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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On Spencer's cohomology theory for linear partial differential operators


Author: Joseph Johnson
Journal: Trans. Amer. Math. Soc. 154 (1971), 137-149
MSC: Primary 57.50
DOI: https://doi.org/10.1090/S0002-9947-1971-0283826-3
MathSciNet review: 0283826
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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

$\displaystyle 0 \to {R_{hq}} \to {\Lambda ^1}{T^\ast}{ \otimes _\mathcal{O}}{R_... ... \to \cdots \to {\Lambda ^n}{T^\ast}{ \otimes _\mathcal{O}}{R_{h - n,q}} \to 0.$

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.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0283826-3
Keywords: Linear partial differential operator, formally integrable operator, over-determined system of linear partial differential equations, Spencer cohomology
Article copyright: © Copyright 1971 American Mathematical Society