Differential forms on general commutative algebras

Abstract

Introduction. Let K be a commutative ring with ulnit, 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 K-derivations of R, and let E(TR) be the exterior algebra over R of TR. We can form HoomR(E(TR), R) and define on it the usual formal differentiation. If R is the ring of functions on a C?-rmanifold 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 HoMR(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 R-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(E(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 Poincare-BirkhoffWitt theorem for V. In ?4 we show that if Lis R-projective then for any V-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 C manifold is thus identified with an Extv(R, R).