Eivind Dahl
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 Jul 5 comment What happens geometrically when the Jacobson radical is non-zero? Well I think that answers my question pretty well to be honest. So thanks! Jun 18 comment What happens geometrically when the Jacobson radical is non-zero? Naturally I do not think there is actually anything wrong with it. I removed the comment on separation; oops. I also tweaked the question, since we're not actually looking at the ring of real-valued functions on an affine scheme. I'm looking for intuition, and phenomena which can be directly attributed to a non-vanishing Jacobson radical $J$. How does for instance the space $Spec(A/J)$ look compared to $Spec(A)$. Hopefully the question is in better shape now. Thanks for the feedback! Jun 18 comment What happens geometrically when the Jacobson radical is non-zero? Old habit from writing mails, but I might as well stop if anyone feels it worth commenting on. Jun 13 comment etale space v. covering space So covering map $\Leftrightarrow$ étale map? This is a surprise to me. I think my attempts to locally trivialize an arbitrary étale map rely on tacit assumptions. Jun 11 comment How to study math to really understand it and have a healthy lifestyle with free time? I'd try to get an overview of whatever I'm learning first. I'd like to think of it as a big canvas; fill in the details according to whatever piques your interest. You absolutely positively can't learn everything (not even close), but you can learn top-down instead of bottom-up. For your comments on rigour, you might find this interesting: cheng.staff.shef.ac.uk/morality/morality.pdf May 25 comment Is there a universal property of $\text{Spec}(-)$? Exactly; thank you! May 25 comment Is there a universal property of $\text{Spec}(-)$? I clarified this assumption in the OP. May 25 comment Is there a universal property of $\text{Spec}(-)$? I knew about that, but I'm thinking about $\text{Spec}$ as a functor $\text{Ring}^{op}\to\text{LRSpaces}$. This question may still be silly though :) Apr 28 comment Natural Isomorphism $\mbox{Hom}(\oplus_\alpha A_\alpha,G) \simeq \prod_\alpha \mbox{Hom}(A_\alpha,G)$ @Qwirk: That's precicely what happens in this case as well, but with different words in it. By definition, a morphism from the coproduct is the same thing as a morphism from each of its factors. Apr 25 comment What is a natural isomorphism? Nice meta-observations :) Apr 25 comment A question on differentiability of the inverse of strictly monotonically increasing functions Yup, the answers down below cover it. Just to be sure you're not missing out on anything; the inverse $f^{-1}$ of a real to real function can aquired from the graph of $f$ by just flipping the xy plane along the diagonal. If you haven't already you might find it worthwhile to connect the algebraic picture below with this geometric picture. Apr 9 comment How to “grok all the major pieces” of math +1 for category theory. Apr 9 comment How can there be alternatives for the foundations of mathematics? Interesting! I've been hoping you could get around it somehow. Apr 9 comment How can there be alternatives for the foundations of mathematics? You talking about the appendix which I removed? Sheaf topoi are Grothendieck topoi. I'm not entirely sure how this all ties together. But it's a huge and interesting field no less. Apr 7 comment Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ Oh gee of course! :p Apr 7 comment Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ Bravo, bravo! I quite like it. I want to read about "because the equality holds for infinitely many values of x (and we are working over Z...)". Do you have a reference? It's probably in my algebra book, and it does make sense. Apr 6 comment Same symbol “$\partial$” - different things ( the boundary $\partial A$ / partial derivative $\frac{\partial f}{\partial x}$)? The integral $\int_a^b f(x) dx$ of a map $f:[a,b]\to\mathbb{R}$ is the same thing as the difference $F(b)-F(a)$ where $F$ is the antiderivative of $f$. Notice that $a,b$ are the boundary points of $[a,b]$, and that $f=F'$. This connection between integration over a cell and its boundary gives us Stokes' theorem. When integration happens in several dimensions, we need the partial derivatives in there. I'm not sure about if this choice of notation was intentional. Apr 3 comment Is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$? @Ryan Sounds good to me. Thank you! Apr 3 comment Two definitions of exactness Thanks a lot! I suspected it would have something to do with building the limit from products and equalizers. That fact will probably prove a good exercise :) Mar 24 comment Is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$? Yeah I noticed that one too. If there is such a homotopy-equivalence ever (unless $X$ is $n$-connected or something) it seems to be highly non-obvious. I've not read that, no. Do you have a good reference for it? Thanks!