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Free logarithmic derivation modules over factorial domains. (English) Zbl 1390.13079
Summary: 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.

13N15 Derivations and commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
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