Sorry. I'm saying that the notation prevents confusion between the dual of a vector space and the group of units of a ring. The question is why the "\vee" is used.
I've been reading about algebraic geometry lately. I see that a lot of authors use ##V^\vee## to denote the dual space of a vector space ##V##. Is there any particular reason for this?
The only reason I could think of is that this notation leaves us free to use ##R^*## to denote the units of...
How would one construct a graph, like the Petersen graph, in LaTeX? Is there a good package to use for it?
Edit: I should mention that I'm having trouble using instructions (which are not always consistent) that I'm finding on Google.
I only just recently picked up on some of these jokes.
https://www.youtube.com/watch?v=L9cVCaOsBSA
(27:00-ish)
Rocky: "What kind of game can you play with girls?"
[pause]
Bullwinkle: "Boy! This really is a children's show, isn't it?" :smile:
I think I misinterpreted what was being said. Is there an example where the structure sheaf of a (real) manifold is not naturally isomorphic to a sheaf of real-valued functions?
Oh. I thought I was clear when I said "sheaves of functions." By this, I meant sheaves that have whose sections that are functions. Sorry for being confusing.
Once again, thank you!
Indeed, being cocky with notation has the added effect of not being clear.
I implicitly assumed that the morphism of sheaves is given by ##f^\sharp:\mathcal{O}_N\to f_*\mathcal{O}_M##, where on each open ##U\subseteq N##, ##f^\sharp_U:\mathcal{O}_N(U)\to f_*\mathcal{O}_M(U)## is defined by...
I recently bought a copy of S. Ramanan's Global Calculus. I skimmed around a bit. Naturally, I was confused when it defined a differentiable function ##f:M\to N## between differentiable manifolds as a continuous map such that, for each ##x\in M## and for each ##\phi\in\mathcal{O}_N(V)##, where...
At the time, I thought, for some reason, that this would collapse to simply the product of topological spaces with a nice sheaf. Clearly, this is not necessarily the case.
I think I'm just lacking intuition for why this is the product. The prime ideal is a little foreign looking to me. Could...
Let me see if I understand what you're saying. Let ##\pi_i: \prod A_j\to A_i## be the natural projection maps from the universal property defining the product. Then, ##\pi_i\circ w=w_i##. But, since the ##w_i## commute with the induced maps ##\mathcal{F}(\phi_{ik})##, and ##A## is defined as...
I think this looks like a homework problem, so I'll just put it here.
Homework Statement
Demonstrate that, for any index category ##\mathscr{J}## and any diagram ##\mathcal{F}:\mathscr{J}\to\mathbf{Sets}##,
$$\varprojlim_{\mathscr{J}}A_j=\left\{a\in \prod_{j\in \operatorname{obj}(...
Wouldn't ##\mathfrak{P}## be unique for each pair ##(x,y)##?
Are specifying a basis element ##U##, each of which comes from ##U_1##, ##U_2##, and ##g##?
This is the localization, correct?
Where can I find this (preferably in English)? I'm having difficulty finding it with Google.