Presentation depth and the Lipman-Sathaye Jacobian theorem

Abstract

We give a version of the theorem of Lipman and Sathaye on Jacobian ideals but with substantially weaker hypotheses. Both their version and the result here are very useful in providing explicit test elements in tight closure theory. There are two separate ways in which the hypothesis in the theorem is weakened here: one is that the larger ring is not required to be a domain, although it will be reduced. Second, the regularity condition on the smaller ring is weakened to the point where one need not assume that it is Cohen-Macaulay. Instead, a condition on the ring homomorphism is imposed that may be viewed as a relative analogue of the Serre conditon S$_2$: a family of such conditions is introduced and studied here. The definition is made in terms of a presentation of an algebra, but is independent of the presentation.