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Jun
21
comment Matrix of linear forms
It might be good to explain what the notation $I_a(M)$ precisely means. Is it the ideal generated by the entries of $M$? Why is there an $a$ in the index, if it already depends on $M$?
Jun
16
comment When $f(I)S=S$ for each ideal $I$ of $R$?
@user1: I tried to make my answer more comprehensible. Is this better?
Jun
11
comment When regularity of $A$ implies regularity of $A[w]$?
@user237522: I added an explanation of "unramified". I am not sure how easy that is to check, though.
Jun
4
comment A categorical approach to algebraic geometry
I haven't read it, but you might have a look at Categories and Sheaves by Masaki Kashiwara and Pierre Schapira. It seems to contain lots of the stuff you want.
May
21
comment Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)
You are, of course, completely correct. Regarding your second thing, I missed the "different" ... so yea, you are right about that, too.
May
20
comment Can someone illustrate the definition of manifold with a simple example?
The most classical example would be the sphere in $\mathbb R^3$. For any point $x$, you can pick the point opposite to it $y$ and use the stereographic projection map to identify $M\setminus\{y\}$ with $\mathbb R^2$.
May
19
comment Dimension of affine part equals to the dimension of the variety
The dimension of $\mathbb P^n$ is not equal to the Krull dimension of its projective coordinate ring $k[x_0,\ldots,x_n]$, which is $n+1$. That's because $\mathbb P^n$ is not an affine variety. The problem with your question is that it is ill-posed: What is your definition of the dimension of $\mathbb P^n$?
May
18
comment Inclusion of Tori induces surjection of character groups?
This is true, but I have too little time for anything but a reference right now: Tauvel & Yu, Lie algebras and algebraic groups, 22.5.4 (iii) is a slightly more general statement for diagonalizable algebraic groups. Any torus is diagonalizable, of course.
May
12
comment Inclusionwise maximal linear subvarieties of a projective variety
@DanielMcLaury: Interesting, I didn't see that angle. If I am not somehow mistaken, this is the image of a projective morphism, so it should be closed. Any flaws in that argument?
May
6
comment what is the precise definition of a morphism defined over $k$?
@ZhenLin: That's true, but along with your comment above this should be enough for the OP, I hope.
May
4
comment A confusion on the definition of morphism between varieties
I believe the answer to your question might be Lemma I.3.6 in Hartshorne.
Apr
22
comment Every variety contains open affine normal subvariety
@DanielMcLaury: Good point, fixed.
Apr
21
comment Is the preimage of the non-normal locus a divisor?
@AsalBeagDubh: A very good nitpick, I fixed that.
Apr
20
comment Multiplicity of point as a zero of polynomial.
Also, how do you define $I_p(C,L)$?
Mar
19
comment Are unipotent algebraic groups connected?
this answer on mathoverflow strongly suggests that your argument is solid.
Mar
16
comment An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$
These two groups are not isomorphic. $\operatorname{GL}_3(\mathbb F_2)$ has at least $7$ elements, because there are $7$ nonzero vectors in $\mathbb F_2^3$. On the other hand, $S_3$ only contains $6$ elements.
Mar
11
comment Resolution of singularities of the determinant hypersurface
This is indeed a very nice resolution. I would have preferred an embedded one, but this is quite a nice start, so +1 and accept. Thanks!
Mar
11
comment Can a birational morphism surject from an affine to a projective variety?
+1 and accept, also thank you, this is a nice proof.
Mar
7
comment True or False: $f$ is injective if and only if $f^*$ is surjective where $f^*$ is corresponding to the pullback to $f$.
Of course, you can easily turn this into a direct proof, which is then more elegant. But it is late, and I will leave you with this.
Mar
6
comment Can a birational morphism surject from an affine to a projective variety?
@AlexYoucis: What is the argument for the fact that it has an inverse on some $U\subseteq Y$ whose complement is codim. 2? The approach sounds interesting!