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Local Cohomology, Second Edition by R. Y. Sharp, M. P. Brodmann

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7

Artinian local cohomology modules

In this chapter, we shall show that certain local cohomology modules are Artinian, that is, satisfy the descending chain condition on submodules, and we shall use our results to provide a different proof of a theorem in Chapter 6.

Suppose, temporarily, that (R,image) is local. In the Non-vanishing Theorem 6.1.4, we proved that, if M is a non-zero, finitely generated R-module of dimension n, then image(M) ≠ 0. One consequence of our work in this chapter is that we can give an alternative proof of this result, and, at the ...

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