Homological Theory of Regular Local Rings
In this section we shall examine some further uses of the homological tools. After preliminaries on minimal free resolutions, we shall use the Koszul complex to prove the Hilbert syzygy theorem and the basic results of Auslander-Buchsbaum and Serre about regular local rings and polynomial rings: the characterization of regular local rings as the rings of finite global dimension, and the consequences that localizations of regular local rings are regular and that regular local rings are factorial. We also derive some important relations of depth to homology in the Auslander-Buchsbaum formula and the formula connecting the vanishing of Ext i R (M, N) with the depth of the annihilator of M on N. This “explains” in a certain sense the relations of depth and the homology of the Koszul complex that we saw in Chapter 18.
KeywordsLocal Ring Maximal Ideal Polynomial Ring Free Module Projective Dimension
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