On Spencer’s cohomology theory for linear partial differential operators

Abstract:

Let D be a linear partial differential operator between vector bundles on a differentiable manifold X of dimension n. Let $\mathcal {D}$ be the sheaf of germs of differentiable functions on X. For every $h \in Z$ a spectral sequence ${(^h}{E^{pq}})$ is associated to D. When D satisfies appropriate regularity conditions these spectral sequences degenerate for all sufficiently large h and $^hE_2^{p0}$ is the pth Spencer cohomology for D. One can compute $^hE_2^{pq}$ as the cohomology at ${\Lambda ^p}{T^\ast }{ \otimes _\mathcal {O}}{R_{h - p,q}}$ of a complex \[ 0 \to {R_{hq}} \to {\Lambda ^1}{T^\ast }{ \otimes _\mathcal {O}}{R_{h - 1,q}} \to \cdots \to {\Lambda ^n}{T^\ast }{ \otimes _\mathcal {O}}{R_{h - n,q}} \to 0.\] When q = 0 this complex coincides with the usual (first) Spencer complex for D. These results give a generalization of Spencer’s theory. The principal importance of this generalization is that it greatly clarifies the role played by homological algebra in the theory of overdetermined systems of linear partial differential equations.