So if we have an exact sequence of $A$-modules $0 \to M' \to M \to M'' \to 0$, and $M$ is finitely generated as an $A$-module, apparently we have $M''$ is also finitely generated as an $A$-module. The map $g : M \to M''$ takes the generators of $M$ to the would-be generators of $M''$.
My questions: $1.$ Some of these generators could be mapped to zero right? What prevents all of them from going to zero? probably exactness somehow I bet. $2.$ What are some examples of $M$ being finitely generated and then $M'$ not being finitely generated?
2. A non-noetherian ring has ideals that are not finitely generated. A minimal example is $k[x_1,\dots,x_n,\dots]$ and the ideal $(x_1,\dots,x_n,\dots)$.
A more interesting example may be a valuation ring of dimension $>1$. Any finitely generated ideal is principal, but it must have ideals that are not finitely generated.
1. The fact that things map to $0$ isn't really a problem.
$1.$ But say the generators for the middle one are $\{a_1, ..., a_n \}$. apparently the generators for $M''$ are $\{ g(a_1) , ... , g(a_n) \}$, but what if they all get sent to $0$? Can that even happen?
and for the valuation example what is an example of a not finitely generated ideal of a valuation ring of dimension $>1$
@user2103480 Speaking of partial fractions, there's a nice elementary proof that any (finite-dimensional) linear map whose minimal polynomial has distinct roots is diagonalizable ... using partial fractions, nothing fancy.
@BigSocks I rarely think about the ring structure of valuation rings. The maximal ideal can be principal, as Matsumura tells his poor readers, but then it is the height $1$ prime ideal that is not finitely generated. Defies my Noetherian upbringing.
So suppose $A$ satisfies $f(A)=0$ for $f(t)=(t-\lambda_1)\dots (t-\lambda_k)$. Write the partial fraction decomposition for $1/f(t)$. When you clear the denominator and substitute $A$, you get $I = \sum p_j(A)$ where $(A-\lambda_j I)p_j(A)=0$. This gives a decomposition of your vector space into eigenspaces.
@BigSocks if you have an exact sequence $M\rightarrow M^{\prime\prime}\rightarrow 0$, the first map is surjective. if a surjective map maps a generating set to $0$, what can you say about the codomain of that map?
Someone cast an undeserved downvote, they commented why they downvoted, but the reason was wrong. I explained this in a comment and even updated the answer because the information in my comment was helpful (though not necessary). Of course, they have not removed their downvote (since I've edited, they can) or commented to the contrary. :-( I guess I should be happy they commented rather than drive-by downvoting.
I'm going to suggest that when someone downvotes, and the question or answer is edited, they are notified of the edit. That might promote people to actually improve their post in response to a downvote.
They know that their edit won't be sent into a vacuum.
the irritating thing i came across earlier was someone who put up a question which attracted an answer...and then deleted the question and put it up again. (there was an algebra error in their original question, so they're not identical in text. but it's the same problem...)
@Semiclassical They should have simply corrected the question, unless that would invalidate the answer. In which case, they can post again, but they should not have deleted the earlier question.
I need to derive the taylor series for ln(1+x) about 1 then create an inequality that gives the number of terms that must be taken to yield ln4 with error terms less that $2^{-10}$. I got the taylor series to be $$f(x) = ln(2) + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n \cdot 2^n}$$ I was wo...
@Semiclassical hi again, re majorana representation, what about the scenario where its not "mere philosophy" or has implications outside of philosophy?
eh. the trouble with the Bell analogy is that, on the QM side of things, the Bell inequality is pretty trivial. the interesting side of the inequality is showing that "local realism" has specific implications
it's perhaps a failure of imagination to not see any interesting implications of the majorana representation, but i see little point to pretending that I see such when I don't
yes was going in that direction, theres a 100+ page paper on that, managed to track it down again with some trouble wink... my idea is that some kind similar analysis applies to or could be applied to the majorana formulation...
one thing I particularly don't see anything philosophically-suggestive of is the 2D Coulomb gas aspect. that's very useful from a visualization purpose, but it's effectively just an implication of going from the unit sphere to the plane via stereographic projection
and the unit sphere, in turn, is basically just a matter of "you can orient a stern-gerlach device to any direction in space"
dont grasp it yet myself, but fundamentally its fluid dynamics. theres a huge trend of papers look at fluid dynamics connections to QM. have you looked into "emergent QM" some?
bohm himself was considering fluid dynamics ideas but never wrote them up much, have found it in some paper(s)
did you have any reaction/ opinion on this? think it shows that relegating deep questions of QM foundations to mere "philosophy" is no longer tenable to say the least. Quantum Leaps, Long Assumed to Be Instantaneous, Take Timequantamagazine.org/…
Ok so same scenario with the exact sequence $0 \to M' \to M \to M'' \to 0$. Apparently it is really obvious that $M$ being noetherian makes $M'$ noetherian (by definition)
@robjohn: Have you noticed a bunch more editing (recently than before) of very old posts? I guess people are trying to earn rep by editing, but I find it very annoying. I've complained to a few of them.
