Čech and Local Duality

Abstract

In this chapter we provide some duality formulas for the Čech cohomology of an unbounded complex, which involve the general Matlis dual ${\check{C}}_{\underline{x}}^{\vee }$ of the Čech complex. When the sequence $\underline{x}$ is a system of parameters of a Noetherian local ring our formulas provide a version of the Grothendieck Local Duality for Cohen–Macaulay or Gorenstein local rings. As a byproduct we obtain new characterizations of Gorenstein local rings in terms of local homology. As another byproduct there are some characterizations of $\mathfrak {m}$-torsion and $\mathfrak {m}$-pseudo complete modules over a Gorenstein local ring. When the sequence $\underline{x}$ is a system of parameters of a complete Noetherian local ring, it turns out that the complex ${\check{C}}_{\underline{x}}^{\vee }$ is a bounded complex of injective modules with finitely generated cohomology. For that reason we start the chapter with an investigation of such complexes.