Recent content by Mandelbroth

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...

Helpful as ever. Thank you!

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 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!

Could you please explain why?

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.

I'm relatively new to locally ringed spaces and sheaves. I was aware of them before, but I lacked the mathematical maturity to understand them. Let ##(X,\mathcal{O}_X)## and ##(Y,\mathcal{O}_Y)## be locally ringed spaces. If I were to take the product of two, what would the corresponding...

Alright. That makes more sense. Thank you!