On the Cohen-Macaulay property of the Rees algebra of the module of differentials

Abstract

Let R R be an algebra essentially of finite type over a field k k and let Ω k ( R ) \Omega _k(R) be its module of Kähler differentials over k k . If R R is a homogeneous complete intersection and c h a r ( k ) = 0 \mathrm {char}(k)=0 , we prove that Ω k ( R ) \Omega _k(R) is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every non-maximal homogeneous prime p \mathfrak {p} the embedding dimension of R p R_{\mathfrak {p}} is at most twice its dimension. This improves a previous result of Simis, Ulrich and Vasconcelos for the module of differentials of a ring that is locally a complete intersection, in which case the latter condition is assumed locally at every prime.