BCnrd
  • Member for 11 years, 10 months
7 answers
39 votes
11k views
Classification of (compact) Lie groups
63 votes

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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5 answers
58 votes
8k views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
59 votes

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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2 answers
13 votes
3k views
Example of connected-etale sequence for group schemes over a Henselian field?
48 votes

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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3 answers
18 votes
5k views
The algebraic fundamental group of a reductive algebraic group
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47 votes

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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1 answers
21 votes
2k views
Does formally etale imply flat for noetherian schemes?
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40 votes

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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1 answers
22 votes
3k views
fpqc covers of stacks
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37 votes

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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4 answers
33 votes
9k views
Flatness and local freeness
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35 votes

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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3 answers
31 votes
5k views
What is the difference between PSL_2 and PGL_2?
29 votes

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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4 answers
16 votes
3k views
Moduli stack of principally polarized abelian varieties
29 votes

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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2 answers
12 votes
2k views
Is the fixed locus of a group action always a scheme?
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28 votes

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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2 answers
23 votes
2k views
Why are Tamagawa numbers equal to Pic/Sha?
26 votes

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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2 answers
20 votes
2k views
Standard reduction to the artinian local case?
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26 votes

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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4 answers
17 votes
4k views
Why do automorphism groups of algebraic varieties have natural algebraic group structure?
25 votes

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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2 answers
18 votes
2k views
Are Jacobians principally polarized over non-algebraically closed fields?
24 votes

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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5 answers
23 votes
5k views
To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field?
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23 votes

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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6 answers
36 votes
6k views
Why does non-abelian group cohomology exist?
22 votes

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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6 answers
2 votes
2k views
How do we study the theory of reductive groups?
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22 votes

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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4 answers
18 votes
2k views
Is projectiveness a Zariski-local property of modules? (Answered: Yes!)
22 votes

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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13 answers
39 votes
13k views
What is a good introductory text for moduli theory?
21 votes

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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4 answers
22 votes
5k views
When is an irreducible scheme quasi-compact?
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21 votes

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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2 answers
17 votes
2k views
Extra principal Cartier divisors on non-Noetherian rings? (answered: no!)
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21 votes

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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2 answers
19 votes
3k views
non principally polarized complex abelian varieties
20 votes

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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4 answers
11 votes
2k views
The central role of varieties (a comment from Mumford's Red Book)
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19 votes

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 $\...

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1 answers
8 votes
914 views
Non-representable functor, representable on locally Noetherian schemes?
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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 ...

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6 answers
19 votes
4k views
Is a quotient of a reductive group reductive?
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. ...

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5 answers
16 votes
2k views
Comparing algebraic group orbits over big and small algebraically closed fields
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,...

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1 answers
9 votes
1k views
Realizations and pinnings (épinglages) of reductive groups
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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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2 answers
10 votes
1k views
Complete intersections and flat families
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17 votes

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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3 answers
12 votes
3k views
Are complex semisimple Lie groups matrix groups?
17 votes

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 ...

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5 answers
19 votes
1k views
Equivalence of ordered and unordered cech cohomology.
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17 votes

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...).

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