# Is “locally of finite type” affine-local on the source?

Hartshorne exercise II.3.3(c) asks the reader to prove that if $$f:X\to Y$$ is finite type, then for any $$\mathrm{Spec}A\subset Y$$ and $$\mathrm{Spec}B\subset X$$ with $$\mathrm{Spec}B\subset f^{-1} \mathrm{Spec}A$$, we have $$A\to B$$ is of finite type.

However, the proof I produced seems to work in the case that $$f$$ is only locally of finite type, but exercise 2 of that section of Hartshorne asks to prove a strictly weaker condition in that case. (That is, for any affine of $$Y$$, the preimage can be covered by affines giving us finite-type homomorphisms.)

Is the property of local finite-typeness affine-local on the source? My proof seemed very straightforward:

Take any affine open $$\mathrm{Spec}\ B$$ of $$X$$, and also choose a cover $$\mathrm{Spec}\ B_i$$ of $$X$$ by affine opens so that each $$A\to B_i$$ is finite type. (This cover is guaranteed to exist by part II.3.2.(b).) Then by the "Affine Communication Lemma" (5.3.2 of Vakil's FOAG - page 155 in the Feb 25, 2013 version) it suffices to show that if $$A\to B$$ is finite type and $$f\in B$$, so is $$A\to B_f$$, and if $$f_1,\dots,f_n\in B$$ generate the unit and each $$A\to B_{f_i}$$ is finite type, then so is $$A\to B$$.

The first claim is trivial. As for the second, take $$b\in B$$, and choose $$b_{i1},\dots,b_{ik}\in B$$ which generate $$B_{f_i}$$ over $$A$$. Then we can choose $$n$$ and $$g_{ij}\in A$$ for each $$i$$ we have $$f_i^nb=\sum_j g_{ij}b_{ij}$$. Some linear combination of the $$f_i^n$$ is equal to one, and so we can write $$b$$ as an $$A$$-linear combination of all the $$g_{ij}$$.

Does this reasoning make sense?

• "Is “locally of finite type” affine-local on the source?" Yes, see any good introduction to algebraic geometry (and Hartshorne is not good, although standard). – Martin Brandenburg Jun 7 '13 at 10:05
• Okay thanks! I've been using Hartshorne, Qing Lui, and Vakil together. I don't usually even read the text of Hartshorne other than to get definitions, and I otherwise just use it for the exercises, but I couldn't seem to find anything about it in Lui or Vakil. Vakil doesn't seem to do it, but I did find it in Lui, page 87, Definition 2.1 and Proposition 2.2, for those interested. – Xander Flood Jun 7 '13 at 15:13
• Hartshorne isn't a good idea even for definitions (some are simply "wrong"). – user314 Jun 8 '13 at 21:07
• I've noticed that; I only get definitions from Hartshorne for purposes of Hartshorne's exercises (and unfortunately they are "right" in that context). – Xander Flood Jun 11 '13 at 1:44