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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.)
Software:
Macaulay2
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