It has been a while, but finally, the next version of the notes are here. (For convenience, all earlier versions are posted at the usual place. Added Sept. 4, 2022: a version with an imperfect index is here.)
It took a while because I split the quasicoherent sheaves chapter in two (one is much earlier, and is quasicoherent sheaves ~ “modules over a ring”, and the later one is quasicoherent sheaves ~ “vector bundles”), which required a cascading series of changes in logic and ordering and references.
But it is in much better shape, and I intend to post updates periodically this academic year. People have given me a large number of very useful comments, and they are all in the process of being considered (and often implemented). I think comments made on this blog have been mostly responded (even if to say “it is now on the list”).
Please continue to let me know about any suggestions, corrections, typos, etc., no matter how small. The only exceptions are stylistic things (margins, fonts, low-quality figures).
At this point the figures are supposed to now be substantively correct (except of course where there are things on my “to do list” to fix). Everything before the chapter on dimension now has essentially no remaining known issues (except for a few pages on geometric fibers that I am still rewriting). Later chapters are in different stages of editing.
Here are some random notable changes that I can remember off the top of my head — the things mentioned in earlier posts (associated points; cohomology and base change; etc.) are now in. I like the proof of Chevalley’s Theorem much better (it feels now very easy modulo the fact that finite morphisms are closed).
Math 216: Foundations of Algebraic Geometry 
September 2, 2022 at 11:00 am
The posted version lacks an index – will that be forthcoming?
September 2, 2022 at 11:33 am
Yes! It is “in process”, which means it is a disorganized mess. It will likely stay a mess for much of the year, because it makes sense to rationalize it only when it is clear what I am rationalizing. I’m hoping that a “searchable pdf” will be an okay (but not great) substitute. But if a weird index is better than no index, I’m happy to just post it with the weird index. (Currently, I try to index as much as possible, and worry later about what should be under what heading, and what merits a listing.)
September 2, 2022 at 2:31 pm
Hi Ravi and Mark,
I made an HTML version of the current notes.
https://wanminliu.github.io/Ravi_AG/202208/Ravi_AG.html
Since it is an HTML webpage, so it is easy to make a link in its “outline” (last line of the outline) to the previous HTML version (which has index). I hope it would be useful.
The purpose of the HTML version (via the tool pdf2htmlEX) is for a quick preview of the book by a web browser without downloading the whole PDF.
September 2, 2022 at 8:41 pm
Hi Wanmin, As with your previous version, this is just brilliant!
September 3, 2022 at 10:18 am
I should also say: because the page numbers have changed from the previous version, just using the previous index won’t help; but I’d be happy to include the new (highly imperfect) index if it wouldn’t be distracting.
September 3, 2022 at 11:19 am
Speaking personally I would find an imperfect index better than nothing at all.
September 4, 2022 at 8:29 am
Great, in that case I’ll post a version soon (when I get a chance in the next couple of days) with an index.
September 4, 2022 at 5:04 pm
I’ve now posted the version with index (see the edited first paragraph of this post). It is also here: http://math.stanford.edu/~vakil/216blog/FOAGaug2922publici.pdf
September 3, 2022 at 11:18 am
Wow, that’s awesome – thanks!
September 4, 2022 at 5:05 pm
(for the record, this was in response to Wanmin Liu, not anything I did…)
November 19, 2022 at 6:22 pm
Hi, I am writing just to ask a question. Version is august 29. 2022. On page 151, definition of vanishing set of ideal, is homogenous ideal I required to satisfy I<S_+ as in earlier version of notes? It seems equivalent, but for somebody learning for the first time can be confusing.
June 20, 2023 at 1:05 pm
I think that’s not necessary, and that’s why I removed it. I hope I don’t eat my words!
September 2, 2022 at 1:47 pm
Thank you very much!
September 3, 2022 at 10:29 am
Typo in Example 1.4.3 on page 43: “For a prime number
” should be “for a ring 
“. Probably the fault of example 1.4.4.
September 3, 2022 at 10:32 am
Also in examples 1.4.3 and 1.4.4 you omit the set notations, writing![A[[x]] = a_0 + a_1 x + a_2 x^2 +\cdots](https://s0.wp.com/latex.php?latex=A%5B%5Bx%5D%5D+%3D+a_0+%2B+a_1+x+%2B+a_2+x%5E2+%2B%5Ccdots&bg=ffffff&fg=29303b&s=0&c=20201002)
instead of ![A[[x]] = \{ a_0 + a_1 x + a_2 x^2 +\cdots\}](https://s0.wp.com/latex.php?latex=A%5B%5Bx%5D%5D+%3D+%5C%7B+a_0+%2B+a_1+x+%2B+a_2+x%5E2+%2B%5Ccdots%5C%7D+&bg=ffffff&fg=29303b&s=0&c=20201002)
.
September 4, 2022 at 5:07 pm
Thanks Lior! Both now fixed. I thought to myself “how did that happen?!” but you actually are right that it is the fault of example 1.4.4.
September 4, 2022 at 10:41 am
Possible edit to 3.2.G : You write in italics “the prime ideals of k[x1, x2, …, xn] with finite residue ring…” . Possibly write “with residue ring finite dimensional over k” so that people don’t have to remember what “finite” means in this context.
Or maybe not; you do recall it earlier in the paragraph. It just stood out to me as I was skimming: “wait, the finite residue rings? If k is infinite, there aren’t any …”
September 4, 2022 at 11:02 am
+1 I know this is standard terminology in AG, but I think “finitely generated as a k-algebra” and “finitely-generated as a k-module” are precise, crystal-clear, and worth the modest effort for the additional clarity.
September 4, 2022 at 5:14 pm
I’m with you both on this. I’ve changed it to “finite-dimensional residue ring”, under the understanding that it is clear from context that it is finite-dimensional over the field k, as that’s the only possible meaning. (But I’m open to changing it further.)
I don’t like the phrase “finite A-algebra” even though it is standard. (Perhaps instead of “finite A-algebra vs. finitely generated A-algebra” it could be “module-finite A-algebra vs. finitely-generated A-algebra”? Then “finite type quasicoherent sheaf” could instead be “module-finite quasicoherent sheaf” in some alternate universe.)
September 4, 2022 at 2:19 pm
Paragraph 2 of 14.1 (pg 391): missing a “with” in the parenthetical remark “This agrees its rank as an O-module at a point”.
Thanks for the notes!
September 4, 2022 at 5:11 pm
Thanks, Matthew, now fixed! And this bit of exposition is new, so I’m glad to have you catching things like this.
September 8, 2022 at 12:05 pm
I’m glad to help – these phenomenal notes really inspired me to start learning the subject seriously, so I can’t thank you enough for them.
By the way, another super tiny typo in a new section in chapter 2: Definition 2.7.4 on page 92 says “sheaes on Y” rather than “sheaves”. In addition, in this same section, the upper star notation is used, but in these notes I think that notation is reserved for pullback of an Ox-module. Should that be the inverse image sheaf instead? I don’t think that notation was introduced previously.
September 5, 2022 at 3:41 pm
A few comments on chapter 19. I hope they are not frivolous.
1. At the start of the proof of 19.1.4, you say that we can pass to the case for X=P^n. For part b though, you would want that pushforward of (F(n)) is isomorphic to (pushforward of F)(n). This is true by the projection formula, but I feel like this should be mentioned.
2. Your definition of Cech cohomology in 19.2 depends on an order of the open cover (as the differentials’ signs change when changing the order). And when proving the independence of open cover, you only add in an open set at the beginning, as opposed to adding open in at an arbitrary index, so this cannot directly be used to show that it is independent of order, and hence, I think that makes the proof of 19.2.2 incomplete. I was able to come up with a proof that changing the order yields isomorphic complexes, by proving it just for transpositions (and it seems like the isomorphism for general permutations may be hard to describe).
3. For sections 19.4.5, 19.4.6, it felt ambiguous to me which hypothesis were being used. First, degree is defined for projective integral curves, and then it is said that for locally free sheaves the irreducible hypothesis can be dropped. But are we still requiring reduced? I.e. are we defining degree of a line bundle on a nonreduced curve? I would have though not, but in 19.4.M, it says to prove the degree of an ample line bundle on a projective curve is positive. Is there an underlying integral hypothesis for this? Or should be take the curve to just be reduced?
(Side note on 19.4.M: I was trying to do this on the 2017 version, and was only able to get it in the regular case. After thinking about it for a while, I looked it up to see if someone had asked about this on Stack Exchange before, and someone had. The answer basically said that you need to develop a lot of machinery, including some sort of RR for nonreduced curves, like in 19.4.S. But then you added this new version, and I was able to use the new 19.4.K to show that nontrivial line bundles on integral curves with a nonzero global section have positive degree, which shows things for very ample bundles. Going from very ample to ample was hard for me, as we only proved deg(L\otimes M)=deg(L)+deg(M) for regular curves, but I was eventually able to do it by showing this equality when either L or M has a nonzero global section. I feel like my path here is not intended, as what I did shows the result for semi ample bundles.)
4. I saw you said somewhere you are wanting to do changes to 19.4.S. One thing I noticed is that you took out the discussion talking about the length of a module, but you still use the notion of length in the problem statement of 19.4.S.
Thank you for these notes and for your time reading this.
June 21, 2023 at 2:19 pm
Thanks William! The issues around 19.4 should now be fixed, and the new version should be out fairly soon (by end of June 2023?), but let me know if you are not happy. Everything else is also hopefully dealt with!
September 9, 2022 at 12:05 pm
Some possible corrections and suggestions that I found while going through the notes (the notes were very helpful, thank you!):
1) In proposition 20.10.3, I think it might be better to replace $C^{reg}$ with the smooth locus of $C$. This is because the former does not behave well with base change, so point b) does not make much sense. It’s not hard to show that in this case, the smooth locus is non-empty, even though the ground field may not be perfect: Indeed, the smooth locus $V$ of $C_{\bar{k}}$ is non-empty (by 22.3.5), and its image $U$ in $C$ is an open set which must be smooth over $k$ (because $V\to U$ is faithfully flat, hence $\Omega_{U/k}$ is invertible as its pullback to $V$ is invertible). Moreover, if p, q are points on the smooth locus of C, and the line pq intersects C again at r, it’s possible to show that r is also in the smooth locus. Indeed, choose lifts p’, q’, r’ of p,q, r in $C_{\bar{k}}$ which are still collinear. By the first paragraph of the proof of the proposition, r’ is also regular. Hence r is smooth (it is in the image of $V$).
2) In the proof of Proposition 21.1.4, I don’t think enough explanation is given as to why we may assume $k$ is infinite. While it is explained that intersection numbers are preserved by a field extension, it is not at all obvious why *numerical equivalence* is also preserved. Indeed, I don’t know if this is true for any arbitrary non-algebraic extension $K/k$. I thought about how to prove this and came up with the following argument which works for algebraic $K/k$ (and in particular allows one to assume $k$ is algebraically closed): Let $\mathscr{L}$ be numerically trivial on $X$, and fix any curve $C\subset X_{K}$. The ideal sheaf of $C$ can be defined by finitely many elements of $K$ over $k$, so it is actually the base-change of a subscheme $C’$ of $X_{k’}$, for some finite extension $k’/k$ contained in $K$. The morphism $X_{k’}\to X$ is proper (indeed, finite) so by exercise 19.4.T (b), the degree of $\mathscr{L}$ pulled-back to $C’$ is naught. This implies the degree of the pullback to $C$ is also 0, because the degree is preserved by the flat base change $K/k’$.
3) In the hint for Exercise 21.2.I, the notion of the normal bundle is used, although this is defined only later (in chapter 22).
4) Exercise 25.4.N is false as stated: the element $tym\in A$ is equal to $t^2x(x+1)\in A$, because
\[(ty)m=(mx)m=(m^2)x=(t^2(x+1))x=t^2x(x+1)\]
But tym-t^2x(x+1) has no monomial divisible by $m^2$, $mx$, or $y^2$. Actually, the morphism $A\to k[t]$ is not even flat because $t$ is a zero-divisor in $A$ (as $ym\neq tx(x+1)$). To make $A$ flat, you need to add the relation $ym=tx(x+1)$, which holds in $A_t$. The real flat limit then turns out to be
\[\text{Spec}(k[x,y,m]/(m^2,mx,my,y^2-x^3-x^2))\]
5) Exercise 25.6.F is false as stated: a counter-example is given in this MSE post:
https://math.stackexchange.com/questions/4188790/slicing-criterion-for-flatness-in-the-source-vakil-24-6-f/4401615#4401615
The essence of this example is as follows: Its possible to show that $M/fM$ is flat at some $p\in \text{Spec}(A)$, if the condition holds for $n$ being the image of $p$ in $\text{Spec}(B)$. Thus we might need the condition to hold for primes $n$ other than maximal ones. If $B\to A$ is a local map of local rings, the condition suffices for maximal ideals, because it implies $M$ is flat at the maximal ideal of $A$ which implies it is flat. In general its enough for the condition to hold at the images of all closed points of $\text{Spec}(A)$.
6) Exercise 25.7.A (d) is false as stated. For a counter-example, take $Y=A^1$, and $X$ to be the subscheme of $P^1_Y given bt the union of the two axes (the scheme cut out by $xy=0$). Let $\mathscr{F}=\mathscr{O}_X$. Then $X\to Y$ is projective, and the Euler characteristic of $\mathscr{F}$ on the fibers is always 1$, but $X\to Y$ is obviously not flat. The exercise becomes true if the Hilbert polynomial of $\mathscr{F}$ on the fibers is constant: Indeed, we can assume $X=P^n_Y$. Now for large $m$ the hilbert polynomial is just $H^0(F|_q(m))$. By the “stable base change theorem”, this equals the fiber of $H^0(F(m))$ at $q$ for large $m$. If this is constant as a function of $q$, the sheaf $H^0(F(m))$ must be locally free for large $m$, which implies $F$ is flat by 25.7.A (c).
7) Exercise 26.2.F (b) is false as stated. For a counter-example, take $X=Y=A^1$, and $X\to Y$ given by $u\to t^2$. In characteristic not 2, the corresponding map on differentials is injective ($du $ goes to $2tdt$). Maybe the intention was to assume the map is injective on the *fibers*, not the stalks.
8) Exercise 26.3.C is not well-stated, because no definition was given for homogeneous spaces over non-algebraically closed fields. I assume transitivity is no longer the right notion, because different closed points have different residue fields.
9) In exercise 28.1.N, it’s not correct to state theorem 26.2.2, because the fibers are not necessarily smooth, only regular (there is a difference!). This poses the additional problem of $\Omega_{X/Y}$ not necessarily being flat over $Y$, thus withholding us the use of the Cohomology and Base-Change theorem on it. This could all be fixed by replacing the word “regular” with “smooth”.
10) The proof of Theorem 19.1.3 (ii) is flawed in the non-noetherian case. Indeed, it isn’t necessarily true that $\mathscr{G}$ is coherent, as $O_X$ need not be coherent in general. Thus you can’t use the induction hypothesis $H^{i+1}(G(n))=0$ to show $H^i(F(n))=0$.
According to a post I found on MSE, Serre vanishing holds for all finitely presented sheaves $F$ on $P^n_A$, although I could not find a proof for this. Maybe you can use Grothendieck’s theory of limits of schemes to show this.
11) The proof of Theorem 19.2.2 is flawed because the sets $U_i$ do not play symmetric roles in the definition of the Cech complex. Thus, as you cannot assume that you only remove/add sets on the first index, the proof is not complete. In the Stacks project, the Cech complex is defined in a more symmetric fashion and the proof there is correct.
12) Related to the previous point, the proof of 19.1.2 is flawed, again in the sense that assuming which indices have negative exponents on the variables loses generality. Both the Stacks project and Hartshorne have correct proofs.
13) Exercise 30.5.B is false as stated: for a counter-example, take $Y=A_1$, and $X=Y\sqcup \text{Spec}\ k$. Define $X\to Y$ by identity on $Y$ and $\text{Spec}\ k\to 0$.
The exercise becomes correct if you additionally assume $X$ is integral (even just irreducibility is not enough, as given by $\text{Spec}\ k[x,y]/(xy, y^2)\to \text{Spec}\ k[x]$)
14) 30.5.C is false as stated: for a counter-example, take $X=\text{Spec}\ k[x,y]/(xy,y^2)$ and $Y=Y’=\text{Spec}\ k[t]$. Define $\pi$ by $t\to x$ and $\pi’$ by $t\to x+y$.
15) In the proof of Theorem 30.7.1, it is not shown that the resulting $Y$ is smooth, only *regular*. I’ve found a way to fix this as follows. Hartshorne shows in his proof of the theorem, that assuming $k=\bar{k}$, the normalization is not actually needed. But the constructions involved apart from the normalization respect base change. Thus, $Y’_{\bar{k}}$ is exactly the contraction resulting from the same construction on $X_{\bar{k}}$. This is smooth (according to Hartshorne), and hence the original $Y’$ is also smooth. Thus normalization is not needed in this case either.
16) Exercise 29.8.B is false. For a counter-example, take $B=k[x]$, $X=V(x)\subset \text{Spec}\ B$, $\mathscr{F}=\mathscr{O}_X$, $I=(x)$, and $i=0$.
This can be fixed as follows: Instead of identifying the required sum with $R^i\rho_* \sigma^*\mathscr{F}$, instead show that it is isomorphic to $R^i\rho_* \mathscr{G}$, where $\mathscr{G}$ is a certain quotient of $\sigma^*\mathscr{F}$ (and hence coherent). This $\mathscr{G}$ is constructed as “$\bigosum_n I^n\mathscr{F}$”. Formally, for each affine open $U\subset X$, if $\mathscr{F}(U)=M$, we define $\mathscr{G}(U_R)=\bigoplus_n I^nM$, which can be easily seen to be an $\Gamma(U_R)$-module. It’s not hard to check that these glue, and thus define a quasi-coherent sheaf $\mathscr{G}$ on $X_R$. Finally, it’s easy to define a surjection $\sigma^*\mathscr{F}\to \mathscr{G}$ locally as summing $I^n\otimes_B M\to I^nM$.
17) In 31.3, the functor $\pi^!$ does not necessarily take quasi-coherent sheaves to quasi-coherent sheaves, because sheaf-Hom is only necessarily quasi-coherent when the first argument is coherent. But $\pi_*\mathscr{O_X}$ is rarely coherent. This does not matter so much in the discussion of Serre duality though, as for a finite morphism $\pi$ we have $\pi_*\mathscr{O}_X$ coherent and there are no problems.
This concludes the list of errors that I think are important to fix.
As an appendix, I’ll add a list of exercises that have no “errors” per se but are stated in a way that makes them impossible to my knowledge to solve using the material learned up to that point. If anyone has found a solution to one of these exercises that fits in at the respective point in the book, I’d be glad to see it.
I tried so solve all the exercises in the book, and these exercises are the ones I had the most trouble and frustration with. In the end I couldn’t deal with most of the following myself, so I looked online or asked colleagues for assistance.
a) 13.4.C: The only way I know how to solve this is by bounding the dimension of the “bad” collections of hyperplanes, which only works given that $X$ has a dense smooth open subset. This is used to show that for a certain point $p\in X$, generally chosen hyperplanes containing $p$ will cut out a scheme which around $p$ is only one reduced point. For this it is crucial that $p$ is smooth, to bound the dimension of the cotangent space.
While it is true that $X$ contains a dense smooth subset, it is non-trivial, and only proved much later (in chapter 22).
b) Exercise 18.3.B cannot be solved as far as I know without the addition assumption on quasi-separateness of $Z$. Indeed, both EGA and the Stacks project assume this in their respective proofs of the projectivity of a composition of projective morphisms, and crucially use this condition. I’d be glad to see a proof without this extra assumption.
c) Theorem 18.3.9 and Exercise 18.3.F have the same problem, without quasi-separateness of the target I have not found a proof anywhere online (the corresponding statements in the Stacks project assume quasi-separateness of the target).
d) Exercises 19.4.M and 19.4.N are both doable in the case of regular curves using the theory developed in the chapter, but the general case, as developed in the Stacks project, requires much stronger machinery and background.
e) I could only solve 23.2.A under the additional assumption that the category involved had enough projectives. The direction $P$ projective implies every sequence splits is easy, but I know of no proof for the converse in general. This is the one I’m the least sure of; there might be a simple solution I’m missing.
f) While exercise 26.2.E (b) can be reasonably done in the case where the ground field is perfect, I know of no solution in the general case without the use of Miracle Flatness (only developed in later chapters). The same logic applies to exercise 26.2.F (a). I found a solution to both of these using miracle flatness here:
https://math.stackexchange.com/questions/4067677/source-is-smooth-if-target-is-smooth-and-fibers-are-smooth-of-constant-dimension
Thanks for the notes!!!! I loved them and they helped me learn a lot. Hope these corrections help.
September 10, 2022 at 12:04 pm
For d), I believe I was able to come up with a reasonable solution to 19.4.M (and 19.4.N follows directly from 19.4.M, I believe). I do not believe it is what was intended, but I am not sure. I posted in on stack exchange here: https://math.stackexchange.com/questions/838732/degree-of-ample-bundle-over-projective-curve-is-positive/4528815#4528815
September 11, 2022 at 5:58 am
Thank you!! This solution is indeed simple, exactly the kind I was looking for.
I noticed your method also deals with Exercise 19.4.L, which also causes problems in the singular case. Indeed, just replace M in your proof by the required vector bundle, and get deg(M (x) L)=deg(M)+(rank M)*deg(L) when L is very ample. This implies we only need to solve the case when M has many sections (which is easier).
February 10, 2024 at 5:34 pm
I never get around to replying to this, but a revised version of these comments are here.
September 15, 2022 at 9:47 am
In 2.7.4 (push-pull map), by $\beta^*$ etc do you mean $\beta^{-1}$?
February 10, 2024 at 5:34 pm
Thanks, finally fixed!
September 19, 2022 at 3:28 pm
Typo on page 39: “If this exercise too hard now…” Missing “is”?
November 25, 2022 at 6:43 pm
Thanks, now fixed!
November 21, 2022 at 10:17 am
Thank you for coming back to the project!
Here is a suggestion for the area around 2.6.D (page 91). You say “exactness of a sequence of sheaves may be checked at the level of stalks”. However the following 2.6.D only showes that an exact sequence of sheaves induces an exact sequence of stalks. We have not proved that exactness at stalk level for all points implies exactness of sheaves. I was confused by this and after googling I quickly found out many other people are confused by this too:
https://math.stackexchange.com/questions/2521822/exactness-can-be-checked-on-stalks
So I would suggest to insert an exercise showing that exactness of stalks for all points implies exactness of sheaves. (Or a sequence of exercises leading to this.)
Thank you!
July 1, 2023 at 9:37 am
Thanks! I didn’t even notice that I never spelled this out. I’ve now added an exercise as you suggested, in the version I just posted a couple of minutes ago. If that’s not the right wording, of course, let me know.
December 16, 2022 at 3:09 am
Typo in exercise 1.4.H. the relation should be m_j – F(n)(m_i) instead of m_i – F(n)(m_i).
Thanks a lot for the notes!
December 16, 2022 at 8:21 am
Please ignore this, it didn’t make sense. (Except the last line)
December 19, 2022 at 11:59 am
I could have deleted your comments, but I decided to keep them — earlier I thought precisely the same things as you did (see some earlier version of the notes), and then someone else caught it. I just think it is praiseworthy to speak up like this, even if you are sometimes not right. Because if you speak up only when you are certain, your understanding grows much more slowly.
Separately, thanks for the thanks!
February 21, 2023 at 11:05 pm
Definition 7.3.3 (pg. 201 – scheme morphism):
“We then have notions of isomorphism – just the same as before, §4.3.7 – …”
But §4.3.7 is the definition of locally ringed space. Should be §4.3.1 (pg. 137 – isomorphism of ringed spaces).
Thanks a million for your work all these years! Words really can’t express our gratitude ❤
March 14, 2023 at 7:02 am
Thanks, that makes sense! I’ve patched that as you suggested.
February 26, 2023 at 6:23 pm
Typo in exercise 22.2.U: should the exercise instead be to describe an isomorphism from $\Omega_{X\times_Z Y/Z}$ to $pr_1^*\Omega_{X/Z}\oplus pr_2^*\Omega_{Y/Z}$? (Instead of describing an isomorphism from $\Omega_{X\times_Z Y/Z}$ to $\alpha^*\Omega_{X/Z}\oplus \beta^*\Omega_{Y/Z}$)
Also, thank you for the notes! I’ve very much enjoyed them.
March 14, 2023 at 7:00 am
Thanks, now fixed!