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

Hochschild, G.; Kostant, Bertram; Rosenberg, Alex

Trans. Am. Math. Soc. 102, 383-408 (1962).

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.)

Reviewer: Masayoshi Nagata (Kyoto)

MSC:

13D07	Homological functors on modules of commutative rings (Tor, Ext, etc.)
13H05	Regular local rings
13N05	Modules of differentials

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