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Dec 26 
awarded  Nice Answer 
Nov 17 
awarded  Yearling 
Jun 10 
comment 
Reference request: Affine Grassmannian and Gbundles
@S. Carnahan: Where in the MirkovicVilonen paper is there anything resembling a proof or precise description about these matters? All I can find is a very terse discussion in section 1, which doesn't even have precise definitions. Is the matter discussed somewhere else in the paper? 
Jun 10 
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Reference request: Affine Grassmannian and Gbundles
If you provide contact information (e.g., a webpage which lists your email address) then I can send you something. But perhaps someone else will provide a more detailed literature reference (I am not aware of any). 
Jun 10 
revised 
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
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Jun 10 
revised 
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
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Jun 10 
revised 
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
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Jun 10 
answered  Levi decomposition in disconnected linear algebraic group (characteristic 0)? 
Jun 10 
comment 
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
@Theo: Model theory seems inappropriate to the situation. One needs only basic algebraic geometry at a level known to Zariski and Weil and Borel to show that an affirmative answer to Jim's question over $\mathbf{C}$ implies the same over any algebraically closed field of characteristic 0. It is the same exact algebraic geometry needed to prove that definitions in Borel's textbook over an algebraically closed field (e.g., reductive, or connected) are insensitive to extension to a bigger such field. 
May 30 
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etale cohomology of an abelian variety and its dual
Perhaps it should be said that $\overline{A}$ means $A_{\overline{k}}$ (and similarly in Joel's answer below, one has to take cohomology of the geometric fiber). 
May 30 
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Wrongway Frobenius reciprocity for finite groups representations
Can't one just use that restriction commutes with duality and consider the dual of the induction of the dual? Or more conceptually, for general groups $G$ and $H$ there is induction and there is "compactly supported" induction, the latter contained in the former and equality when $H$ has finite index in $G$. In general these functors have opposite adjointness properties with respect to restriction (much like direct sum versus direct product), so when the two functors agree one sees this common functor having two adjointeness properties relative to restriction. 
May 30 
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Affine neighborhood of an $S$valued point
If the base $S$ is local then any open affine $U$ around the closed point of the identity section $e$ will contain $e$ (since the only open subscheme of a local scheme that contains the closed point is the entire space), so by passing to a local base you get such an affine open. It may be unnecessary to have such a $U$, depending on the goal. If the aim is to discussion formal completion along the identity via a power series ring then one wants the base to be local, or at least the relative tangent space along $e$ to be globally free (of rank 1). Passing to a local base is usually harmless. 
May 28 
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Flattening techniques of Raynaud and Gruson
Have you looked in the early parts of the paper "Formal and Rigid Geometry I" by Bosch and L\"utkebohmert, or perhaps the part II sequel? It is addressed in one or both of those papers for sure. 
May 17 
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Lattices as invertible modules.
No. The case of quadratic fields is misleading, or rather has a special property that fails in higher degree: the ring of integers is monogenic over $\mathbf{Z}$. Once you drop that property, the invertibility can fail (and all orders in rings of integers of number fields are CohenMacaulay). I think there are counterexamples given in Shimura's introductory book on modular forms, around where he discusses the invertibility in the quadratic case. 
May 17 
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extending truncated BarsottiTate group
Yes; see Illusie's paper that surveys Grothendieck's work on the deformation theory of pdivisible groups. Early in there he gives Dieudonnemodule arguments of Gabber and Ekedahl to handle the extension problem over any perfect field. 
May 17 
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Algebraic variety geometrically reduced
Another proof: $\Omega^1_{K(X)/k}$ has $K)X)$dimension $d$ by computing after extension to the perfect closure of $k$ (preserves topology  irreducibility  and reducedness by hypothesis), using existence of separating transcendence bases over perfect fields. Since $\Omega^1_{K(X)/k}$ is $K(X)$spanned by elements ${\rm{d}}f$ for $f \in K(X)$, there is $\{f_j\}$ so ${\rm{d}}f_j$'s are a $K(X)$basis. The $f_j$'s are algebraically independent over the perfect $k$ (differentiate a potential irreducible relation) and $L=k(t_1,\dots,t_d)\rightarrow K(X)$ has $\Omega^1_{K(X)/L}=0$. QED

May 7 
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Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field
Yes. Let $U$ be the open affine curve away from $S$ over the constant field $k$. For any elliptic curve $E \rightarrow U$ there's a finite etale cover $U' \rightarrow U$ of universally bounded degree over which $E$ acquires a point of order 1728. Since $\pi_1(U)$ has only finitely many quotients of any given size, there are only finitely many possibilities for $U'$ up to $U$isomorphism. Each connected component $U'_i$ of $U'$ has a specified map to $Y_1(1728)$ and it is dominant if $j(E) \not\in k$. There are only finitely many such maps since $X_1(1728)$ has genus $> 1$. QED 
May 7 
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examples of “exotic” moduli problems for elliptic curves?
Aside from the fact that the phrase "doesn't have to do with torsion data" is vague, if you consider a $\mathbf{Z}[1/N]$schemes $S$ that is sufficiently disconnected then the set of level$N$ structures on a fixed elliptic curve $E$ over $S$ can be arbitrarily large and in particular not finite (akin to global sections of a constant sheaf on a disconnected space). So do you mean just that for elliptic curves over algebraically closed fields the associated set should be finite? If so, then how about assigning to any $E \rightarrow S$ the automorphism group? 
May 2 
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Reason for studying coherent sheaves on complex manifolds.
It's the same reason that one develops a general theory of finitely generated modules over noetherian rings rather than focus exclusively on locally free modules (akin to vector bundles): we get more robust notions of kernel and cokernel, and a wider framework in which to prove results about ideals of holomorphic functions by viewing them as instance of coherent sheaves (for which many operations are available which cannot be expressed purely within the setting of ideal sheaves: tensor product, Homsheaves, etc.). And one can perform normalization of coherent sheaves of algebras, etc., etc. 
May 1 
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Example of codim 1 regular embedding that is not an effective Cartier divisor?
Stripping away the geometric terminology, it sounds like you seek a commutative ring $A$ and ideal $I$ such $I_{\mathfrak{p}}$ is invertible as an $A_{\mathfrak{p}}$module for all prime ideals $\mathfrak{p}$ of $A$ but $I$ is not invertible as an $A$module (equivalently, $I$ is not finitely presented as an $A$module). Is that correct? 