## 51.1 Introduction

This chapter continues the study of local cohomology. A reference is [SGA2]. The definition of local cohomology can be found in Dualizing Complexes, Section 47.9. For Noetherian rings taking local cohomology is the same as deriving a suitable torsion functor as is shown in Dualizing Complexes, Section 47.10. The relationship with depth can be found in Dualizing Complexes, Section 47.11.

We discuss finiteness properties of local cohomology leading to a proof of a fairly general version of Grothendieck's finiteness theorem, see Theorem 51.11.6 and Lemma 51.12.1 (higher direct images of coherent modules under open immersions). Our methods incorporate a few very slick arguments the reader can find in papers of Faltings, see [Faltings-annulators] and [Faltings-finiteness].

As applications we offer a discussion of Hartshorne-Lichtenbaum vanishing. We also discuss the action of Frobenius and of differential operators on local cohomology.

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