Covariant functors and asymptotic stability (Q2357078)

The goal of this work is on asymptotic stability of the sets \(\mathrm{Ass}_{R}F(M/I^{n}M)\), \(\mathrm{Ass}_{R}F(I^{n-1}M/I^{n}M)\) and the values \(\mathrm{depth}_{J}F(M/I^{n}M)\) and \(\mathrm{depth}_{J}F(I^{n-1}M/I^{n}M)\), where \(M\) is finitely generated \(R\)-module, \(I\), \(J\) ideals of \(R\), \(F\) is a covariant \(R\)-linear function from the category of finitely generated \(R\)-modules to itself, and \(R\) is a commutative Noetherian ring. The author considers two covariant \(R\)-linear functors, the zeroth local cohomology functor \(\Gamma_{I}\) where \(I\) is an ideal of \(R\), and the torsion functor \(\tau_{S}\) where \(S\) is a multiplicative closed subset of \(R\), he shows that if \(F=id/\Gamma_{I}\), \(id/\tau_{S}\), \(\Gamma_{I}\) or \(\tau_{S}\) then the sets \(\mathrm{Ass}_{R}F(M/I^{n}M)\) and \(\mathrm{Ass}_{R}F(I^{n-1}M/I^{n}M)\) always stabilize, in spite of the functors \(F=id/\Gamma_{I}\) and \(id/\tau_{S}\) are finitely generated but not coherent. The main result of this work is Theorem. Let \(R\) be a Noetherian ring, \(I\), \(J\) ideals of \(R\), \(M\) be a finitely generated \(R\) module and \(F\) be a coherent functor. Then the sets \(\mathrm{Ass}_{R}F(M/I^{n}M)\), \(\mathrm{Ass}_{R}F(I^{n-1}M/I^{n}M)\) and the values \(\mathrm{depth}_{J}F(M/I^{n}M)\) and \(\mathrm{depth}_{J}F(I^{n-1}M/I^{n}M)\) stabilize. To give example when asymptotic stability does not hold, the author considers the case where \(R\) is a Dedekind domain. Finally, the author gives an example of \(R\)-linear covariant functors which are non-finitely generated, he shows that if \(R\) is a one-dimensional Noetherian domain, then the sets \(\mathrm{Ass}_{R}F(M/I^{n}M)\) stabilize.