Differential Forms on Regular Affine Algebras
DIFFERENTIAL FORMS ON REGULAR AFFINE ALGEBRAS BY G. HOCHSCHILD, BERTRAM ROSTANT AND ALEX ROSENBERG(') 1. Introduction. The formal apparatus of the algebra of differential forms appears as a rather special amalgam of multilinear and homological algebra, which has not been satisfactorily absorbed in the general theory of derived functors. It is our main purpose here to identify the exterior algebra of differ- ential forms as a certain canonical graded algebra based on the Tor functor and to obtain the cohomology of differential forms from the Ext functor of a universal algebra of differential operators similar to the universal enveloping algebra of a Lie algebra. Let K be a field, E a commutative E-algebra, Tr the E-module of all E-derivations of E, Dr the E-module of the formal K-differentials (see §4) on E. It is an immediate consequence of the definitions that Tr may be identified with HomR(DR, R). However, in general, Dr is not identifiable with Homie(Fie, E). The algebra of the formal differentials is the exterior E- algebra E(Dr) built over the E-module Dr. The algebra of the differential forms is the E-algebra HomÄ(£(Fie), E), where E(Tr) is the exterior E-algebra built over Tr and where the product is the usual "shuffle" product of alter- nating multilinear maps. The point of departure of our investigation lies in the well-known and elementary observation that Tr and Dr are naturally isomorphic with Extfi.(E, E) and Torf(E, E), respectively, where Ee = E<8>x E. Moreover, both Extfl«(E, E) and Tors'(E, E) can be equipped in a natural fashion with the structure of a graded skew-commutative E-algebra, and there is a natural duality homomorphism h: Extfl«(E, E)—>HomR(TorR'(R, E), E), which ex- tends the natural isomorphism of Tr onto HomÄ(T>Ä, E).
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