Vol. 21, No. 2, 1967

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Existence of a homotopy operator for Spencer’s sequence in the analytic case

C. Buttin

Vol. 21 (1967), No. 2, 219–240

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.

Mathematical Subject Classification
Primary: 57.60
Secondary: 35.00
Received: 9 September 1965
Published: 1 May 1967
C. Buttin