Commutative Algebra: An Introduction by Hoffman J. William & Jia Xiaohong & Wang Haohao

Author:Hoffman, J. William & Jia, Xiaohong & Wang, Haohao [Hoffman, J. William]
Language: eng
Format: azw3
ISBN: 9781944534608
Publisher: Mercury Learning and Information
Published: 2016-04-06T16:00:00+00:00

of irreducible subvarieties of X. This corresponding to a strictly increasing chain of prime ideals in ,

The geometric idea behind this definition is that making an irreducible subvariety smaller is only possible by reducing its dimension, so that in a maximal chain as above the dimension of Xi should be dim X − i, with Xn being a point, and X0 an irreducible component of X.

Similarly, the codimension of an irreducible subvariety Y ⊆ X is the greatest length n of a maximal chain , where X0 should be an irreducible component of X, and the dimension should drop by 1 in each inclusion in the chain. Moreover, Xn = Y in a maximal chain, so that we can think of n as dim X − dim Y, and hence as what one would expect geometrically to be the codimension of Y in X.

One of the main obstacles when dealing with dimension is that, in general, the maximal chains of prime ideals may have different length. This can be observed by the corresponding geometry.

For example, is the union of a line (the intersection of the planes x = 0, y = 0) and a plane z = 0. Then there are two chains of irreducible varieties in X of length one or two:

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