I have recently posted the latest pdf version (dated October 21, 2025) in the usual place. This preprint version has (with trivial exceptions) the same content as the published version, although the pagination and formatting are different, and it is not as beautiful. Because people may use different versions, I strongly encourage everyone to refer not to page numbers, but to chapters and section numbers.
I will likely not post another pdf version, at least any time soon, because it will be too confusing if there are two diverging versions out in the world. The comments and errata should then be applicable to both the published and the “preprint” pdf versions.
As usual, there are a number of comments here and in emails I owe replies to. Also as usual I am gradually replying.
Some errata (temporarily parked here):
I want the published version and the online one to be close to each other, so I subscribe to a “less is more” philosophy of errata. I thus am arbitrarily putting potential errata into three categories.
(1) substantive errata that I think may potentially seriously confuse a significant portion of readers.
(2) less essential errata that I would still like to correct.
(3) potential errata that don’t meet the high bar, but which I still may make note of. Many imperfections will be allowed to slide by. Some I save for a (very hypothetical) revised edition in the indefinite future.
I will list Substantive and Less Essential errata here. Other errata may be mentioned elsewhere, for example in response to comments to posts.
Substantive Errata (please copy into your version)
1.5.2, end of par 1: “preserve products” -> “preserve finite products”
6.1.B(b): delete “in the distinguished topology of X”
6.3 par 3: “check that a subcategory” -> “check that a full subcategory”
2 lines before (6.5.0.1): “if its” -> “if M \neq 0″ and its”
at end of paragraph befoer 6.5.1: “A composition series with “n links in the chain”, sch as (65.0.1), is said to have length n.” [I almost want to say “A composition series (or more generally a filtration)” but I don’t want to distract the reader.]
6.5.L: “simple objects” -> “finite type simple objects”. But this problem is not currently well-placed or well-chosen. For now, just read and know the statement, and show the trivial direction (that the structure sheaf of a closed point is simple). As a consequence, in 6.5.M, “finite length” -> “finite type finite length”, and delete “F is finite type, and”. Possibly the end of 6.5 from 6.5.8 onwards should be moved into 6.6.34.
6.6.H: “some multiple” -> “some nonzero multiple”
6.6.26: Proof, last line of first paragraph: “M” –> “S^{-1} M”. Then in the starred unimportant remark, “A/Ann_A m” -> “Ann_A m” in l. 3, and “P” should be a fraktur “p” in l. 4.
6.6.38 “said to be p-primary” -> “said to be p-primary where p=\sqrt{I}”. “associated prime ideals A/I” -> “associated prime ideals of A/I”.
sentence after 8.4.H: “dense open subset” -> “nonempty open subset”.
13.7.G: “(respectively, Z) have valuation 0” -> “have valuation 0 (respectively, no condition on the valuation)”
21.4.1 l. 5: “Note that finite morphisms” -> “Note that dominant finite morphisms”
21.4.E: add “Assume the characteristic of the base field is zero.” at the start.
21.6.1 line 2 of proof: “rank” -> “rank of \Omega_{X/k}”.
24.4.H equation display (published version only): add “… \rightarrow” at the end of the first line of the equation display.
26.3.D, first sentence: add “in the case where A is an integral domain”.
Less Essential Errata and Minor Edits (issues that shouldn’t fatally derail the reader, including minor notation errors; feel free to copy in to your version)
0.1, two-three pages in: paragraph heading “Chapters 3-5.” should be “Chapters 3-6.”
2.7.F: At the end, add “(I do not recommend trying this exercise unless you are truly motivated. See [SGA4, XVII, Prop 2.1.3] for a proof by Deligne.)”
4.5.B: “4.4.9” –> “\S 4.4.9”
5.4.L: “Exercise 4.4.12 and \S 5.4.D” –> “\S 4.4.12 and Exercise 5.4.D”
6.1.1: “into category” -> “into a category”
6.1.2, line 3 of proof: delete “sheaves of”
6.3 title: “Quasicoherent sheaves” -> “Quasicoherent sheaves on X”
6.5.A: Delete “(by descending induction on i)”. “simple element” -> “simple object”.
6.5.E: “finite length module” -> “finite length object”
6.5.F: “a full subcategory” -> “an abelian subcategory”
6.5.I: “M_i/M_{i-1}” -> “X_i/X_{i-1}”. Equation display shortly after that: add “\times” before the last term.
6.5.M “length of a 𝓕…” -> “length of 𝓕”
6.5.9: delete last sentence.
6.6.A (line after it): “2.7.6” -> “2.7.8”
6.6.K: delete “finite generation or”
6.6.26, 2nd last line of proof: “Spec S^{-1} M” –> “Spec S^{-1} A”.
6.6.U “Exercise 6.6.U” -> “Part (d)”.
6.6.33 delete “(with M=A)”.
6.7 paragraph 3, sentence 3: delete “clearly”, and add at the end of the sentence “(Exercise 6.7.N)”.
7.6.K “(or indeed any category)” -> “(or indeed any category where the necessary products exist)”
8.1.A “Assume that X \times_S X’ and Y \times_S Y’ exist” -> “Assume all relevant fiber products exist”.
8.2.A delete “there is some t and m such that”
8.2.1 proof: four times (between the two equation displays, inclusive) “Id_{n \times n}” –> “Id_n”
(8.4.5.1) prepend “\pi^{-1}(Y_{\vec{e}}) \cap Z =”
10.3.1: (published version only) delete equation label “(10.1.1)”
10.5.R: This is subsumed by 10.5.O.
11.3.G: “scheme-theoretic image” -> “scheme-theoretic closure”
paragraph after 11.4.A: “To motivate the definition of properness for schemes, we remark that a continuous map \pi: X \rightarrow Y of locally compact Hausdorff spaces that have countable bases for their topologies is universally closed if and only if it is proper (i.e., preimages of compact subsets are compact). You are welcome to prove this as an exercise.” -> “To motivate the definition of properness for schemes, we remark that a continuous map of topological spaces with locally compact Hausdorff target is universally closed if and only if it is proper; see [Wedhorn’s “Manifolds, Sheaves, and Cohomology”, Problem 1.20].”
12.4.4 Proof sentence 2: $X = \cup_{i=1}^n U_i$ -> $X = \cup_{i=1}^N U_i$
13.5.13: “locally Noetherian scheme A” -> “locally Noetherian scheme X”
14.3.A(c) Change to: (“Hom is left-exact in the quasicoherent setting in an appropriate sense”, cf. Exercise 2.6.I) Show that Hom(F, .) gives a left-exact covariant functor QCoh_X -> QCoh_X and Coh_X -> Coh_X. (We can’t quite say that Hom(., G) gives a left-exact contravariant functor from finitely presented shears to QCoh_X with our definition of left-exactness, as finite presented sheaves on X do not necessarily form an abelian category, but the situation should be clear to you.)
16.2.L: “projective projective” -> “projective”
17.2.4 paragraph 2: the calligraphic G should be a calligraphic F.
21.4.B: “ramification order” -> “ramification index”
21.7.G: (published version only) delete equation label “(21.0.1)”
24.8.8: In both (iii) and (iv): “k-schemes” -> “schemes”, and before the period at the end of both sentences, add “(over the residue field of the point of Y)”
25.3.I: “(by $\pi$)” -> “(by $\pi: \proj^n_B \rightarrow B$)”
28.6.F: (published version only) delete equation label “(28.1.3)”
, but I did not know of an argument, and couldn’t think of an argument. I think Sándor Kovács told me an argument, but I managed to lose it. Burt Totaro gave me an argument which is very nice. I’ve decided not to include it in the book because I’m trying to hold to a hard line about adding length, but I want to make sure it is public, so I include it here. I will clean this up later; I may have mangled things a little bit in the transcription and translation (so blame me, not Totaro).
be a hypersurface in 
over a field 
. We can assume that ![k[x_1, ... ,x_{n+1}]](https://s0.wp.com/latex.php?latex=k%5Bx_1%2C+...+%2Cx_%7Bn%2B1%7D%5D&bg=ffffff&fg=29303b&s=0&c=20201002)
is a UFD) 
, say) there is a finite locally free morphism 
. (Indeed, in some local coordinates, you can rewrite the equation of X as a monic polynomial, 
.
is free as a module over ![R := k[x_1, ..., x_n]_p](https://s0.wp.com/latex.php?latex=R+%3A%3D+k%5Bx_1%2C+...%2C+x_n%5D_p&bg=ffffff&fg=29303b&s=0&c=20201002)
(for a given point 
in 
.)
be the integral closure of the domain 
. Then we have an inclusion 
of 
-modules, with 
has codimension at least 2 in 
.
of 
(closed subset of codimension at least 2) extends uniquely to an element of 
. We have a commutative square that I don’t know how to do in wordpress, where the top row is 
, where 
is injective since 
Math 216: Foundations of Algebraic Geometry 