anon
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Unregistered User
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Feb 24 |
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Property of lattices in Lie groups The rank one case is exceptional in that there are lots of nonarithmetic irreducible lattices. No one was saying that it isn't important. |
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Feb 22 |
answered | Rigid monoidal abelian category without an exact tensor functor to Vect |
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Feb 15 |
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Extension of unipotent algebraic groups @Jim In order to define the unipotent radical, you need to know enough about unipotent groups to answer the question. And connectedness is not a problem in any characteristic, at least, not if you are talking about algebraic group schemes. |
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Feb 15 |
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Extension of unipotent algebraic groups Yes, see for example Milne's notes AGS, XV, 2.5. |
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Feb 14 |
awarded | ● Commentator |
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Feb 14 |
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field of definition of isogenies of abelian varieties This is in Mumford's book on Abelian Varieties (for a finite subgroup scheme). Mumford assumes that the ground field is algebraically closed, but the proof doesn't need this. Alternatively, if you are willing to assume that the ground field is perfect, then the statement follows from the algebraically closed case + descent. |
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Feb 9 |
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Differential geometry study materials It's a good book, but it takes 327 pages to get to metrics. For someone wanting to learn differential <i>geometry</i>, there are faster routes. |
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Feb 6 |
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intersection cohomology and etale cohomology Intersection cohomology (in the context of the etale topology) generalizes the usual l-adic cohomology. With the appropriate choice of the perversity, intersection cohomology gives the usual l-adic cohomology, with or without compact support. |
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Feb 2 |
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Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why? According to the Tate conjecture, l-adic realization gives an equivalence of categories from motives tensor $Q_l$ to the category of l-adic Galois representations generated by the cohomology of smooth projective algebraic varieties over $Q$. The standard conjectures imply the first is semisimple, hence also the second. |
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Jan 30 |
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A question on the Picard group I'd guess that, by using etale cohomology, you can give an algebraic proof valid over any algebraically closed field (use the same argument you use over $\mathbb{C}$). |
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Jan 12 |
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Primitive Cohomology Useful? The question should be, why do we need to decompose the cohomology into its primitive parts? You answered your own question: in order to be able to state the Hodge index theorem, or, more generally, Grothendieck's standard conjectures. |
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Jan 6 |
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Does every polynomial diophantine equation have solutions modulo p? My recollection is that they worked with projective varieties, but removing a lower dimensional subvariety is not going to change much. I suggest you look at their article --- it is quite short and readable. |
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Jan 1 |
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Why is the Brauer group of a local field is $\mathbb {Q/Z}$? Is it an accident? "When Tate was finding local duality, How could he know cup product make dual relation exactly?" Actually, he didn't. He originally proved a duality theorem for abelian varieties over local fields, observed that it implied a local duality for modules occurring in the abelian varieties, and only later realized that the local duality held for all modules. Concerning your general question of why all this holds. Well, we can prove it. I'm sure there a vague philosophical reasons why it must hold, but they are probably not very helpful. |
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Dec 31 |
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Etale site is useful - examples of using the small fppf site? Crystalline cohomology is a Weil cohomology, which explains its importance, but is not very useful for the things I mentioned in my answer. There are relations between flat cohomology and crystalline cohomology, but they are rather complicated to explain. As far as I know, only for the Zariski topology, the etale topology, and topologies in between, do we have a really explicit description of the "points". |
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Dec 28 |
answered | Etale site is useful - examples of using the small fppf site? |
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Dec 22 |
awarded | ● Teacher |
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Dec 22 |
accepted | Center of the algebraic group G_{\mathbb{R}} for a centerless G |
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Dec 22 |
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Center of the algebraic group G_{\mathbb{R}} for a centerless G Actually, having read Jack's comments, I still haven't a clue what he asking. |
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Dec 22 |
awarded | ● Editor |
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Dec 22 |
revised |
Center of the algebraic group G_{\mathbb{R}} for a centerless G added 113 characters in body |
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Dec 22 |
answered | Center of the algebraic group G_{\mathbb{R}} for a centerless G |
