In 5.1.1, you give the definition of a quasiseparated topological space as follows: “A topological space is quasiseparated if the intersection of any two quasicompact open sets is quasicompact.”

However, Exercise 5.1F states: “Show that a scheme is quasiseparated if and only if the intersection of any two affine open subsets is a finite union of affine open subsets, if and only if the intersection of any two quasicompact open subsets is a quasicompact open subset.” It looks to me like the last if and only if is the definition of quasiseparated you give above; am I missing something?

Thanks very much for looking!

]]>The sentence following the diagram on page 58 (which refers to the definition of a double complex) reads —

“There are variations on this definition, where for example the vertical arrows

go downwards, or some different subset of the E^{p,q} are required to be zero, ….”

However, none of the E^{p,q} were required to be zero in the preceding text, so the allusion to a “different subset” might be confusing.

Perhaps you could just remove the word “different” (and change the following “are” to “is”.)

]]>I just tried another common viewer, Okular, and this does display correctly. ]]>

Can anyone else report (positively or negatively) on whether they experience this?

]]>that appears separately and toggles on/off with F9. Maybe recompile. ]]>

Exercise 17.3.B which shows that the composition of projective morphisms is projective, where the final target is quasi-compact has me a little confused. It seems from the hints that you’re aiming to find a line bundle that is locally O(1) and then use this to show that it is globally O(1). The affine case shows that L x (pi)_* M^m is locally very ample for all large m. Then by quasicompactness we can find an m that works for each patch in a finite affine open cover, so the trick then is to show that L x (pi)_* M^m is globally O(1).

You mention several times throughout 17.3 that in section 17.3.4 we will see that in Noetherian circumstances, projectivity with a choice of O(1) is affine local. It seems that in this exercise we’re trying to prove that it is affine local, but without the Noetherian hypotheses.

I asked about it here:

But so far the only answer seems to be going along the lines of the argument for affine-localness of relative very-ampleness for locally Noetherian schemes in the double starred section of relative very ampleness later in the chapter, which may or may not work out without Noetherian hypotheses. I also saw your original mathoverflow post about this question, where a few people remarked that EGA and the stacks project only seem to prove this proposition for the case where Z is additionally quasi-separated, although no-one disagreed with your claim that quasi-compactness is enough.

I guess the question is, then: is it really true that we don’t need further quasi-separated/locally Noetherian hypotheses? If so, would you mind clarifying how the local to global argument works?

(I’m sorry to bother you with questions asking for clarifications about exercises, but I’ve given them a lot of thought and spent a good deal of time looking for answers externally without much success. It’s therefore hard for me to tell if the intended hints don’t work, or if I’ve just not worked out how to apply them properly.)

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