Krull Dimension and Schemes
It follows readily from the definition of the spectrum of a ring, the space of prime ideals of equipped with the Zariski topology, that the Krull dimension of is precisely equal to the irreducible dimension of its spectrum. This follows immediately from the Galois connection between ideals of and closed subsets of and the observation that, by the definition of, each prime ideal of corresponds to a generic point of the closed subset associated to via the Galois connection.
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