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Apr 16 |
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A question about $R$-points of an complex reductive group. Since $\mathbf{C}$ is algebraically closed, $Z_e$ is a split group of multiplicative type, so $Z_e$ is a split torus. Thus, for any $\mathbf{C}$-algebra $R$, the obstruction to $G(R((t)))/Z_e(R((t))) \rightarrow (G/Z_e)(R((t)))$ being bijective is a class in the etale cohomology set $H^1(R((t)),Z_e)$, which is a power of ${\rm{Pic}}(R((t)))$. For $R$ a field or even artinian local ring, this Pic is trivial and so bijectivity holds. Thus, you have bijectivity on infinitesimal points over $\mathbf{C}$, which probably implies an isomorphism as smooth ind-schemes, yes? |
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Apr 16 |
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Algebraic machinery for algebraic geometry Fulton's "Algebraic Curves" does an excellent job of introducing commutative algebra in a geometric context, and its selection of exercises does an amazing job at conveying the rich interaction of geometry and algebra beyond what is done in the text. I recommend trying Fulton's book alongside Reid's, and then you can decide for yourself which you prefer. |
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Apr 14 |
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Isomorphic maximal commutative semi-simple sub algebras of M_n(C). @ofir: Any such $A$ is a product of copies of $\mathbf{C}$, so a faithful representation $A \hookrightarrow {\rm{M}}_n(\mathbf{C})$ on $\mathbf{C}^n$ from a commutative semisimple $\mathbf{C}$-algebra $A$ sends the primitive idempotents to {\em distinct} pairwise orthogonal commuting nonzero idempotent linear operators on $\mathbf{C}^n$ whose sum is the identity operator. This is exactly a decomposition of $\mathbf{C}^n$ as a direct sum of nonzero subspaces. The "maximal" way to do this is with an ordered $n$-tuple of independent lines, and in a suitable basis all $n$-tuples look the same... |
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Apr 14 |
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Parabolic-type subgroups of GL(V) Dear Hung Nguyen: Thanks for the explanation, though I'm a bit puzzled as to why your friend thinks that the class of subgroups as in your question is a "natural" one. Of course, the notion of a "natural" subgroup of ${\rm{GL}}(V)$ is a matter of taste, but parabolic subgroups and their unipotent radicals and Levi factors seem to be rather more "natural" than the things in your question. But maybe your friend has some good reason to regard the subgroups in your question as "natural"? |
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Apr 12 |
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n-canonical embedding @Artin009: Just because those MIT notes say it that way doesn't mean it is the right way to think about it (and those notes don't explain the exercise or the precise meaning of the terminology; curious that the notes mask all evidence of who wrote them, as far as I can tell). I recommend looking at my above suggested reference in Deligne-Mumford, where you'll see them carry out the actual cohomological computations using duality on the semistable curve, and I hope that should clarify the situation for you. |
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Apr 12 |
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n-canonical embedding @Artin009: In the absence of smoothness, is "separating points and tangent vectors" the right way to think about things? Anyway, have you read the proof of very ampleness for $n > 2$ in Theorem 1.2 of the Deligne-Mumford paper? (Please also consider to use a name other than "Artin009".) @Will: Since $\omega$ is not the pushforward of a line bundle on the normalization, why is RR on the normalization relevant? Isn't duality on the nodal curve a more appropriate argument? For example, that is what Deligne and Mumford do. |
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Apr 10 |
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Elliptic curve over a scheme is a group scheme? @anon: Did you mean to write that there is a down-to-earth proof that there is at most one group law with the given zero when the base is reduced (in which case the proof is immediate from consideration of the generic fibers over the base, due to flatness and separatedness considerations over the base)? To say "there is a unique" (which I read as including an existence assertion) over any base or to say "unique" over a non-reduced base both seem to lie beyond the reach of "more down-to-earth" arguments. |
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Apr 10 |
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Elliptic curve over a scheme is a group scheme? Strictly speaking, the argument in GIT has a projectivity hypothesis on the abelian scheme (due to the role of projectivity for the existence of Hilbert schemes), and there are non-projective abelian schemes over non-normal noetherian domains. If one uses algebraic spaces, which didn't exist at the time that GIT was written, then Hilbert functors are representable without projectivity hypotheses and so the proof of Grothendieck's theorem works without projectivity hypotheses. |
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Apr 9 |
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Elliptic curve over a scheme is a group scheme? There is a general theorem of Grothendieck to the effect that a smooth proper morphism equipped with a section and geometrically connected fibers is an abelian scheme if it is so on a single geometric fiber, but the proof isn't in any sense down-to-earth. I am amazed to hear that there could be a "down-to-earth" proof of the existence of the group scheme structure even just when the base is an artin local ring (for which "generic fiber" is the special fiber). Anon, what method do you have in mind which isn't the deformation-theoretic proof of Grothendieck's version? |
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Apr 6 |
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Base change of affine group schemes with respect to Frobenius map. I recommend ignoring that exercise, since such an assertion is rather misleading. (The point is that the Frobenius is an automorphism of $k$, from which the isomorphism claim in his weak sense is obvious.) |
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Apr 6 |
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Base change of affine group schemes with respect to Frobenius map. [I assume you intend for the isomorphism to be of group schemes over $k$.] |
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Apr 6 |
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Base change of affine group schemes with respect to Frobenius map. No, since they're not generally isomorphic at all. |
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Apr 5 |
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Etale fundamental group of punctured formal neighborhood Whoops, the argument I had in mind only shows (via Abyhankar's Lemma) that the kernel has no nontrivial prime-to-$p$ quotients, which is of course much weaker than being pro-$p$, so I have nothing useful to say about your question. More specifically, assuming regularity and excellence but not "henselian local", the maximal prime-to-$p$ quotient of the kernel is the quotient $\prod_{\ell} {\mathbf{Z}}_{\ell}(1)$ that you know. Please talk to experts near you in Princeton to learn about Abhykanar's Lemma beyond the 1-dimensional case (as higher dimensions is where the real strength of it lies). |
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Apr 4 |
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Finite-type Artin Stack over $\mathbb C$ Any map $f:X \rightarrow Y$ between quasi-separated algebraic spaces locally of finite type over an affine scheme $B$ with $X$ of finite type over $B$ is itself of finite type. Indeed, $Y$ is covered by q-c opens $U_i$, preimages of which are open in $X$, so finitely many preimages cover $X$. Hence, $f$ lands inside a quasi-compact open $U \subset Y$, and $U \rightarrow B$ is finite type (being q-c and lft), so $X \rightarrow U$ is finite type. Thus, we want $U \hookrightarrow Y$ to be finite type. For $V \rightarrow Y$ with $V$ affine, $U \times_Y V$ is q-c since $Y$ is quasi-separated. |
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Apr 4 |
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Finite-type Artin Stack over $\mathbb C$ There is generally no such factorization (think about the case when $\mathfrak{M}$ is a scheme and $S = \mathfrak{M}$!). The argument you've seen should have the source replaced by $S \times_{\mathfrak{M}} M$ (which is, strictly speaking, an algebraic space and not necessarily a scheme, but nonetheless is of finite type over $M$ and has the "expected" set of $\mathbf{C}$-points, so you're good to go). |
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Apr 4 |
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Etale fundamental group of punctured formal neighborhood If $R$ is a henselian regular local ring (in particular, noetherian) then this follows from Abyhankar's Lemma. The next case to wonder about is a normal noetherian domain (maybe henselian), since then "normalization" makes sense and $x$-adic separatedness and completeness is inherited by finitely generated $R[[x]]$-modules (as $R$ is noetherian). Perhaps try to understand that before the case of more general $R$. |
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Apr 1 |
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Parabolic-type subgroups of GL(V) @Amritanshu, Hung Nguyen: Oops, so I also misread the question in the same way (as being about $V_i/V_{i'}$ for "consecutive" $i, i' \in I$, rather than about $V_i/V_{i+1}$ with $i \in I$ but possibly $i+1 \not\in I$). It would be helpful to know how Hung Nguyen become interested in this particular condition. The name "uni-parabolic" sounds bad to me, but knowing where it comes from might clarify matters. |
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Mar 31 |
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Parabolic-type subgroups of GL(V) Dear Hung Nguyen: Amritanshu is completely correct. Try to think about some examples. |
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Mar 31 |
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Naive question about the representation theory of algebraic groups and hopf algebras By the way, of course Hochschild has to first prove that $A(\mathfrak{g})$ is actually finitely generated over $k$, which he gets from considerations with fundamental weights (if I remember correctly). |
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Mar 31 |
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On mentioning recommenders' names in cover letter for postdoctoral applications @Michael: I have no idea why MathJobs asks for cover letters, though it could be that for some purposes it is more relevant and the system is "one size fits all". For instance, Alexander Woo describes such a situation that never would have occurred to me (since I always read the actual research and teaching statements of any file that I look at, expecting the first page of each to include some kind of synopsis for the non-expert -- I never would have imagined that such synopses should be found in a cover letter!). |
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Mar 31 |
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Naive question about the representation theory of algebraic groups and hopf algebras @Qiaochu: the smallest counterexample is $G = \mathbf{Z}/(2)$ since there has been some sloppiness about connectedness hypotheses (a genuine issue in view of Instance 1), though $S^1$ is the smallest "interesting" counterexample. Actually, the compact form $PSU(2)$ of $PGL_2$ is the smallest genuinely interesting counterexample. |
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Mar 31 |
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The octonions on a bad day Google "octonion algebra" and "split octonion algebra" (analogy: special orthogonal groups for isotropic quadratic forms, or even for a split quadratic form). |
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Mar 31 |
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Naive question about the representation theory of algebraic groups and hopf algebras For semisimple $\mathfrak{g}$ in char. 0, Hochschild's 1959 paper "Algebraic Lie algebras and representative functions" addresses the algebra structure of $U(\mathfrak{g})^{\ast}$, and studies the $k$-subspace $A(\mathfrak{g}) = \varinjlim (U(\mathfrak{g})/J)^{\ast}$ for the 2-sided ideals $J$ of finite codimension. He shows $A(\mathfrak{g})$ consists of the "matrix coefficients" of finite-dimensional representations of $\mathfrak{g}$ and so is a $k$-subalgebra, and that Spec($A(\mathfrak{g})$) is simply connected semisimple with Lie algebra $\mathfrak{g}$. |
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Mar 31 |
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On mentioning recommenders' names in cover letter for postdoctoral applications Yes, since MathJobs lists all recommenders who have sent or even will be sending (in an ideal world...) recommendation letters, and which ones have arrived or not, it is a complete waste of time provide this information in cover letters for applications through MathJobs. In over 10 years of reading postdoc job files, I never looked at a cover letter. The choice of "primary subject area" is much more important, as that affects who actually looks at your file (e.g., the subject area "[0] General Math" should be banished, as surely all it does is ensure career suicide). |
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Mar 20 |
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intersection of all borel subalgebras Since $\mathfrak{sl}_2$ in characteristic 2 is solvable, perhaps you wish to avoid some "bad" characteristics (if considering Lie algebras of semisimple groups)? |
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Mar 17 |
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Volume of PGL(2,F) \ PGL(2, A) Oesterle's 1984 Inventiones paper on Tamagawa numbers (see Inv. Math. 78) has a very elegant discussion of the normalization issues to define the Tamagawa measure and Tamagawa number for rather general smooth connected affine groups over global fields. His normalizations are not governed by maximal compact subgroups (as they ought not be, since typically there are a lot of non-conjugate such subgroups!). He also cleaned up some points of possible confusion that arose in earlier work by others in the case of tori. |
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Mar 17 |
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Algebraicity of the canonical projection $X(\Gamma)\to X(1)$ and of $X(\Gamma)$ This answer is certainly correct, but wccanard makes an important point: the viewpoint of the original question may be "mistaken" (depending on the motivation). The relationship between certain maps among algebro-geometric moduli spaces and maps between their "uniformized analytifications" rests on arguments that have a lot to do with GAGA and moduli functors (and nothing to do with Grothendieck-Weil). For many applications in number theory (work of Mazur, Gross-Zagier, etc.), one needs this alternative viewpoint in order to work with points over specific number fields, etc. |
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Mar 15 |
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Definition of relative Picard functor Justin, it doesn't seem particularly natural to consider the relative Picard functor away from the proper setting. Are you aware of any non-proper examples? (Note that generalized Jacobians are Picard schemes of proper curves with singularities.) The only "natural" hypotheses I'm aware of which ensure that $f_{\ast}O_X = O_S$ universally are that $f$ is proper and finitely presented with geometrically reduced and geometrically connected fibers. |
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Mar 10 |
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What do formal group laws of height $\geq 3$ look like? Dear Will: Thanks for clarifying the intent behind your query concerning "algebraic functions". I recommend nonetheless looking in Hazewinkel's book; you might find some useful examples and constructions in there. |
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Mar 10 |
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one “big” Hilbert scheme? Dear IMeasy: My above comment can stand on its own merits without needing "validation". :) |
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Jan 28 |
awarded | ● Critic |
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Jan 28 |
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Galois action on special fiber of a stable model added 155 characters in body |
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Jan 27 |
answered | Galois action on special fiber of a stable model |
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Jan 6 |
awarded | ● Commentator |
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Jan 6 |
awarded | ● Nice Answer |
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Jan 5 |
awarded | ● Editor |
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Jan 5 |
revised |
Does every polynomial diophantine equation have solutions modulo p? added 568 characters in body |
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Jan 5 |
answered | Does every polynomial diophantine equation have solutions modulo p? |
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Jan 5 |
awarded | ● Supporter |
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Jan 4 |
awarded | ● Nice Answer |
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Jan 3 |
answered | Laurent Polynomials |
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Jan 3 |
awarded | ● Teacher |
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Jan 3 |
answered | The fibres of smooth projective families over all geometric points have isomorphic cohomology; are these isomorphisms ‘functorial’? |
