Currently learning about projective varieties in the classical setting (over algebraically closed fields). I've seen several sources mention that "quasi-projective varieties have the same geometric dimension as their closure", but without really explaining/proving why this is the case. Any references?
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Partly... how is the function field of an open subset defined? – E. Variste May 04 '20 at 11:36
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The same way the function field of anything that has a function field is defined: you take formal quotients of regular functions. It might be helpful for you to add more context to your problem in order to help answers be targeted at the right level (and not use things you don't yet know). Are you working out of a certain book or set of notes? (Approximately) How much algebraic geometry do you know so far? – KReiser May 04 '20 at 18:29
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I'm working my way through Silverman's Arithmetic of Elliptic Curves, and I find some of his treatment of varieties in chapter 1 unsatisfactory, so trying to look for other sources. For example, he defines the dimension of a projective variety as the dimension of any of its affine patches, without mentioning the topological definition of dimension. While going through Silverman, I'm going through Harthshorne chapter 1 to understand some of this material better. – E. Variste May 04 '20 at 21:11