A new version of the notes is available here. (It, and all older versions, are available at the usual place.)
Supravat Sarkar kindly pointed out that my Chevalley’s Theorem proof was still flawed at the very last step. I have now, I think (again!) fixed it. See Section 8.4.4 for that.
Math 216: Foundations of Algebraic Geometry 
April 2, 2023 at 4:29 pm
Thanks for the update, Ravi!
In 0.1 of the Preface, there are now two dependency diagrams, but AFAICT you only refer to the second one, which is isomorphic to the one in the previous draft.
Also, on p. 14 of the new draft, in the last sentence of the 4th-last paragraph, compared to the previous draft, the word “19” has been replaced by “c:curves”, which looks like a bare reference.
April 2, 2023 at 4:40 pm
Well that’s embarrassing. Thanks! Indeed, I meant to replace the old dependency diagram, and also I forgot a \ref. I’ve now fixed them. I’m tempted to repost, but feel like I should just wait until the next update…
April 2, 2023 at 4:47 pm
Hmmm, a repost would be nice because it will line up most of the page numbers with the previous edition (and most likely the next one as well), which might be convenient for some.
April 2, 2023 at 5:14 pm
Okay, I’ve just reposted it, with the same filename, so future readers will wonder what this thread is all about. Hopefully the new version has these fixed, but let me know if not!
April 2, 2023 at 6:12 pm
Looks good – thanks!
April 3, 2023 at 7:43 am
Just a minor thing: Exercise 7.3.M refers to a set {(p in X, local homomorphisms O_{X,p} to A)}. Its indending to represent that a morphism from Spec A to X corresponds to a choice of a point p together with a choice of a local homomorphism O_{X,p} to A. But since the plural of homomorphism is used in the description of the set, it’s technically not really this. For technical correctness one could write the set as {(p in X, some local homomorphism O_{X,p} to A)}. On the other hand I think everybody knows already what you mean.
June 12, 2023 at 6:39 pm
You are right, and it is best to fix it to avoid needlessly confusing people, so I’ve now made the change you suggested. Thanks!
April 4, 2023 at 12:01 pm
On page 53, in 1.6.6, you define homotopic maps. But your d’s given previously are degree-increasing, so your w’s should be degree-decreasing, in order to have dw + wd be degree-preserving, like f – g.
June 12, 2023 at 6:38 pm
You are right! The source of my cognitive error is that I wrote those complexes with subscripts rather than superscripts (for which the convention is that the d’s would be *decreasing*, but I didn’t say that). I’ve now fixed it, by (i) making the subscripts superscripts as they should have been, and then (ii) having w degree-decreasing as you said. Thanks!
April 4, 2023 at 9:34 pm
In the new section 1.6.6, in the definition of homotopic maps, the homotopy $w$ should go from $C_i \to C’_{i-1}$, not to $C’_{i+1}$.
April 5, 2023 at 8:31 am
I don’t think so, it’s in homological notations
April 5, 2023 at 4:05 pm
The boundary maps go in the direction of increasing index; the chain homotopy maps have to go in the opposite direction in order for their composite to preserve index, as chain maps must.
June 12, 2023 at 6:38 pm
See above; you are both in some sense right!
April 11, 2023 at 12:39 pm
(This is the same as David’s previous comment, I didn’t notice that he had already posted this.)
April 22, 2023 at 12:37 pm
There seems to be a small typo in the commutative diagram for the proof of Proposition 7.3.2 (p206). We are taking an element g of the ring so the restriction maps should be D(g) and D(\pi^{\sharp}g) instead of D(b) and D(\pi^{\sharp}b).
June 3, 2023 at 9:38 pm
Thanks! That was subtle, and resulted from some much-earlier changing of notation of “b” to “g”. And then your comment made me realized that there were several related infelicities near there too, caused by the same change. Now fixed.
April 24, 2023 at 12:17 am
Was idly browsing the bibliography and noticed that the link provided for [BP] is dead: http://www.maths.bangor.ac.uk/research/ftp/rpam/06 08.pdf, The article can still be found on the web, but I don’t know what the canonical URL should be.
June 3, 2023 at 10:22 pm
Hm, I can’t figure out what it should be either. I also found it on the web, and kept a copy in case it disappears. I tried to contact the author via a webform, but it didn’t work. I’ve removed the link from the bibliography, but really wish there could be a canonical source. If I could contact the author, I’d recommend that he post it on the arXiv, where it could be preserved for posterity.
April 24, 2023 at 9:49 am
Proposition 20.1.4 : there is this subtlety that should be stated somewhere. https://math.stackexchange.com/questions/4458152/is-numerical-equivalence-preserved-by-base-change-to-an-algebraically-closed-fie
June 3, 2023 at 9:57 pm
That’s a good point! I’ve answered that question on SE, and will fix it in the notes. Basically, I prefer not to go to the algebraic closure.
May 9, 2023 at 12:51 am
In section 2.1.1. you define a germ as an *object of the form* {(f,U):blah blah} modulo blah blah. But a germ is an element of this set (modulo the relation), not an object of the form of this set.
Sorry if this comment makes no sense.
June 3, 2023 at 10:00 pm
No, that makes sense! You’re right; I’ve removed the “set brackets” and hope that what I mean will be clear. If you think another wording is better, please feel free to suggest something.
May 13, 2023 at 11:54 am
Tiny typo on proof of 11.3.3.
The reasonableness of closed embeddings implies the reasonableness of separated morphisms. The proof says “locally closed” instead.
June 3, 2023 at 9:34 pm
Thanks, nice catch! Now fixed (in the version to be posted soon).
May 22, 2023 at 2:27 am
It seems to me that in the definition and discussion of rank-a-maps of vector bundles (14.3.4. – 14.3.K.), the notation for the ranks of the loc. free sheaves is mixed up:
– In Definition 13.4. you state that E is of rank b and F is of rank c, though in the commutative diagram they are of rank a+b and a+c respectively (upon restriction to some open U).
– In exercise 14.3.J. you refer to the rank of the cokernel of a map of vector bundles. The notation here seems to contradict the diagram in 14.3.5.
June 3, 2023 at 9:26 pm
Thanks for catching that! Now fixed (in the version to be posted next, fairly shortly).
June 19, 2023 at 1:49 am
In section 8.3.1, it is said:
>A morphism π:X→Y is quasiseparated if for every affine open subset U of Y, π⁻¹(U) is a quasiseparated scheme (§5.1.1). (Equivalently, the preimage of any quasicompact open subset is quasiseparated, although we won’t worry about proving this. This is the definition that extends to other parts of geometry.)
The sentence between parentheses is wrong https://math.stackexchange.com/a/4721092/394668
Instead, it should be “the preimage of any quasiseparated open is quasiseparated” https://math.stackexchange.com/a/4421658/394668
June 19, 2023 at 8:50 pm
You are right, thanks! I can see the error I made: I added this comment at the same time as adding the corresponding statement for quasicompact morphisms, and I was typing carelessly. (Aside: I actually don’t know if this definition is at all used or useful in other parts of geometry…)
June 19, 2023 at 11:09 pm
April Grimoire wrote to me (in an email): “In FOAGapr0123public.pdf, at the beginning of p.57, in the following paragraph
Here is a counterexample. Because the axioms of abelian
categories are self-dual, it suffices to give an example in which a filtered limit fails
to be exact, and we do this.
It seems that ‘filtered’ should have been ‘cofiltered’.”
That sounds right to me (and the example is both filtered and cofiltered in any case), but I’m mentioning this here so someone might stop me if I am doing something silly!
(And thank you to April Grimoire…)