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Compact Lie groups may not be connected, and the question did not assume connectedness whereas all of the other answers did. If $G$ is a linear algebraic group over $\mathbf{R}$ then $G(\mathbf{R})$ ...

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There's a famous quote, I think due to Szego, that a technique which can be used once is a trick, but if you can use it twice then it is a method. In that spirit, here is the EGA method which is very ...

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For concepts related to algebraic geometry when the base is not a field, it can be difficult for a beginner to reconcile the approach in Silverman with the approach via schemes. I wasted a lot of time ...

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At Jim's request, here's an expanded version of my comments above. I will have to use some facts from the topological theory of complex algebraic varieties, but out of stubbornness I will not use any ...

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Every formally smooth morphism between locally noetherian schemes is flat; this is a deep result of Grothendieck. Indeed, the formal smoothness is preserved by localization on target and then ...

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It is false. I'm not sure what the comment about algebraic spaces has to do with the question, since algebraic spaces do admit an fpqc (even \'etale) cover by a scheme. This is analogous to the fact ...

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By request, my earlier comments are being upgraded to an answer, as follows. For finitely generated modules over any local ring $A$, flat implies free (i.e., Theorem 7.10 of Matsumura's CRT book is ...

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As Kevin says, the "right" definition of ${\rm{PSL}}_n$ is as representing the quotient sheaf ${\rm{SL}}_n/\mu_n$, just as one defines ${\rm{PSO}}(q) = {\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$ (with $Z_G$ ...

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The method is really the same as what Deligne-Mumford do to handle the moduli space of curves (creating a smooth cover from a part of a Hilbert scheme), except facts about curves (e.g., Riemann-Roch ...

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The question gives the "wrong" definition of Fix(T), hence the resulting confusion. A more natural definition of the subfunctor X^G of "G-fixed points in X" is (X^G)(T) = {x in X(T) | G_T-action on ...

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I assume $G$ is affine. The quick answer is that in the simply connected case it says $1 = 1/1$ by various hard ingredients, and then it is a kind of (not easy) game with Galois cohomology and ...

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Dear Workitout: The list of comments above is getting unwieldy, so let me post an answer here, now that you have finally identified 1.10.1 in Katz-Mazur as (at least one) source of the question. As I ...

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This is really a comment on Pete's comment for Mikhail's answer, but I am making it an answer because it raises a question which I think should be more widely known. The construction of Aut-scheme ...

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There's a more down to earth way to deal with this, which is already explained in Mumford's GIT: make an fppf (or even etale) surjective base change to acquire a section, use that to define the ...

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These logic/ZFC/model theory arguments seem out of proportion to the task at hand. Let $k$ be a field and $A$ a finitely generated $k$-algebra over a field $k$. We want to prove that there is a $k$-...

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A concrete and arithmetically useful way to interpret it without appeal to explicit cocycle formulas is to express everything in the language of torsors. More specifically, for arithmetic purposes if ...

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Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops ...

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A point worth noting: the proof of fpqc descent for projectivity in Raynaud-Gruson is apparently incorrect (as I learned today from Gabber in connection with something else), but the result is ...

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Read Katz-Mazur, "Arithmetic moduli of elliptic curves" (and for your purposes you can ignore the last chapter, even though it was their motivation for writing the book).

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There are smooth counterexamples. Let $S_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $d > 1$, and $s_0 \in S_0(k)$. Blow up $s_0$ to get another such scheme $S_1$ ...

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In the setup in the question, it should really say "we could have invertible meromorphic functions on Spec($A$) that don't come Frac($A)^{\times}$", since those are what give rise to "extra principal ...

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I've always meant to sit down and figure out some examples. OK, got it. I think the following works over any field (including finite fields and numbers fields) and so must be standard (unless I've ...

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Here is a really cool illustration of the principle which Emerton was outlining. We know that the Picard group of projective $(n-1)$-space over a field $k$ is $\mathbf{Z}$ ($n \ge 2$), generated by $\... View answer 1 answers 8 votes 914 views Accepted answer 18 votes Define$F(X) = {\rm{Hom}}_{\mathbf{C}}(X,{\rm{Spec}}(R/tR))$where$R$is the valuation ring of an algebraic closure of$\mathbf{C}((t))$. Note that every element of the maximal ideal of$R/tR$is ... View answer 6 answers 19 votes 4k views 18 votes It is important in answering this question that one can extend scalars to a perfect (e.g., algebraically closed) ground field, as was implicit in many of the other answers even if not said explicitly. ... View answer 5 answers 16 votes 2k views 17 votes Since you ask about other situations where this sort of thing occurs, let me describe a general principle (applied to the context of the original question) which is widely applied in EGA and elsewhere,... View answer 1 answers 9 votes 1k views Accepted answer 17 votes OK, here's the deal. I. First, the setup for the benefit of those who don't have books lying at their side. Let$(G,T)$be a split connected reductive group over a field$k$, and choose$a \in \...

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EGA IV$_4$, 19.3.8 (and 19.3.6); this addresses openness upstairs without properness, and (as an immediate consequence) the openness downstairs if $f$ is proper (which I assume you meant to require). ...

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In the spirit of the title of the question, the argument doesn't quite prove that $G$ is a matrix group, since more input is needed to prove that the faithful representation has closed image which is ...
I wrote it up for my algebraic geometry course as a 2-page handout, inspired by EGA $0_{\rm{III}}$, 11.8.7 (which isn't to say this is a canonical reference; just some written reference...).