Cotilting modules over commutative Noetherian rings

Recently, tilting and cotilting classes over commutative Noetherian rings have been classified in [2]. We proceed and, for each n  -cotilting class CC, construct an n  -cotilting module inducing CC by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n  -cotilting module inducing CC. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.