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According to E.
Cartan’s prolongation theorem, an analytic system of linear partial differential
equations becomes an involutive system, after prolongation in a finite number of
steps, and an involutive system has local solutions, by the Cartan-Kähler
theorem.
Recently, a homological procedure has been developed, in terms of which the
notion of involution is equivalent to the vanishing of a certain type of cohomology
(so-called δ-cohomology”). Moreover, the local solvability of a linear system of partial
dfflerential equations has been shown by Quillen to be equivalent to the exactness, at
degree one, of a certain resolution introduced originally by Spencer, which is
canonically associated with the given system. The terms of the resolution
are sheaves of germs of jet forms, i.e., differential forms with values in jet
spaces.
The exactness of this resolution, providing a replacement for the Cartan-Kähler
theorem in the linear case, in the analytic case is known. We shall have given another
proof, based on the construction of a homotopy operator which is a natural
generalization, to jet forms, of the well-known homotopy operator used in proving the
Poincaré lemma for the exterior derivative d.
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