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The module of logarithmic derivations of a generic determinantal ideal. (English) Zbl 07243336
Summary: An important problem in algebra and related fields (such as algebraic and complex analytic geometry) is to find an explicit, well-structured, minimal set of generators for the module of logarithmic derivations of classes of homogeneous ideals in polynomial rings. In this note we settle the case of the ideal $$P\subset R=K[\{X_{i,j}\}]$$ generated by the maximal minors of an $$(n+1)\times n$$ generic matrix $$(X_{i,j})$$ over an arbitrary field $$K$$ with $$n\geq 2$$. We also characterize when the derivation module of $$R/P$$ is Ulrich, and we investigate this property if we replace $$R/P$$ by determinantal rings arising from simple degenerations of the generic case.
##### MSC:
 13C05 Structure, classification theorems for modules and ideals in commutative rings 13N15 Derivations and commutative rings 13C40 Linkage, complete intersections and determinantal ideals 14M12 Determinantal varieties 13C14 Cohen-Macaulay modules 13E15 Commutative rings and modules of finite generation or presentation; number of generators 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Macaulay2
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