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bio website math.berkeley.edu/~pablo
location Berkeley
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visits member for 4 years, 11 months
seen Sep 13 at 8:32

Aug
16
awarded  Nice Question
Aug
14
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 $\mathbb{C}[x,y,z]/(x y - z^k)$
Aug
13
comment vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$
If I understand correctly then the conclusion should be the same for higher rank: any vector bundle on $V$ is trivial so $Spec\ A$ you will see $B_{r_1} \oplus ... \oplus B_{r_n}$ and the fiber at $(u^k,v^k, u v)$ will be bigger than $n$ unless all $r_i = 0$. So $Spec\ A$ has only trivial vector bundles.
Aug
13
comment vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$
Great! The point I was missing is if a locally free on $U$ is to extend to a locally free on $Spec A$ it must be the reflexive extension.
Aug
13
asked vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$
Jul
2
awarded  Inquisitive
Jul
2
awarded  Curious
Apr
15
comment Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$
Oh and $T \subset G$ is a maximal torus.
Apr
15
accepted Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
Apr
14
awarded  Critic
Apr
14
comment Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$
There is an obvious thing that breaks in passing from $\mathbb{C}[z^\pm]$ to $\mathbb{C}((z))$: the former has ring homomorphisms to $\mathbb{C}$ and the latter doesn't. On the other hand $G(\mathbb{C}((z))) = G(\mathbb{C}[z^{-1}])T(\mathbb{C}[z^\pm])G(\mathbb{C}[[z]])$. You can talk about convergence for $G(\mathbb{C}[[z]])$ in the sense that you can ask if $g \in GL_n(\mathbb{C}[[z]])$ lands in $G(\mathbb{C}[z]/z^n) \subset GL_n(\mathbb{C}[z]/z^n)$ for all $n$.
Apr
14
asked Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
Mar
5
comment The localization of a regular local ring is regular
It seems to me $\sum_i x^n/y^{n^2}$ is an element of $k((y))[[x]]$ that will not be in $k[[x]]((y))$. In the latter ring any element can be brought into $k[[x,y]]$ by multiplying by a finite power of $y$. See mathoverflow.net/questions/34010/… for related differences.
Feb
20
awarded  Popular Question
Feb
8
awarded  Popular Question
Dec
17
comment Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected
This seems to rely on working over $\mathbb{R}$. I'd like an example over $\mathbb{C}$.
Dec
15
asked Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected
Oct
5
awarded  Nice Question