4,105 reputation
833
bio website nullteilerfrei.de
location Germany
age 29
visits member for 3 years, 6 months
seen 14 hours ago

I'm a PhD student with research interests in Algebraic Geometry, Algebraic Complexity Theory and Geometric Invariant Theory.


Oct
15
answered Divisibility of polynomials in a subfield of a field.
Oct
11
comment Is a coherent locally free sheaf isomorphic it's dual?
No, a locally free sheaf is not in general isomorphic to its dual, not even in the very special case of line bundles. Here, the dual $\mathcal L^\ast$ of a line bundle $\mathcal L$ serves as an inverse to $\mathcal L$ in the Picard group. This might be a good case to study to understand what's going on.
Oct
5
comment Why is the multiplicative subgroup of a field an affine algebraic group?
No, I am relying on the fact that (2) is true. They are isomorphic as varieties and the group structure on $G_m$ induces a group structure on the hyperbola via this exact isomorphism.
Oct
4
comment Why is the multiplicative subgroup of a field an affine algebraic group?
Regarding 1), you just gave the description in the sentence before: Take $n=2$ and observe that $f=x_1x_2-1$ is irreducible. The ideal generated by $f$ is a prime ideal in $K[x_1,x_2]$ and its zero locus is (isomorphic to) the multiplicative group. Regarding 2), simply consider the projection $K\times K\to K$. You should be able to show that the restriction to $G$ is injective and has image $G_m$.
Oct
1
answered How to show that $GL_n/U$ is birationally isomorphic to $B^-$?
Sep
30
awarded  Explainer
Sep
28
answered Lang Category Theory
Sep
25
accepted Rational Points, classical versus modern notion
Sep
25
comment Rational Points, classical versus modern notion
Thanks, that's a good answer.
Sep
24
comment About the fixed part of a linear system
If your surface is nonsingular in codimension one, the answer is easy: For any $Z$ satisfying that property, all elements of your linear system considered as sections of the line bundle $\mathcal L(D)$ vanish at $Z$. Hence, $Z$ is contained in the base locus of your linear system $\mathcal S$. I'd have to ponder a bit if anything goes wrong in the singular case, but I have only ever heard the term "linear system" used in the context of "sufficiently nice" varieties.
Sep
22
comment Rational Points, classical versus modern notion
The problem with this is that I'd like to start with an $L$-scheme in the first place, not with a $K$-scheme.
Sep
21
comment Can we say “commutative ring = field”?
clearly, that was supposed to be a "but". thx =)
Sep
21
revised Can we say “commutative ring = field”?
deleted 3 characters in body
Sep
21
answered Can we say “commutative ring = field”?
Sep
20
asked Rational Points, classical versus modern notion
Sep
20
comment Is the maximal ideal of a localization at a prime ideal principal?
Also: A good place to read about this, in my opinion, is 11.1 of David Eisenbud's book Commutative Algebra with a View (Toward Algebraic Geometry).
Sep
20
revised Restriction of sheaf via inclusion induces isomorphism on stalks
added 17 characters in body
Sep
20
answered Restriction of sheaf via inclusion induces isomorphism on stalks
Sep
13
comment The greatest common divisor of homogeneous polynomials
If you choose $a_{ij}$ equal to the same polynomial $g$, the greatest common divisor of the $F_j$ will always be associated to $g$. Seems like this will heavily depend on the matrix $M$.
Sep
9
comment Local complete intersection ring
What is "the" maximal ideal of $R$? Is $R$ local or graded?