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Jul
28
comment Finding a path in a graph by its hash value
At this point you'd need to make the question more formal, I think. What are your exact requirements and what properties does the hash function have?
Jul
27
comment Finding a path in a graph by its hash value
This is trivially possible if there are finitely many vertices, so I assume there is an infinite number of vertices - but in this case, there could be infinitely many paths of length $n$, how are you "given" all these hash values?
Jul
17
comment Algorithmic question about algebraic varieties and affinely independence
What precisely is your input and what is your algorithm allowed to do? If you are allowed to use randomization for example, just pick the $u^{(i)}$ randomly and you'll be fine.
Jul
13
comment Fiber dimension theorem for locally closed sets
@guest_09072015: Yes, this is true, but simply because a nonempty, open subset of an irreducible algebraic variety is again an irreducible algebraic variety. It might not be affine, but it is a variety.
Jul
12
comment Fiber dimension theorem for locally closed sets
It is open in $\mathbb C^n$, but it can be given the structure of an affine variety. Inside $\mathbb C^{n}\times\mathbb C$, it correponds to the points $(z,t)$ subject to the polynomial condition $f(z)\cdot t = 1$.
Jul
5
comment Projection is an open map
@GFR: True & done. Thanks for the comment btw, I realize now that tomasz already pointed it out before but I completely overlooked that in his comment.
Jul
5
comment Projection is an open map
@GFR: Indeed, that was a serious blunder. Should be fixed now.
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 does the regularity of $A$ implies the 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.