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Jul
2
awarded  Curious
Feb
12
comment graph of the compostion of morphisms category-theoretically
Dear Qiaochu, thanks a lot for this answer. I am afraid for me the fact that in a span $X \leftarrow F \to Y$ the "graph" $F$ embeds into $X \times Y$ is crucial. In this case you need to use the push forward in the definition of the composition of graphs and that's what causes confusion in the proof (at least for me), I think. (Actually, not so much confusion, I think I have an idea now about using some kind of projection formula, I am figuring out the details.)
Feb
11
comment Scheme theoretic image of a base change of a morphism of schemes
is $g^{-1}(f(X))$ a closed subscheme of $Y'$? what is its scheme structure and how is the closed embedding defined?
Feb
11
revised graph of the compostion of morphisms category-theoretically
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Feb
11
comment graph of the compostion of morphisms category-theoretically
oh, I should have remarked that all morphisms are proper and image is the minimal closed subcheme through which a morphism factors.
Feb
11
asked graph of the compostion of morphisms category-theoretically
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
as per the bounty, I hope that the discussion is not over, so let us wait.
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
For non one-dimensional $Z$ one uses the fact that Albanese is of dimension $h^{0,1}$ and this equals $\mathrm{dim} Z$ if the canonical bundle is trivial. I think the statement also works for vector groups and one-dimensional subsets (it looks plausible that there are counterexamples with $Z$ of dimension $> 1$). It would be nice to know what is the most general statement that is still true.
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
Jesko, I also have some other considerations. For example, the statement is still true for Abelian varieties. The simplest case is when $Z$ is one-dimensional, then we can just notice that a curve with a trivial tangent bundle is an elliptic curve, and by a little argument involving functoriality of taking Albanese, an ellpitic curve embedded into an Abelian variety is a coset.
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
these considerations in principle should lead to a counterexample, but I still cannot construct one.
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
... moreover, if it turns out that such manifold does not touch any constant vector field, then it obviously cannot be a closed subgroup.
Jan
29
comment a closed subset of an algebraic group with a constant tangent space is a coset
After thinking on this question I have realised that I in fact cannot answer it even over complex numbers. Here is the problem. Suppose we are given a subalgebra $\mathfrak z$ of the Lie algebra of our group $G$, so we have an involutive subbundle of the tangent bundle; taking its non-vanishing sections gives us various vector fields, and solving the corresponding ODE we can find a submanifold passing through the identitiy and touching such a vector field. Now I am not sure which of these manifolds will be a subgroup; ...
Jan
18
awarded  Nice Question
Jan
14
revised a closed subset of an algebraic group with a constant tangent space is a coset
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Jan
14
revised a closed subset of an algebraic group with a constant tangent space is a coset
added 192 characters in body
Jan
14
revised a closed subset of an algebraic group with a constant tangent space is a coset
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Jan
13
asked a closed subset of an algebraic group with a constant tangent space is a coset
Dec
4
comment why is Borel subgroup not nilpotent?
great! thanks for the reference.
Dec
4
comment why is Borel subgroup not nilpotent?
could you explain how one can see that, please?
Dec
4
revised why is Borel subgroup not nilpotent?
edited title