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 Dec 4 awarded Yearling Sep 23 awarded Yearling Jun 21 awarded Nice Answer Apr 22 comment Analytic isomorphisms above two etale maps For proofs, see: Grauert and Remmert, Math. Ann. (1958), 245--318; SGA 4, XI.4.3; and SGA 1, XII. If you are willing to use resolution of singularities, the proof is not too hard (see SGA 1). Mar 25 comment Any local algebraic group is birationally equivalent to an algebraic group It's not a textbook, but the theorem is proved in the setting of schemes in an expose of Artin in SGA 3. Artin also gives a very sketchy proof of the theorem in his first article in Arithmetic Geometry 1986 (Cornell/Silverman) --- I think I would look there. Mar 23 comment Derived Functors Versus Spectral Sequences "This is usually proved using the Leray spectral sequence" I hope not (Example 1). The hypotheses imply that $f_*$ takes an injective resolution of $F$ to an acyclic resolution of $f_*F$, which can be used to compute its cohomology. This gives the result. Mar 18 comment Why do twists of an algebraic group over k correspond to k-torsors over G Look at Hom from G to the twist (as a functor). This is a G-torsor. Mar 10 comment Soft(?) algebraic groups question Well first you need to do it over Q. If H and G are semisimple and G is split over Q, you can take the split form of H. If G is not split, it's a twist of the split form, so you need the cocycle to come from one on H. Looks dubious to me. Mar 7 comment Decomposition of semisimple Lie group into almost simple factors A semisimple Lie group is a covering of a semisimple algebraic group .... Alternatively, use that its Lie algebra is a product of simple Lie algebras. Feb 18 comment Lie algebras with abelian Cartan subalgebras. Even simpler, an abelian Lie algebra is a Cartan subalgebra of itself. Feb 15 comment Role of fiber functor monoidal structure in Tannakian bialgebra reconstruction My guess is the answer is no. Specifically, I'd guess that there exist really different fibre functors that become isomorphic when you forget their monoidal structures. For example, two fibre functors send an object to vector spaces of the same dimension, so they become equal on objects when you replace the category of vector spaces with its skeleton. Perhaps this can be pushed further to show that sometimes (often? always?) two fibre functors will become isomorphic when you forget their monoidal structures. Jan 19 comment Should I write to the referee? It shouldn't "restart the review procedure" --- the editor will just send the new version of the article to his current referee. Jan 13 comment The historical development of automorphic geometry "Langlands did quite a good job of suggesting that the Jugendtraum was some sort of wrong turning." Where did he do that? Jan 5 revised Permission to use Online Notes added 351 characters in body Jan 5 answered Permission to use Online Notes Jan 5 awarded Commentator Jan 5 comment finite non-commutative local group schemes Over a field of characteristic $p$, there is an obvious action of $\mathbb{G}_m$ on $\alpha_p$, and hence an action of $\mu_p$ on $\alpha_p$. The semi-direct product is a noncommutative connected group scheme of order $p^2$. Jan 3 comment Topological examples of profinite groups "I would like to exclude Galois groups". Actually, all profinite groups are Galois groups, so you may be in trouble. More seriously, I agree with KConrad: for most us of a profinite group is a projective limit of finite groups, so it's better to start with that as the definition. Dec 20 comment understanding Milne's article “Duality in the flat cohomology of a surface” Have you tried looking at the exposition of the theorem in: Berthelot, P. Le théorème de dualité plate pour les surfaces (d'après J. S. Milne). (French) [The flat duality theorem for surfaces (according to J. S. Milne)] Algebraic surfaces (Orsay, 1976--78), pp. 203--237, Lecture Notes in Math., 868, Springer, Berlin-New York, 1981. MR0638601? Dec 16 comment Why is the definition of l-adic sheaves so complicated? With the naive definition, $H^{1}(X,\mathbb{Z}_{\ell))=\Hom_{\text{continuous}}(\pi_{1}(X),\mathbb{Z}_{\ell‌​})$ with the discrete topology on $\mathbb{Z}_{\ell}$. This is generally zero, because (for nice schemes) $\pi_{1}(X)$ is profinite. With the nonnaive definition, it is Hom with the natural $\ell$-adic topology on $\mathbb{Z}_{\ell}$, which is what you want.