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Test ideals in local rings. (English) Zbl 0849.13003
This paper deals with the theory of tight closure. The author proves that the property of being a test ideal is preserved by localizations in case of a complete local Cohen-Macaulay ring of prime characteristic. The author develops a theory of \(F\)-ideals and \(F\)-submodules of the canonical module over a Cohen-Macaulay local ring. In particular, parameter test ideals are \(F\)-ideals. The parameter test ideal is never contained in a parameter ideal. Under certain assumptions, every \(F\)-ideal is radical.
Reviewer: M.Roitman (Haifa)

MSC:
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13B22 Integral closure of commutative rings and ideals
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13J10 Complete rings, completion
13B30 Rings of fractions and localization for commutative rings
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