Finite Grassmann algebra and top-coefficient integral

Let $R$ be a commutative unital $\mathbb{C}$-algebra and let $d\ge 1$. Let ${\mathrm{\Lambda }}_d(R)$ be the exterior algebra over $R$ on $2d$ distinct generators ${\psi }_1,\dots ,{\psi }_d,{\bar{\psi }}_1,\dots ,{\bar{\psi }}_d$. Thus every pair of generators anticommutes, each generator squares to zero, and coefficients from $R$ commute with every generator. The bar is part of a generator's name; no complex-conjugation relation is imposed.

Order the generators as ${\bar{\psi }}_1,\dots ,{\bar{\psi }}_d,{\psi }_1,\dots ,{\psi }_d$. The corresponding ordered monomials form the $R$-basis of ${\mathrm{\Lambda }}_d(R)$. Let $M_d:={\bar{\psi }}_1\cdots {\bar{\psi }}_d{\psi }_1\cdots {\psi }_d$ and let $[M_d]P\in R$ denote the coefficient of $M_d$ in $P\in {\mathrm{\Lambda }}_d(R)$. Define the finite Grassmann integral by the $R$-linear functional ${\mathcal{I}}_d(P):=(-1{)}^{d(d+1)\mathrm{/}2}[M_d]P\mathrm{.}$ For $K\in M_d(R)$, put $B_K:={\sum }_{i,j=1}^d{\bar{\psi }}_i{K}_{ij}{\psi }_j$. Since every monomial of degree greater than $2d$ vanishes, its exponential is the finite sum $\exp (-B_K):={\sum }_{n=0}^d\frac{(-B_K{)}^n}{n!}\mathrm{.}$ The sign factor in ${\mathcal{I}}_d$ is the chosen normalization. Its Gaussian determinant identity, including the value at $K=I_d$, is proved separately.