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Closure-interior duality over complete local rings. (English) Zbl 1477.13047

Different types of closure operations have been successfully applied to various homological conjectures in commutative algebra in the past. For instance in equicharacteristic \(p>0,\) tight closure was introduced and applied to prove the existence of balanced big Cohen-Macaulay algebras for rings containing a field by Mel Hochster and Craig Huneke. The paper under review introduces a new duality operation which connects closure operations, interior operations, and test ideals. Two new notions, submodule selectors and residual operations, which serve as the basis for the duality operation, are also introduced in the paper. Residual operations and submodule selectors generalize the notions of closure operation and interior operation. It is then proved that there is a one-to-one correspondence between residual operations on pairs of module \(N\subseteq M\) and submodule selectors on modules (see Proposition 2.4). A relation detailing how properties of submodule selectors correspond to properties of residual operations under this bijection is also established (see Proposition 2.6).
For categories of modules over a complete Noetherian local ring that satisfy Matlis duality, smile duality is introduced and a duality between residual operations and submodule selectors is established. The duality between residual closure operations on the one hand and interior operations on the other can then be recovered by suitable restriction of the smile dual. The action of smile duality on several commonly used constructions is also investigated. For intstance it is shown that the notions of trace and module torsion are smile-duals to each other (Theorem 8.18), \(W\)-torsion preradical is smile-dual to the \(W\)-divisible preradical (Proposition 9.3), tight closure is dual to tight interior and good localization is dual to good colocalization. In addition to this, connections of the theory to almost ring theory and Heitmann closures are also investigated.

MSC:

13J10 Complete rings, completion
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13B22 Integral closure of commutative rings and ideals
13C12 Torsion modules and ideals in commutative rings
13C60 Module categories and commutative rings

Citations:

Zbl 1315.13016
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References:

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