Nl = 'Homo(wx,M) , x) x). called the left conversion functor . It is obvious that M when ME Mod(V x ) and N E Mod(V x)). 4 The adjoint involution. Let (U, xl, . ,x n ) be a chart in X.

Identifying ex with a subsheaf of Dx we construct the commutator [{j, {j'] in the ring Dx. In a chart we write Then the commutator is j ,v From this expression in local coordinates we recognize that [{j, (j'] is the usual Lie bracket of the two vector fields. 9 Integrable connections. Let M be some Ox-Module. Put L(M) = 'Homcx(M , M) If cp E L(M) and f E Ox we get fcp E L(M) defined by m L(M) is an Ox-Module. Now we consider the sheaf 'Homo (ex , L(M)) f-> f· cp(m). Hence THE SHEAF 'Dx AND ITS MODULES 19 where the index X in Ox is dropped to simplify the notations.

R(An). r(An). r(An) is isomorphic with An. So the Fuchsian filtration gives an associated graded ring isomorphic to the ring itself! r(An)(m)} and for every m one has: where Vk(m) = Vk(m) n An(m) . r(An)(m - 1) m~O is bigraded and denoted by B. The C-algebra B is commutative. Consider the the bihomogeneous elements {O'I(Pk)} and {O'-I(Tv)}. These elements generate a sub algebra of B which is isomorphic to the polynomial ring in 2n-variables. )] is not equal to B. For example, if n = 1 this subring does not contain bihomogeneous elements of the form O'o(p(x)), where p(x) is a polynomial identified with an element of VO/ V- 1 .