Differential forms on general commutative algebras

Introduction.Let K be a commutative ring with unit, and let R be a commutative unitary K-algebra.We shall be concerned with variously defined cohomology theories based on algebras of differential forms, where R plays the role of a ring of functions.Let TR be the Lie algebra of the X-derivations of R, and let E(TR) be the exterior algebra over R of TR.We can form HomR(E(TR), R) and define on it the usual formal differentiation.If R is the ring of functions on a C""-manifold then the elements of TR are the differentiable tangent vector fields, and the complex HomR(E(TR), R) is naturally isomorphic to the usual de Rham complex of differential forms.In 5, 6-9 the complex UomR(E(TR),R) is studied.It is shown that if K is a field contained in R, and if either R is an integral domain finitely ring-generated over K and TR is R-projective, or R is a field, then the homology of this complex may be identified with Extv(R,R), for a suitably defined ring V. 1-6 of the present work are primarily a straightforward generalization of the results of this portion of 5 to the case in which K and R are arbitrary (commutative) rings.In making this generalization we are led naturally to replace TR by an arbitrary Lie algebra with an B-module structure which is represented as derivations of R and which satisfies certain additional properties satisfied by TR.We give these properties in 2.Lis essentially a quasi-Lie algebra as defined in 3.The precise definition given corresponds to that of a d-Lie ring given in 8, where also the cohomology based on HomR((L), A) is defined.In 2 we define an associative algebra V of universal differential operators generated by R and L. In case L operates trivially on R, V is the usual universal enveloping algebra of the R-Lie algebra L. In 3 we prove a Poincar-Birkhoff-Witt theorem for V.In 4 we show that if Lis B-projective then for any F-module A we may identify the cohomology based on HomR(E(L),A) with Extv(R,A), which we denote by HR(L,A).In particular, the de Rham cohomology of a C00manifold is thus identified with an Bxtv(R,R).