Alex Youcis

Berkeley, CA

ayoucis.wordpress.com

Age: 24

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

My interests are generally in the field of arithmetic geometry. More specifically, I am interested in various aspects of the Langlands program. My advisor is Sug Woo Shin.

23h
comment class field theory via schemes?
Thanks for the blog post. :) If I'm not mistaken that is geometric class field theory. Not class field theory for number fields, no? Of course one would expect a geometric way of discussing geometric class field theory!
Jan
23
comment Topology on $Z_p$
@RobertGreen I see now that this is technically true, but I thought this was the more 'obvious' (should be read 'well known' part) of the question--the inverse limit topology is just the one inherited from the product topology.
Jan
22
awarded Nice Answer
Jan
22
answered Topology on $Z_p$
Jan
11
comment Is $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$?
Also, the scheme theoretic image of $\mathbb{A}^1\times\mathbb{P}^1\to\mathbb{A}^1$ is, well, $\mathbb{A}^1$. The image of proper things are proper. $\mathbb{A}^1$ is not proper.
Jan
10
comment Image of a maximal torus via epimorphism
Are you thinking in classical language, or in scheme theoretic language? If in classical, this should follow quite easily from the fourth isomorphism theorem (and, in fact, it should also work for scheme theoretic language).
Jan
10
comment $K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$?
he speaks of showing $H^2(X,k^\times)=0$ and discusses the application to showing $H^1(X,\text{GL}_n(k))\to H^1(X,\text{PGL}_{n-1}(k))$ is surjective. What he denotes by $k^\times$ is short hand for the sheaf of regular maps to $k^\times$, which is just $\mathcal{O}_X^\times$ (the same goes for the other two sheaves)! I think this notation is also explained in the Tohoku paper. Anywhoo, nice answer Georges! +1
Jan
10
comment $K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$?
Yeah, I am moderately sure that is the first appearance. Indeed, in a letter to Serre (found in the famous Correspondence) Grothendieck discusses the subject with much excitement (dare I say pride). This is for the above theorems applciation to show that the canonical map $H^1(X,\text{GL}_n)\to H^1(X,\text{PGL}_{n-1})$ is surjective (since it's cokernel is $H^2(X,\mathcal{O}_X^\times)$!), and thus that every projective bundle comes from a vector bundle (at least on a factorial scheme). Beware though, if you go look for this letter: Grothendieck has it couched in non-modern language
Jan
9
comment $K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$?
This is all nicely exposited on in the Tohoku paper, if anyone is curious.
Dec
27
comment Identity of dimensions related to cohomology group of projective space
This should follow from a standard Cech cohomology computation. Have you looked at, uhm, 5.4 (I think?) of Hartshorne?
Dec
27
revised $k$-point after base change
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Dec
27
revised $k$-point after base change
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Dec
27
comment Solvable algebraic groups and base-change
What's your definition of solvable? For me, $G$ is solvable if its derived series $\mathcal{D}^kG$ terminates. But, $\mathcal{D}^k(G_\ell)=(\mathcal{D}^k G)_\ell$ for all extensions $\ell/k$. Doesn't that answer your question?
Dec
27
revised $k$-point after base change
deleted 29 characters in body
Dec
27
answered $k$-point after base change
Dec
22
comment Meaning of "local equation" of a divisor.
I think the easier way to see such a thing exists, is to think about the local equations $f,g$ at $p$ as being the generators of $C_p$ and $D_p$ in $\mathcal{O}_{X,p}$. Since $C$ and $D$ are divisors, $C_p$ and $D_p$ are divisors of $\mathcal{O}_{X,p}$, and so height one primes. But, since $\mathcal{O}_{X,p}$ is a UFD (by regularity assumptions), $C_p=(f)$, and $D_p=(g)$ for some $f,g\in\mathcal{O}_{X,p}$. Then, using standard 'geometric Nakayama lemma' business, you can show that this is equivalent to what you said.
Dec
18
awarded Good Answer
Dec
17
answered Describing $Spec(\mathcal{O}_K[X])$
Dec
14
awarded Enlightened
Dec
12
comment Module of differentials in the functorial approach to schemes and quasi-coherent modules
Serious question (please don't take this facetiously)--what is the use of this point of view? Thanks! :)
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