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bio website ayoucis.wordpress.com
location Berkeley, CA
age 23
visits member for 2 years, 10 months
seen 17 hours ago

I am a first year graduate student at the University of California, Berkeley.


17h
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@ZhenLin No, because that only discounts the quasicoherent case.
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@ZhenLin I disagree with you, in so much as the category of torsors is equivalent to the category of PHSs (when you require these to be 'nice' [e.g. flat and locally of finite presentation over the base])--this is not superficial, as far as I can see. Also, of course if one moves to any setting there are many ways to understand $\mathbb{G}_a$-torsors. For example, as I said above, you can think of them as extensions. So, while I appreciate your suggestion, I'd like to know if there is an answer to my specific question. :)
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@ZhenLin Torsors and PHS aren't always the same though, for example, torsors of an algebraic group $G$ in the fppf topology needn't be equivalent to PHSs for $G$ (when you define the latter to be 'geometrically nice'). Since I, tacitly, was hoping that the above would extend to other geometric sites, I'd prefer to stick to the purely sheaf theoretic definition of torsor, and look for sheaves with the condition I stated. I appreciate the suggestion though.
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@ZhenLin That sounds like a principal homogenous space, correct? I don't quite know if that answers my question.
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@ZhenLin Could you explain a little bit more, please?
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@QiaochuYuan Although, truthfully, we should have been restricting to the commutative case to begin with :)
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@QiaochuYuan Hmm, actually now that you say that, I have doubts as well. If $\text{Aut}(G)$ is cyclic, then $G/Z(G)$ would be cyclic, so $G$ abelian, and then clearly this can't happen.
1d
comment Geometric Interepretation of $\mathbb{G}_a$-torsors
@QiaochuYuan I agree, there is no obvious way. Do you think there is actually an obstruction to such a module existing (in general)? I actually hadn't even considered that possibility--I had thought, at worst, it might just be unenlightening.
1d
asked Geometric Interepretation of $\mathbb{G}_a$-torsors
1d
comment The difference between vector space and group
It might be worth mentioning, if it is non-obvious, that there are (many!!) groups which cannot be given the structure of a vector space. For example, $\mathbb{Z}$, or $\mathbb{Z}/(6)$.
2d
comment Invertible sheaves on affine varieties
@LiYutong Yes. Depending your definition, a Dedekind domain is a dimension 1, Noetherian, normal domain.
2d
comment Invertible sheaves on affine varieties
@LiYutong I was talking about for $A$ a dimension one domain, and Noetherian, as in Bruno's example. In that case, normality is the same as regularity.
2d
comment Cohomology Calculation
@JC574 Right :). The comparison theorem is one of the main reasons why algebraic de Rham cohomology is considered a 'nice' cohomology theory, for varieties in characteristic $0$.
Jul
25
comment Factorization of ideals in a coordinate ring (Dedekind domain)
What does the bar denote here?
Jul
25
answered Cohomology Calculation
Jul
25
answered Show that $D \cong {\rm End}_A(D^n)$ where $D$ is a division algebra and $A\cong M_n(D)$
Jul
25
answered do the homomorphisms between two group schemes form a sheaf in the (whatever)-topology?
Jul
25
comment A not free $\mathbb{Z}$-module
Did you click the link in your link?
Jul
25
comment Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.
What do you mean by connection? The (relative) cotangent sheaf is the algebro-geometric analogue of the tangent bundle. For example, if you believe that $\mathfrak{m}_x/\mathfrak{m}_x^2$ is the 'cotangent space' of $X$ at $x$, and if $X\to S$ is finite type, with $k(x)/k(s)$ separable, then $\mathfrak{m}_x/\mathfrak{m}_x^2$ is isomorphic to the fiber of $\Omega^1_{X/S}$ at $x$. I can give you other geometric reasons why $\Omega^1$ is like a 'relative cotangent bundle', but I'm not sure what you mean. Are you wondering if you can literally phrase $T_M^\ast$ as some sheaf like this? (you can).
Jul
25
comment Invertible sheaves on affine varieties
To add to this, assuming that $A$ is normal, then such a line bundle will exist if and only if $A$ is not a UFD. Since, in this case $\text{Pic}(A)=\text{Cl}(A)$, and $\text{Cl}(A)=0$ if and only if $A$ is a UFD>