The space of unital algebra homomorphisms $C^\infty(S)\to\Lambda_2$: a singular determinantal model

Abstract

Let $\Lambda_2$ be the real Grassmann algebra on two odd generators. Define $\mathcal{J}(S)$ as the set of unital algebra homomorphisms $C^\infty(S)\to \Lambda_2$ that are not required to preserve parity. This space is motivated by replacing the commutative local test algebra appearing in Weil's near-point construction with $\Lambda_2$. Each homomorphism expands to give a base point and three tangent vectors. The homomorphism condition, together with the noncommutativity of $\Lambda_2$, forces two of these tangent vectors to be linearly dependent. The resulting parameter space is $\mathcal{C}(TS) \times_S TS$, where $\mathcal{C}(TS)$ is the fibrewise determinantal cone of pairs of linearly dependent tangent vectors. When $\dim S \ge 2$, $\mathcal{J}(S)$ carries fibrewise conical singularities along its zero section; when $\dim S = 1$, $\mathcal{J}(S)$ is a smooth manifold.