We now investigate, in earnest, the tractability of the second layer of tame indecomposable injectives over Noetherian rings.

The main problem facing us is easy to discern. Suppose we wish to determine the second layer of a given tame indecomposable injective E. We observe that, at the moment, our only hold on E is in terms of the ‘abstract nonsense’,asserting the existence and, up to an isomorphism, the uniqueness of E. Since the usual method of producing injective hulls is too abstruse to yield much by way of structural details about E, and since the structural details about E is precisely what its second layer is all about, it seems futile to try to use E itself to discover what there is in its second layer. But then, how do we find out what is there? That, clearly, is an important, if nettlesome, problem; and the extent to which the second layer can live up to our expectations is dependent on how satisfactorily we can solve it.

In this chapter, we address the various technical aspects of this problem. Our main result, proved in the first section, provides a powerful technique of investigating the second layer. It shows, in part, that the links studied in chapter 5 virtually determine the tame part of the second layer. In the second section, the study of the tame part is then continued under the guise of local finiteness of the link graph. The last section is devoted to yet another aspect of the tame part: evaluation of multiplicity.