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seen Dec 27 '13 at 2:36

Jul
21
comment What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?
The $\mathbb{C}[[x]]$ case is much easier: this is a $1$ dimensional ring with only two prime ideals. The geometric point $Spec(\mathbb{C})\rightarrow Spec(\mathbb{C}[[x]])$ must go to (possibly a base extension of) the generic point of some subscheme, and so it must either go to the maximal point (what you called the trivial section) or the generic one. But the latter thing cannot happen since that would mean that the section $Spec(\mathbb{C})\rightarrow Spec(\mathbb{C}[[x]])$ would have to factor through $Spec(\overline{\mathbb{C}((x))})$ which it does not.
Jul
21
revised What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?
added 23 characters in body
Jul
21
revised What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?
added 14 characters in body
Jul
21
asked What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?
Jul
11
awarded  Teacher
Jul
11
awarded  Commentator
Jul
6
comment Comparing Category Theory and Model Theory (with examples from Group Theory).
My description of model theory is also outdated (Shelah's work is far beyond my comprehension, for example). But those were the original applications of those theories. It was important to note that they are not competitors.
Jul
6
comment Comparing Category Theory and Model Theory (with examples from Group Theory).
isomorphism in that category will not imply a homeomorphism. But as you see from this example, the whole point is that "usual technique" is not a well defined term. What you're asking is: is a categorical isomorphism always what you expect? But you can expect many things. As I said in the answer, if you are in a concrete category (as groups) and you want to define an isomorphism as a morphism which is 1-1 and onto (as do want to define for groups) then the categorical isomorphism is indeed what you expect.
Jul
6
comment Comparing Category Theory and Model Theory (with examples from Group Theory).
Regarding Q1 you are right, if you care about categories up equivalence rather than up to isomorphism of categories (I have never met anyone who cared about iso.'s of categories) then you can pick however many of any copy. Strictly speaking, when people say "all finite groups" you take all of them. About Q2: surely if you take a non-concrete category then there is no "usual technique". But for concrete examples (maybe someone can come up with an easier one) if you look at the category of topological spaces, with morphisms continuous maps where you identify any two homotopic ones, then an
Jul
5
answered Comparing Category Theory and Model Theory (with examples from Group Theory).
Jul
5
comment What is the precise relationship between connections in differential geometry and Kähler differentials?
I wanted to put this as a comment, but I don't have any reputation points yet. I'm a little confused by John's answer. It is true (in differential geometry) that you can think of $\nabla$ as a map from $M$ to $M \otimes T*(M)$ (the cotangent bundle). How does the cotangent bundle relate to the module of Kahler differentials? I'm finding it hard to connect the dots..
May
25
awarded  Nice Question
Dec
25
accepted Does the Langlands program preserve CFT's distinction between local and global theories?
Dec
25
asked Does the Langlands program preserve CFT's distinction between local and global theories?
Dec
22
comment Is an unramified cover of the p-adics determined by its degree?
Right you are. I guess the right definition of degree here should be profinite size.
Dec
21
comment Is an unramified cover of the p-adics determined by its degree?
Well, all infinite extensions are just unions of finite ones. This would imply the infinite case, unless I'm mistaken.
Dec
21
accepted Is an unramified cover of the p-adics determined by its degree?
Dec
21
comment Is an unramified cover of the p-adics determined by its degree?
Good enough for me. Thanks!
Dec
21
asked Is an unramified cover of the p-adics determined by its degree?
Dec
20
awarded  Scholar