath}
\Omega^\bullet(\mathbb{R}^{p|q})
  = 
  C^\infty(\mathbb{R}^{p|q})
  \left[
   \left\{
     \mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k} 
     \wedge 
     \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}
   \right\}
  \right]
  \,.
\end{displaymath}
This [[de Rham complex]] now carries the structure of a

\begin{itemize}%
\item differential graded-commutative graded commutative superalgebra

\end{itemize}
which should be thought of as bracketet as follows

\begin{itemize}%
\item differential graded-commutative graded (commutative superalgebra).

\end{itemize}
This means in effect that elements of $\Omega^\bullet(\mathbb{R}^{p|q})$ carry a $\mathbb{Z} \times \mathbb{Z}_2$-[[graded object|grading]], where we may say that

\begin{itemize}%
\item $\mathbb{Z}$ corresponds to the ``cohomological grading'';


\item $\mathbb{Z}_2$ corresponds to the super-grading.



\end{itemize}
We write

\begin{displaymath}
(n,\sigma)\in \mathbb{Z} \times \mathbb{Z}_2
\end{displaymath}
for elements in this grading group.

In this notation the grading of the elements in $\Omega^\bullet(\mathbb{R}^{p|q})$ is all induced by the fact that the de Rham differential $\mathbf{d}$ itself is a [[derivation]] of degree $(1,even)$.

\begin{tabular}{l|l}
generator&bi-degree\\
\hline 
$x^a$&(0,even)\\
$\theta^\alpha$&(0,odd)\\
$\mathbf{d}$&(1,even)\\
\end{tabular}

Here the last line means that we have

\begin{tabular}{l|l}
generator&bi-degree\\
\hline 
$x^a$&(0,even)\\
$\theta^\alpha$&(0,odd)\\
$\mathbf{d}x^a$&(1,even)\\
$\mathbf{d}\theta^\alpha$&(1,odd)\\
\end{tabular}

The formula for the ``cohomologically- and super-graded commutativity'' in $\Omega^\bullet(\mathbb{R}^{p|q})$ is

\begin{displaymath}
\alpha \wedge \beta 
  = 
  \;
  (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
  \;
  \beta \wedge \alpha
\end{displaymath}
for all $\alpha,  \beta \in \Omega^\bullet(\mathbb{R}^{p|q})$ of homogeneous $\mathbb{Z}\times \mathbb{Z}_2$-degree. Hence there are \emph{two} contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

\begin{enumerate}%
\item there is a ``cohomological sign'' which for commuting a $p_1$-forms past a $p_2$-form is $(-1)^{p_1 p_2}$;


\item in addition there is a ``super-grading'' sich which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.



\end{enumerate}
Some examples:

\begin{displaymath}
x^{a_1} (\mathbf{d}x^{a_2}) = + (\mathbf{d}x^{a_2}) x^{a_1}
\end{displaymath}
\begin{displaymath}
\theta^\alpha (\mathbf{d}x^a) = + (\mathbf{d}x^a) \theta^\alpha
\end{displaymath}
\begin{displaymath}
\theta^{\alpha_1} (\mathbf{d}\theta^{\alpha_2}) 
   = 
  - (\mathbf{d}\theta^{\alpha_2}) \theta^{\alpha_1}
\end{displaymath}
\begin{displaymath}
\mathbf{d}x^{a_1}
  \wedge
  \mathbf{d} x^{a_2}
  =
  -
  \mathbf{d} x^{a_2}
  \wedge
  \mathbf{d} x^{a_1}
\end{displaymath}
\begin{displaymath}
\mathbf{d}x^a
  \wedge
  \mathbf{d} \theta^{\alpha}
  =
  -
  \mathbf{d}\theta^{\alpha}
  \wedge
  \mathbf{d} x^a
\end{displaymath}
\begin{displaymath}
\mathbf{d}\theta^{\alpha_1}
  \wedge
  \mathbf{d} \theta^{\alpha_2}
  =
  +
  \mathbf{d}\theta^{\alpha_2}
  \wedge
  \mathbf{d} \theta^{\alpha_1}
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Euclidean space]]


\item [[Euclidean G-space]]



\end{itemize}
[[!include geometries of physics -- table]]

\begin{itemize}%
\item [[super-Cartan geometry]]

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

\begin{itemize}%
\item Anatoly Konechny and [[Albert Schwarz]],

\emph{On $(k \oplus l|q)$-dimensional supermanifolds} in \emph{Supersymmetry and Quantum Field Theory} ([[Dmitry Volkov]] memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(\href{http://arxiv.org/abs/hep-th/9706003}{arXiv:hep-th/9706003})

\emph{Theory of $(k \oplus l|q)$-dimensional supermanifolds} Sel. math., New ser. 6 (2000) 471 - 486



\end{itemize}
[[!redirects super Cartesian spaces]]

[[!redirects Cartesian superspace]] [[!redirects Cartesian superspaces]]

[[!redirects super cartesian space]] [[!redirects super cartesian spaces]]

[[!redirects super-Cartesian space]]

[[!redirects super Euclidean space]] [[!redirects super Euclidean spaces]] [[!redirects super-Euclidean space]] [[!redirects super-Euclidean spaces]]

[[!redirects super formal Cartesian space]] [[!redirects super formal Cartesian spaces]]

[[!redirects SuperCartSp]]

[[!redirects SuperFormalCartSp]] [[!redirects SuperFormalCartSpace]]



\end{document}