The main application of the
theorem proved here is to establish the local solvability of a system of linear partial
differential equations, in the analytic case, by a homological procedure based on the
associated Spencer resolution and δ-cohomology. The theorem states that the
δ-cohomology associated with an involutive system of partial differential equations
vanishes in a normed sense. From this one can show that the Spencer resolution
associated with an involutive system is exact for analytic data, and thus by a result
of D. G. Quillen the corresponding inhomogeneous system has local solutions,
provided the inhomogeneous term is analytic and satisfies the appropriate
compatibility conditions in the overdetermined case. It is well known that if an
arbitrary system is prolonged a sufficient number of times, the resulting system will
have vanishing δ-cohomology. According to a result of J. P. Serre this is equivalent to
the resulting system being involutive. Thus the question of local solvability
reduces to the involutive case, and we obtain the classical existence theorem of
Cartan-Kähler.
