Ethan MacBrough

first year PhD student at UW・he/him・all geometry is beautiful・last name pronounced "mack bro" (mækˈbroʊ)

Last Notes

2025-02-10 19:51:08 UTC

- reply

Oh, I see, I forgot the state w is supposed to be a linear combination of all permutations of RGB when moving from the first post to the second. Thanks for the clarification.

2025-02-10 17:22:37 UTC

- reply

Thanks for the post, I found it very enlightening. But I'm having trouble understanding
"The vector ... will contain a component in the RRR direction of U_{11} U_{12} U_{13}, but all six terms will only differ by a sign, and the sum will be zero."
Isn't the U_{11} U_{12} U_{13} term the only thing contributing to the RRR coefficient? What are the six terms you reference here?

2025-02-05 01:07:37 UTC

- reply

No, I'm the one who is wrong. I guess what I was calling finite-type is typically called "locally finite-type", and finite-type = locally finite-type + quasi-compact. So indeed finite-type implies finitely many points for schemes.
Also I figured out how to reduce the algebraic space claim to the scheme claim:
By definition we have a surjective étale map phi: U ->> X with U a scheme. To show desired finiteness for X, it suffices to show U is finite type. Since étale maps are locally finite-type, we know U is locally finite-type. Now let U_i be an open cover of U by finite-type schemes. Since étale maps are open we obtain an open cover of X by phi(U_i)'s. Since X is quasi-compact we cover X by phi(U_1) ... phi(U_n). Then the restriction of phi to U' := U_1 cup ... cup U_n gives a surjective étale cover of X by U', and U' is finite-type by construction.

2025-02-04 21:29:57 UTC

- reply

I don't think that really fixes the example, just take a disjoint union of arbitrarily many Spec(Z)'s.

2025-02-04 18:20:59 UTC

- reply

Actually I can't figure out how to ensure quasi-compactness pulls back to U. I think you might need to directly show that there exists some étale cover U ->> X with U a quasi-compact scheme.

2025-02-04 17:48:09 UTC

- reply

As stated this isn't even true for schemes, e.g. the disjoint union of arbitrarily many Spec(k)'s is finite-type over k. You need to add some topological finiteness criterion like quasi-compact.
With this additional assumption your claim definitely holds for schemes (cover by finitely many affines, each of which embeds in some A^n), and it should also hold true for algebraic spaces: by definition you have an étale covering U ->> X with U a scheme, and then adjective calculus implies U will also be finite-type and quasi-compact, so you can reduce to the scheme case.

2024-09-15 20:07:56 UTC

- reply

Ah, got it! For some reason when I read "the liquid in the raindrops affects the size of the rainbow" my brain jumbled it up and I came away thinking the size of the raindrop affects the rainbow. Oops!

2024-09-15 19:22:45 UTC

- reply

Does this account for the size difference of (m)ethane raindrops vs water raindrops?
Naively I would expect (m)ethane raindrops should be larger due to lower surface tension (although maybe this is counteracted by higher air pressure?), which might counteract the lower refractive index.