ah yeah, if $Y\subseteq X$ is arbitrary and $A,B\subseteq Y$ are disjoint and closed in $Y$, then you can consider the open, hence normal by assumption, subspace $U=X\setminus(\overline{A}\cup\overline{B})$ and the closures of $A$ and $B$ in this space are still disjoint, so you can separate them by neighborhoods in there and then restrict back down to $Y$
If $f:M\rihghtarrow N$ is a surjective submersion then does it follows that there exists for each $p\in N$ a neighborhood $U$ of $p$ and a smooth map $s:U\rightarrow M$ such that $f\circ s=1_{U}$?
$f$ is locally projection, so since $f$ surjective, we can define a smooth map that is locally inclusion, no?
@LukasHeger Ok thanks. Now assuming that R is commutative, we can say that I= RI=IR, which means that R/IR = R/I. Suppose that I is a maximal ideal. So we have field R/I = R/IR. Now the isomorphism f: R^m --> R^n gives isomorphism f': R^m/IR^m ---> R^n/IR^n, where R^m/IR^m is isomorphic to $R/I\times R/I\times...$ (m-times), i.e., $(R/I)^m$. So we have $(R/I)^m= (R/I)^n$ whence m=n.
@LukasHeger Actually I am trying to solve an exercise problem and the hint is to use an another exercise from a previous chapter. So I was just trying to figure why the hint makes sense.
@LukasHeger Ok thanks. Now assuming that R is commutative, we can say that I= RI=IR, which means that R/IR = R/I. Suppose that I is a maximal ideal. So we have field R/I = R/IR. Now the isomorphism f: R^m --> R^n gives isomorphism f': R^m/IR^m ---> R^n/IR^n, where R^m/IR^m is isomorphic to $R/I\times R/I\times...$ (m-times), i.e., $(R/I)^m$. So we have $(R/I)^m= (R/I)^n$ whence m=n.
I think that this answer is different than the rest of the answers there.
your answer is the same as the accepted answer in the linked dupe, except that that you didn't use tensor products, but just stated that an isomorphism $R^m \to R^n$ gives you an isomorphism $(R/I)^m \to (R/I)^n$
tensor products are a quick way to see this
you can show it directly, too
but technically that's still something you need to show
If a function is strictly monotonic increasing after the derivative test for $(-\infty, 0)$ and $(0, \infty)$, does that imply that it's strictly monotonic increasing for $(-\infty, \infty)$?
As an example, $f(x) = x^3$, $f'(x) = 3x^2$, so it's increasing for all $x \in \mathbb{R}$, $x \neq 0$ after the derivative test, but it's actually for all $x \in \mathbb{R}$
ok trying to understand your comment now: Suppose M and N are isomorphic (as left R-modules). And let L be a right R- module, then $L\otimes_R M\simeq L\otimes_R N$?
the best english translation is by heath. it is available in a cheap 3 volume dover reprint, which includes a lot of annotation that is fairly clear about what is in the original vs. added.
@Jakobian I've also only seen it recently. I think it's an exercise in Munkres somewhere. The same argument also implies that separated sets can be separated by neighborhood in a completely normal space (that's what I was looking for originally). I think that's actually an iff, too.
@LukasHeger The derived functor of a composite does not necessarily coincide with the composite of respective derived functors in general, if I remember correctly. There are technical assumptions.
some math has happened since euclid, you might want to read at least some modern commentaries that point out gaps or missing case analysis in the original
hartshorne's book is a good resource in english for that
I definitely see value in learning an ancient tongue to better understand our history as humans.. But I would prefer to observe and decipher the original texts in that case.
As a sort of rich art collector lol
@leslietownes When I learned calc1 in high school, I tried to find the original works by leibniz and newton because I thought I'd gain more insight and learn better. Boy was I wrong
@LukasHeger Yeah that's something I wonder is when we learn ancient languages in modern day, does our usage of the language make sense to those that spoke it back then. The way we communicate is vastly different from even 100+ years ago
@Yai0Phah certainly is sufficient for bounded derived categories, but yeah not sure about unbounded either
@Obliv of course it depends on how good you are and I'm not sure about the pronounciation. But I think the written sentences by advanced students who have taken three courses on writing in Latin/ ancient Greek would be understandable to those back then
there are very strict rules on only using words and grammatical constructions that are in the literature of a specific time
all these mentions of algebra and analysis, algebra and geometry, etc., have me wondering, what exactly changes between approaching a subject as is, and approaching it through algebra?
@shintuku I'm biased, but I'd say algebra is a very powerful und useful language and theory and it has transformed (and in some cases, clarified) some subjects when it was applied to them. Including: (algebraic) geometry, number theory, topology, function theory of severabel variables. But of course, algebra can't solve everything (yet!)
of course, algebra was also invented and expanded through being applied in these subjects
you won't find much algebra in numerical analysis of PDEs probably
but sometimes, algebraic machinery can obscure the underlying geometry
that's not a fault of the algebraic machinery, it's just a fault of blindly trusting it
but algebraization isn't a one-way street, through scheme-theory commutative algebra was geometrified in some sense and when you do homotopical algebra you can transfer intuitions from topology/homotopy theory to algebraic settings
she's a postdoc, no wait, actually junior professor now, sorry in Frankfurt, was a postdoc in Heidelberg before and still lives in Heidelberg and has an office here
Let $G$ be a compact Lie group acting freely and smoothly on $M$. Then, $\pi:M\rightarrow M/G$ is a smooth principle G-bundle.
Proof goes as follows:
By general theory, $\pi$ is a smooth submersion. Hence each point in $M/G$, there exists a neighborhood $U$ of $p$ and a local section $s:U\rightar...
Does anyone have a reference on what should be covered in an undergrad calc 3 course? I have seen widely varying coverages of topics on syllabi online…it seems like some courses don’t touch divergence, curl at all while others focus on it—is this up to the instructor or just a bad calc 3 course?
