px, yt, zb means page x, line y from top, line z from bottom. Incorrect statements of theorems, propositions, ... are marked with an asterisk.
pxiii. The only reason I assumed all the schemes are locally Noetherian was that I didn't want to keep saying finite presentation etc. for finite type. I can't believe it really matters anywhere. The fibre product schemes on p.70, line 2, and p.80 are not automatically Noetherian. If I remember correctly, Grothendieck's example of naturally arising nonnoetherian rings is the tensor product of two completions of ring. Products of Henselizations are probably only about as Noetherian as completions.
p6, 18b. It does not "follow easily". In fact, passing from the affine result to the global result is not easy.
p7, 1.12. Cf. Hartshorne, Algebraic Geometry, Ex. 3.7. (Hartshorne's book appeared just before I sent the final version of the manuscript to the publisher, in time only for me to change some of the references in the manuscript.)
p7, 12b. Atiyah-Macdonald [1, 2.19]
p9.* Correct statement of Proposition 2.5: Let $\alpha\:A\to B$ be a flat $A$-algebra, and consider $b\in B$. If the image of $b$ in $B/\alpha ^{-1}({\mathfrak{n}})$ is not a zero-divisor for any maximal ideal ${\mathfrak{n}}$ of $B$, then $B/(b)$ is a flat $A$-algebra. This necessitates some changes on p.10.
p15. In the proof of (e), an overline is missing from $f(Z)$.
p16.* In the statement of Theorem 2.16, one needs to assume that $f$ is dominant in order for the set to be nonempty (Suchandan Pal). For a proof of Theorem 2.16, see also Mumford [2, p.57] or (9.11) of my Commutative Algebra notes.
p31. In Exercise 3.27, the $Y$ should be $T$.
p40. Remark 5.1c should read: ...states that the functor that...
p42. Chinberg criticizes the statement of Abhyankar's theorem (first paragraph) as being too global to be true.
p47, 5t. Drop the condition: "closed under fiber products". It is not actually needed, and it fails for the small flat site: $U,V,W$ can all be flat over $X$ without $U\times_{V}W$ being flat over $X$ (the point is that $U\rightarrow V$ need not be flat).
p51, 11b. Should be (I, 2.19).
p57. The first line should read … and "$p=\pi^{.}$.", not "$p=\pi$:".
p67. In Exercise 2.19, it is not true in general that (b) $\Rightarrow$ (c). In fact, all the statements in (2.19) should be treated with caution. Lorenzini has pointed out to me that, contrary to what is asserted, (c$^{\prime}$) does imply (b$^{\prime}$), and that this can be proved as follows. Let $\phi\colon G\rightarrow G^{\prime\prime}$ be a morphism of group schemes (not necessarily commutative) flat and locally of finite presentation over a scheme $S$, and assume that $\phi$ is surjective as a map of sheaves for the flat (fppf) topology on $S$. We have to show that the group-scheme kernel $G^{\prime}$ of $\phi$ is flat over $S$. The surjectivity means that, for any $s^{\prime}\in G^{\prime\prime}(U)$ with $U$ flat and locally of finite presentation over $S$, there exists a $W$ faithfully flat and locally of finite presentation over $U$ and an $s\in G(W) $ mapping to $s^{\prime}|W$. When we apply this remark to the identity morphism $G^{\prime\prime}\rightarrow G^{\prime\prime}$, we see that there is a faithfully flat morphism $f\colon W\rightarrow G^{\prime\prime}$ locally of finite presentation and a morphism $s\colon W\rightarrow G$ such that $\phi\circ s=f$. There is an isomorphism of $W$-schemes \[ (g,w)\mapsto(g\cdot s(w)^{-1},w)\colon G\times_{G^{\prime\prime}}W\rightarrow G^{\prime}\times_{S}W. \] Because $G\times_{G^{\prime\prime}}W$ is flat over $G$ and $G$ is flat over $S$, $G\times_{G^{\prime\prime}}W$ is flat over $S$. Thus $G^{\prime}\times_{S}W$ is flat over $S$.\noindent\ Because $G^{\prime}% \times_{S}W\rightarrow G^{\prime}$ is faithfully flat, this implies that $G^{\prime}\rightarrow S$ is flat.
Holger Deppe has pointed out to me that (a$^{\prime}$)$\implies$(c$^{\prime}$) doesn't hold either, i.e., a flat homomorphism $f\colon G\rightarrow G^{\prime\prime}$ commutative flat group schemes locally of finite type over $X$ might not give a surjective map of sheaves on $X_{\mathrm{fl}}$. Consider, for example, the constant group scheme $G$ defined by a finite commutative group $A\neq1$ and let $f\colon G\rightarrow G$ be the zero map; this is flat (even a local isomorphism) but it is not surjective as a map of sheaves. I don't know whether the implication becomes correct when the group schemes are connected.
p77, 3.17. Replace $\pi_{\ast}\colon\mathbf{S}(T_{\mathrm{et}% })\overset{\approx}{\longrightarrow}\mathbf{S}(X_{\mathrm{et}})$ with $\pi_{\ast}\colon\mathbf{S}(Y_{\mathrm{et}})\overset{\approx}{\longrightarrow }\mathbf{S}(X_{\mathrm{et}})$ (Timo Keller).
p90, 7t. In "As any sheaf on $(\text{spec} k)_{\mathrm{et}% }$ is flabby", replace $k$ with $\bar{k}$ (Tiago J. Fonseca).
p93. In the proof of (1.27), we need to show that the two restrictions of $\gamma$ to $X^{\prime}\times_{X}X^{\prime}$ are equal, but this can be seen on the stalks.
p97. Matthieu Romagny points out to me that "universal monomorphism" is redundant: in a category with fibre products, monomorphisms are automatically universal monomorphisms.
p98. Proposition 2.3, $H^{0}(U,-)$ should be $\check{H}^{0}(U,-)$ (Changlong Zhong).
p112. I define a site to be noetherian if every covering has a finite subcovering, and then say "It is not clear to the author why this is called a Noetherian site rather than a compact site". As was pointed out to me by Thomas Wyler, the reason is the following: a topological space is said to be noetherian if it has the ascending chain condition on open sets, and one sees easily that a topological space is noetherian if and only if every open subset is quasi-compact (i.e., every open covering of an open subset has a finite subcovering).
p121. In Theorem 4.3 (b) one needs an additional assumption that G is of finite presentation (i.e., under the omnipresent locally Noetherian assumption, one may need to assume in addition that G is of finite type). This assumption is present in the cited reference Raynaud [2, XI.3.1 1)]. Also, presumably, in the proof of the same theorem, the reference for part (e) to Raynaud [2, XII.2.3] should be replaced by Raynaud [2, XIII.2.3 (iii)]. (Kestutis Cesnavicius).
p125. For clarity, replace $H^1(X,\ldots$ with $H^1(X_{fl},\ldots$ on the fifth row.
p134. "$\underline{\mathrm{Aut}}(Y)$ is the sheaf associated with the presheaf …" The presheaf is actually a sheaf --- there is no need to take the associated sheaf.
p139, 1t. The bar over the $R$ has been omitted.
p143. The last term in the display in Step 3 has a spurious check on the $X$ (Timo Keller).
p151. From Tate April 12, 1992: "It was not at all clear to me, especially for schemes as opposed to rings, that a maximal order is locally maximal. And I think the comment on line -7 (after possibly replacing …) is misleading, since (I hope) if $M_{n}(A)$ is Azumaya, then $A$ is also, and so is maximal."
p153. From Tate: The statement "if an Azumaya algebra..." should refer to classes in the Brauer group. For example, there exists a vector bundle $V$ on ${\mathbb{A}}^{3}-\{(0,0,0)\}$ that doesn't extend, and $\text{End}(V)$ is then an Azumaya algebra on the same variety that doesn't extend.
p153. G. Ofer should be Ofer Gabber.
p155. The heading should have a semicolon, not a colon.
p174. See Gamst and Hoechsmann, Tohoku 1968, for more on pairings.
p190, 4b. Drop the subscript on $\mathbb{Q}$.
p197. Treating only surfaces makes the theory more concrete, but the theory of vanishing cycles is not essentially more difficult in higher dimensions---probably I should have included the general case.
p202. In the diagram at the bottom of the page, the curve $X_{s}^{\prime}$ should not cross itself (think of the picture in $3$-dimensions).
p221, 10b. Should read: $X^{\prime}=X\times_{\text{spec}(k)}\mathbb{A}_{k}^{d}$...
p224.* Corollary 2.8 is (obviously) wrong. For example, if $X$ is $Spec(\mathbb{Q})$ and $k=\mathbb{Q}$, then $H^1(X,\mathbb{Z}/(2))$ is infinite.
p227. Higher direct images with proper support would be better than ... compact support (Ofer Gabber).
p229. Remark 3.3e. For a modern proof of Nagata's theorem, see Lutkebohmert, W., On compactification of schemes, Manuscripta Math. 80 (1993), 95-111.
p244.* Corollary 5.5 is incorrect --- see Corollary 2.8 above.
p253. In the statement of Theorem 7.1, $i:Z\rightarrow X$ is the inclusion of a hypersurface section.
p253. Should add the Lefschetz hypersurface-section theorem: Let $X$ be a projective variety of dimension $d$ over a separably closed field $k$, and let $Z$ be a hypersurface section of $X$ such that $U=X\smallsetminus Z$ is smooth; for any locally constant sheaf $F$ on $X$ with finite stalks whose torsion is prime to the characteristic of $k$, \[ H^{i}(X,F)\rightarrow H^{i}(Z,F) \] is injective for $i=d-1$ and an isomorphism for $i\leq d-2$. To prove this, use the exact sequence \[ \cdots\rightarrow H_{c}^{i}(U,F)\rightarrow H^{i}(X,F)\rightarrow H^{i}(Z,F)\rightarrow\cdots \] (III, 1.30) noting that $H_{c}^{i}(U,F)$ is dual to $H^{2d-i}(U,F^{\vee}(d))$ (VI, 11.2), which is zero for $2d-i>d$, i.e., $d>i$ (VI, 7.1).
p253, 1b. In fact, it's only proved prime to the characteristic.
p256, 9t. $\Gamma(X_{\alpha},\mathcal{O}_{X_{\alpha}})$ not $\Gamma(X_{\alpha},\mathcal{O}_{X\alpha})$.
p256. The reader is invited to rewrite Section 8 using derived categories.
p264. In Lemma 8.17, assume that $\phi$ is a quasi-isomorphism.
p268. At the top of the page, assume that the torsion in $\Lambda$ is prime to the characteristic of $k$.
p268, 15b. Alas, $\sum H^{r}(X,\Lambda([\frac r2]))$ does not become a ring under cup-product. To get a ring, take $\sum H^{2r}(X,\Lambda([ r]))$.
p273. In A4, replace $y$ and $Y$ with $z$ and $Z$.
p274, 13t. Replace $c$ with $c_{t}$.
p275. The proof of Proposition 10.6 is incorrect, because the map at the bottom of the page is not invertible. A correct proof is rather long.
p276, 9t. Replace $\mathbb{Q}_{t}$ with $\mathbb{Q}_{\ell}$.
p292. Add $\frac{t^{n}}{n}$ to the end of the top formula.
p307. In the 4th line, it should say $R^{p}g(f(I))$ rather than $R^{p}g(I)$ (Lucio Guerberoff).
p311, 13t. (f) not $(f)$.
p322. Godement, not Grodement.
Chapter VI. Should add the Lefschetz hypersurface-section theorem in the following form: for any hypersurface section $Z$ of a projective variety $X$ such that $U=X\smallsetminus Z$ is smooth, $H_{l}^{i}(X)\rightarrow H_{l}^{i}(Z)$ is injective if $i=\dim X-1$ and an isomorphism if $i<\dim X-1$. To prove this, use the exact sequence \[ \cdots\rightarrow H_{c}^{i}(U,\mathbb{Q}{}_{l})\rightarrow H^{i}% (X,\mathbb{Q}{}_{l})\rightarrow H^{i}(Z,\mathbb{Q}{}_{l})\rightarrow\cdots \] (III 1.30) noting that $H_{c}^{i}(U,\mathbb{Q}{}_{l})$ is dual to $H^{2d-i}(U,\mathbb{Q}{}_{l}(d))$, $d=\dim X$, (VI, 11.2), which is zero for $2d-i>d$, i.e., $d>i$ (VI 7.1).
dust jacket On the back of the first printing, replace 1971 with 1980.

From Yang Cao, December 14, 2011.
p170. At the end of your proof of Proposition 1.13, you say that "using the fact that $\pi_{!}F^{\prime}\rightarrow\bar{\pi }_{\ast}J^{\bullet}$ is an injective resolution of $\pi_{!}F^{\prime}$, we get the required diagram." But this can not prove that $j_{!}\pi^{\ast}F\rightarrow j_{!}\pi^{\ast}I^{\bullet}[r+s]$ is an injective resolution of $j_{!}\pi^{\ast}F$. So it can not prove that the homotopy class of $\text{Hom}_{\bar{X}^{\prime}}(J^{\bullet}[r],j_{!}\pi^{\ast}I^{\bullet}[r+s])$ is equal to $\text{Ext}_{X^{\prime}}^{s}(F^{\prime},\pi^{\ast}F)=\text{Ext}_{X^{\prime}% }^{s}(j_{!}F^{\prime},j_{!}\pi^{\ast}F)$, and the homotopy class of $\text{Hom}_{\bar{X}^{\prime}}(\mathbb{Z}{},j_{!}\pi^{\ast}I^{\bullet}[r+s])$ is equal to $H^{r+s}(\bar{X}^{\prime},j_{!}\pi^{\ast}F)$.

p242. Line 7 from the top. I think \[ \underline{\text{Ext}}_{X}^{i}(j_{!}j^{\ast}\mathbb{Z}{},F)\approx j_{\ast }\underline{\text{Ext}}_{U}^{i}(j^{\ast}\mathbb{Z}{},j^{\ast}F)\approx R^{i}j_{\ast }(j^{\ast}F) \] is wrong. It should be a spectral sequence \[ R^{s}j_{\ast}\underline{\text{Ext}}_{U}^{i}(j^{\ast}\mathbb{Z}{},j^{\ast}% F)\implies\underline{\text{Ext}}_{X}^{i+s}(j_{!}j^{\ast}\mathbb{Z}{},F), \] and $\underline{\text{Ext}}_{U}^{i}(j^{\ast}\mathbb{Z}{},j^{\ast}F)$ is $j^{\ast}F$ when $i=0$ and $0$ when $i>0$. So we get \[ \underline{\text{Ext}}_{X}^{i}(j_{!}j^{\ast}\mathbb{Z}{},F)\approx R^{i}j_{\ast }(j^{\ast}F). \]

Some typos

From Bhupendra Nath Tiwari
p16. (Proposition 2.18) ... is exact where $B^{\otimes r} = B \otimes_{A} B\otimes_{A}\ldots\otimes_{A} B$
p177. In the proof of Step 1, ... where $\mathrm{tr}\colon\text{Pic}(X^{\prime})\rightarrow\text{Pic}(X)$ is ...
p232. 10b ... is bijective for $0\leq i\leq n$, not $0\leq1\leq n$.
p233. Sublemma 4.7 ... and $(h_{i}^{\prime}\colon X_{i}^{\prime} \rightarrow X^{\prime})$ is a covering of ..., not $(h_i\ldots$.
p250. Proposition 6.5 (b) Let $Z\overset{i_{1}}{\hookrightarrow}% Y\overset{i_{2}}{\hookrightarrow}X$ be a smooth $S$-triple; (the arrows are missing).
p262. Remark 8.12 $\ldots\mathbf{R} h_{\star} ( F \boxtimes G) = C^{\bullet}(F) \boxtimes C^{\bullet}(G)$ should be \[ \mathbf{R} h_{\star} ( F \boxtimes G) = h_{\star}(C^{\bullet}(F) \boxtimes C^{\bullet}(G)) \] (seems so by the symmetry of the canonical pairing ... (you gave, just below it))
p268. The first display should be $\ldots F \boxtimes^{\mathbf{L}} G)$.
p290. Lemma 13.2, replace "identify map" with "identity map".