The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals

In [2] Eisenbud and Evans gave an important generalization of Krull's Principal Ideal Theorem. However, their proof, using maximal Cohen-Macaulay modules, may have limited the validity of their theorem to a proper subclass of all local rings. (Hochster proved the existence of maximal Cohen-Macaulay modules for local rings which contain a field, cf. [4]). In the first section we present a proof which is simpler and guarantees the Generalized Principal Ideal Theorem for all local rings. The main result of the second section was conjectured in [2]. Under a hypothesis typically being satisfied for the most important fitting invariant of a module, it improves the Eagon-Northcott bound [1] on the height of a determinantai ideal considerably. Finally we will discuss the implications of a recent theorem of Fairings [3] on determinantal ideals.