Astérisque N° 219/1994
Les paires duales dans les algèbres

In this series of papers, we investigate the relative index theorem from the point of view of algebraic analysis. On a complex manifold X, let M he a coherent Dx-module and F an IR-mnstruc-tible sheaf. The complex RHompx (M, RHom(F, Ox)) is the complex of solutions of the system of PDE represented by M in the sheaf of generalized holomorphic functions associated to F. A natural problem is to find conditions under which such a complex has finite dimensional global cohomology and then to compute the corresponding Euler-Poincaré characteristic.
We prove the finiteness theorem when (M, F) has compact support and is "elliptic", i.e. : char(M) n SS(F) C TxX where char(M) is the characteristic variety of M, SS(F) is the micro-support of F and TxX is the zero section of the cotangent bundle. In fact, we give a relative version of this finiteness result together with the associated duality theorem and Künneth formula. Our methods rely upon results of functional analysis over a .sheaf of Fréchet algebras obtained by one of the authors.
Next, we attach a "microloeal Order class" µeu(M, F) E H2dx char M+SSF (T X ; D) to any elliptic pair (M, F) and prove that, under natural assumptions, this class is compatible with direct images, inverse images and external products. In particular, we get the index formula : X(RP(X ; RHom(M x F ; Ox))) = f yeu(M, F) T2xx (M) = fTX yeu (M) U yeu (F). When F = CM and X is a complexification of the real analytic manifold M, our salts for the pair (M, F) give an index theorem for elliptic systems and we discuss its relations with the Ariyah-Singer theorem.