On the first hand one can define a superdomain $U^{p|q}$ as the super ringed space $(U^p,\mathcal{C}^{\infty p|q})$ where $U^p\subset\mathbb R^p$ is open and $\mathcal{C}^{\infty p|q}$ is the sheaf of supercommuting rings defined by [ $$\mathcal{C}^{\infty p|q} : V\mapsto C^\infty(V) [\theta_1,\dots,\theta_q], $$ where $V\subset U$ is open and the $\theta_j$ are anticommuting variables satisfying $\theta_i\theta_j=-\theta_j\theta_i$. Its dimension is defined to be $p|q$. $\mathbb R^{p|q}$ is a superdomain.

On the other hand consider the Grassman algebra $G(\mathbb R)$ over $\mathbb R$ with infinitely many generators $1, l_1,l_2,\dots$. Each element of $G(\mathbb R)$ is a finite linear combination of monomials $l_{i_1}\wedge\dots\wedge l_{i_n}$. Such a monomial is said to be even (resp. odd) if $n$ is even (resp. odd). Let $G_0$ and $G_1$ be the subspaces of $G(\mathbb R)$ spanned by the even and odd monomials respectively. One then set $$\mathbb R^{p|q}=\lbrace (x_1,\dots,x_p | \theta_1,\dots,\theta_q) ; x_i\in G_0,\theta_j\in G_1 \rbrace. $$

These are two different definitions of $\mathbb R^{p|q}$ : can anybody explain why they are equivalent or not ?