Mariano Suárez-Alvarez
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Feb
5
comment Number of field homomorphisms
I presume that «homo» in your title is an abbreviation for «homomorphism». Please do not use abbreviations like that: take the time to write out and correctly what you write.
Feb
5
comment Groebner basis and prime ideals.
$G\cap K[X]$ is a groebner basis for the intersection ideal, by the theory on elimination. How are Groebner bases for ideeals in the PID $k[x]$? Also, if $I$ is prime so is its intersection with $k[x]$, as you can easily check.
Feb
5
comment Material on Koszul complex
You could read the book by Alexander Polishchuk, ‎Leonid Positselski called Quadratic Algebras.
Feb
5
comment Material on Koszul complex
What exactly do you want to know about them?
Feb
5
comment When is a diffeomorphism analytic?
You should read en.m.wikipedia.org/wiki/Non-analytic_smooth_function
Feb
5
comment Finite number of maximal ideals of bounded norm
What don't you see, that maximal ideals of a commutative ring are prime?
Feb
5
comment Finite number of maximal ideals of bounded norm
Maximal ideals are prime.
Feb
5
comment When is a diffeomorphism analytic?
Consider the function $f(x)=x+\exp(-1/x)$ for positive x, and $f(x)=x$ for negative ones.
Feb
5
comment When is a diffeomorphism analytic?
That's a very bad convention. An analytic diffeomorphisms should be one which is analytic.
Feb
4
comment Graphs of (un)bounded color valence
If the graph is finite it obviously satisfies the condition for any $d$ larger than the cromattic number.
Feb
4
comment Graphs of (un)bounded color valence
Consider the disjoint union of graphs $K_1$, $K_2$, $K_3$, $\dots$
Feb
4
comment Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?
An important pièce of information to keep in mind —and that Reid proves later in his book— is that birational varieties have the same dimensions.
Feb
4
comment In a Group, is the existence of the left identity equivalent to the existence of the unique two sided identity?
You wrote "right identity" where you meant "right inverse".
Feb
4
comment In a Group, is the existence of the left identity equivalent to the existence of the unique two sided identity?
A set with an associative operation can have at most one identity. Groups, though, can be caracterized in many ways, and sometimes people prefer "minimal" set of conditions (like only imposing things on the left, say)
Feb
4
comment Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$
All this should be explained in any sensible textbook on the subject; for example, Rudin's book on functional analysis does a great job at that. Have you looked into one?
Feb
4
comment Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$
Are you asking if those claims are true? If you really want to know «if we can conclude» all that you have to tell us what argument you have in mind.
Feb
3
comment Darboux coordinates on projective spaces
Anything that depends only on the symplectic structure can be treated locally as if you had $\mathbb R^{2n}$. It is extra structure on top of the symplectic structure that cannot.
Feb
3
comment Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$
I suggest you pick a good calculus book treating functions of many variables to get an intuition. The differential in algebraic geometry is just the one from calculus done algebraically.
Feb
3
comment Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$
For any $g$, the map $x\mapsto gxg^{-1}$ has a differential $dg:T_eG\to T_eG$. That's your map. You need to read on what the differential of a map is.
Feb
3
comment Darboux coordinates on projective spaces
Ah. Then you are not seeing the projective space as a symplectic manifold :-)