Image Theory of Superposed Vector Sets.

Let u e Hs(M, 4r). If 6* u = 0, then u E a* H8(M, 'r-1); that is, there is a covering c1 and a po e Z8( ) (V), (%) * Up = u, such that jo = aw, c e Zs('u, Ur-1). Let T, e r(Fr-1' 8, M) be the image of w in the process describing (2.3). Then a. = 6)Tr is the image of bw in the same process. If M is compact Kdhler, we conclude that o = b54, e Cs-1(clt, Or) (cf. ref. 4). The above remarks are merely intended to suggest the manner in which cohomology with coefficients in a faisceau may be applied to obtain Riemann-Roch theorems. A systematic development will be presented elsewhere.