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.
Ethan MacBrough /
npub178…usvwx
2025-02-05 01:07:37
in reply to nevent1q…hj64
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npub178a94wsk5sexgxa02zyrp9us7l3upzlnuzt4ulp72q3als5snjts2usvwxPublished at
2025-02-05 01:07:37Event JSON
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"content": "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.\n\nAlso I figured out how to reduce the algebraic space claim to the scheme claim:\n\nBy definition we have a surjective étale map phi: U -\u003e\u003e 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.",
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