On the non-linear cohomology of Lie equations. I

Abstract

Let Rkcdk(T) be a differential equation; set Rk_l=dk_l(T), Rk_2=dk_2(T), -~k+z= v-lRk+~cJk+~(T), RO+z=Rk+zf~J~ ~k+~=v-l~+zcJk+z(~7), and set J~(Rk)= V-lJl(Rk)cJ(z.k)(T). For l>~--l, let gk+zcSk+lJo(T)*| ) be the kernel of ~+~-1: (7.3) where 2z: Jk+~(T)-+~(1, k)(T). We remark that the sheaf Sol (Rk) of solutions of R~ is stable under the Lie bracket of vector fields. We say that R k is/ormally transitive if n0: Rk~Jo(T) is surjective. The differential equations Jk(T; ~) and J~(V) considered in w 6 are Lie equa- tions, and Jk(T; ~) is formally transitive. A differentiable sub-groupoid Pk of Qk is a Lie equation (finite form) if it is a fibered submanifold of ~: Qk-'X. For xeX, I~(x)~P~ and V~,~:)(P~) determines a subspace /~.~ of ,)~(T)x. The vector sub-bundle R~,~Ju(T) such that R~.~=v(/~.~) is a Lie equation (infinitesimal form); we say that P~ is a finite form of R~. :For example, the sub-groupoids Q~(~) and Q~(V) of Qz are finite forms of J~(T; ~) and J~(V) respectively. We have/~. F=