By Ernst Kunz (auth.)
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Hence we have in fact ~(I) c I. ~-linear mapping d : nR/ R ~ nR/ R (d(ß+I) = ~ß+I for ß E R[~R]). o 0 In particular, r E R is mapped onto the element, which was denoted dr already at the beginning of the proof. The formulas (2') show that d is Ro-linear,and from the definition of ~ we obtain d(rdr 1 ... •• drn for r,r 1 , ••. ,r n ER. 39 Putting everything together, we have shown that (~R/R is a differential algebra of R/Ro . If P : R ~ 5 and ~ = ,d) are given as in the theorem, there is a ring homomorphism X X(6r) o R[6R] ~ ~ with X(r) = p(r) and 6p(r) for all r E R.
I=1° i 1. For r E R we have d(dr) 2 r ( a = r 0, i For many purposes it is useful to generalize the not ion of universal differential algebra. We fix a pair where R/R o is an algebra and R/Ro ' Let p : R ~ ~ (R/Ro'~)' a differential algebra of S be a homomorphism of Ro-algebras. 6. 11. Definition. (~p'~p)' ~ p : ~ ~ where ~ p ~p ~ is a pair is a differential algebra of S/Ro and a p-homomorphism, such that the following uni- versal property is satisfied: Let a : S ~ T be any morphism in AR ' ~* a differential algebra of T/R o and ~ : ~ ~ o a 0 p-homomorphism.