## Differential forms on regular affine algebras.(English)Zbl 0102.27701

The main result is as follows: Let $$K$$ be a perfect field and let $$R$$ be a regular ring (i. e., every localization of $$R$$ is regular) which is finitely generated over $$K$$. $$T_R=R$$-module of $$K$$-derivations of $$R$$, $$D_R = R$$-module of the formal $$K$$-differentials [i. e., $$S/J$$ where $$S =R\otimes_K R$$, on which $$R$$ acts so that $$x(y\otimes z)=(xy)\otimes z$$ and $$J$$ is the $$R$$-submodule of $$S$$ generated by elements of the form $$1\otimes (xy)-x\otimes y-y\otimes x$$], $$E(D_R) =$$ the algebra of the differential forms $$=$$ the exterior $$R$$-algebra built over $$D_R$$, $$E(T_R) =$$ the exterior $$R$$-algebra built over $$T_R$$. Then $$\mathrm{Tor}^S(R, R)$$ coincides with $$E(D_R)R$$, $$\mathrm{Ext}_S(R, R)$$ coincides with $$E(T_R)$$ and the natural duality homomorphism $$h: \mathrm{Ext}_S(R, R)\rightarrow \operatorname{Hom}_R(\mathrm{Tor}^S(R,R), R)$$ is an isomorphism dualizing into an isomorphism of $$E(D_R)$$ onto $$\operatorname{Hom}_R(E(T_R), R)$$.
The reviewer likes to add here that two theorems in $$\S\,2$$ are immediate consequences of the well known Jacobian criterion of simple points. (These theorems assert: 1. If $$R$$ and $$S$$ are regular affine rings over a perfect field $$K$$, then $$R\otimes S$$ is regular and 2. if $$F$$ is a finitely and separably generated extension of a field $$K$$ and if $$L$$ is a field containing $$K$$, then $$F\otimes L$$ is a regular ring.)
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 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.) 13H05 Regular local rings 13N05 Modules of differentials
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