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Strong \(F\)-regularity and generating morphisms of local cohomology modules. (English) Zbl 1415.13006
Let \(k\) be a field of positive chracteristic \(p>0\) and \(R\) a formal power series ring over \(k.\) Let \(A=R/I\) be a reduced Cohen-Macaulay ring of dimension \(d\geq 1.\) Let also \(\Omega/I\) be a canonical ideal of \(A.\) Assume that \(d\geq 2, \Omega\neq R\) and \(J\) is a radical ideal defining the singular locus of \(R/\Omega.\) The main result of the paper shows that if \(J(\Omega ^{[p]}:\Omega)\nsubseteq \mathfrak{m}^{[p]},\) then \(A\) is a strongly \(F\)-regular ring. Another result gives an explicit description of a generating morphism for the local cohomology module \(H^2_{I_{n-1}(X)}(k[[X]])\), where \(X\) is an \(n\times (n-1)\) matrix of indeterminates over \(k.\) As an application of the above results it is shown that, if \(k\) is perfect and \(p\geq 5,\) the generic determinantal ring \(k[[X]]/I_{n-1}(X)\) is strongly \(F\)-regular.
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13D45 Local cohomology and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
14B15 Local cohomology and algebraic geometry
14M12 Determinantal varieties
Full Text: DOI arXiv
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