Deformation theory and Lie algebra homology .1.

A description of a ring of functions on the base of a universal formal deformation for several moduli problems is given. The answer is given in terms of a homology group of a certain dg Lie algebra associated to a problem. In part I, for any dg Lie algebroid A on a differentially smooth locally ringed space X, a canonical map (connecting morphism) from the sheaf of differential operators Diff(X) to the homology of the dg Lie algebra ker(A --> T-X) is constructed.