Free logarithmic derivation modules over factorial domains

We introduce and characterize the class of tangentially free ideals, which are (not necessarily principal) ideals whose logarithmic derivation module is free, in (not necessarily regular) factorial domains essentially of finite type over a field of characteristic zero. This yields an extension of Saito’s celebrated theory of free divisors in smooth manifolds. Examples are worked out, for instance a non-principal, tangentially free ideal in the coordinate ring of the so-called $E_8$-singularity. Further, we notice a connection to the classical Zariski–Lipman conjecture in the open case of surfaces.

Keywords

derivation, logarithmic derivation, free divisor, Zariski–Lipman conjecture

2010 Mathematics Subject Classification

13C05, 13C10, 13N15, 32M25

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This paper is based on the author’s doctoral thesis (Federal University of Pernambuco, Brazil).

Received 21 May 2014

Accepted 17 September 2015

Published 7 June 2017