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comment Properties of schemes determined by field valued points
I have edited the question to address the comments thus far. Is the phrasing still problematic? Can it be considered for re-opening?
Jun
16
comment Properties of schemes determined by field valued points
@WillSawin I have made edits to state the question more precisely. Mostly I want to consider $X(K)$ as just a topological space. Is there other structure of $X(K)$ from a functorial point of view that come to mind?
Jun
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revised Properties of schemes determined by field valued points
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asked Properties of schemes determined by field valued points
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comment Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
Both of the previous claims can be boiled down to a Lie algebra calculation using that $Gr_{k,n} = GL_n/P$. Factor $P = GL_k\times GL_{n-k}\times U$. Let $U^-$ be the transpose of $U$. Then tangent space of $GL_n/P$ is naturally $Lie(U^-)$ and can check tangent map is injective because $Lie(U^-)$ acts non trivially on highest wt. vector.
Sep
26
comment Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
Closed immersion? The map $GL_n\to \mathbb{P}(\wedge^k \mathbb{C}^n)$ is not even injective so cannot be an immersion. It is a submersion; since $GL_n$ acts transitively on the image so its enough to check map on tangent spaces is surjective at identity. Similarly, the map from $Gr_{k,n} \to \mathbb{P}(\wedge^k \mathbb{C})$ is a closed embedding again because map on tangent spaces is injective in this case.
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comment vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$
The idea is that if $P$ is a nontrivial $G$ bundle on $U$ and $W$ is any etale neighborhood of $(u^k,v^k,u v)$ then $P$ restricted to $W \cap U$ remains nontrivial.
Aug
14
comment vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$
I believe this argument can be adapted to show that on $Spec\ A$ every principal $G$ bundle is trivial whenever $G$ is a split reductive group.
Aug
13
accepted vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$