Starred items have been corrected in the 2022 reprint (both paperback and hardback).

DG=Demazure and Gabriel 1970; CGP=Conrad, Gabber, and Prasad 2015.

Contributors include: Jarod Alper; Michel Brion; Magnus Carlson; Dylon Chow; Brian Conrad; Rostislav Devyatov; Lian Duan; Ofer Gabber; Cédric Pépin; Bjorn Poonen; Matthieu Romagny; Zev Rosengarten; Thierry Stulemeijer; Vladimir Sotirov; Yugo Takanashi; Christian Voigt; Qijun Yan.

Erratum

Chapter 1.

The Notation 3.2 should be placed earlier in the book. For example, it is implicitly used in the proof of 1.49.
*p.18. Proof of 1.40. Multiplication on $G(k)$ is continuous for the Zariski topology, but not necessarily for the product topology, and so we don't know that there exist $A$ and $B$ as claimed. A correct proof is not difficult.
*p.20. Corrected statement of 1.45: Let $G$ be an algebraic group over $k$ and $S$ a closed subgroup of $G(k)$. There is a unique reduced algebraic subgroup $H$ of $G$ such that $|H|$ is the closure $\bar{S}$ of $S$ in $|G|$; it is geometrically reduced and $H(k)=S$. The algebraic subgroups $H$ of $G$ that arise in this way are exactly those for which $H(k)$ is schematically dense in $H$.
*p.21 Definition 1.48. Let $G$ be an algebraic group over $k$ and $S$ a subgroup of $G(k)$. The unique algebraic subgroup $H$ of $G$ such $S$ is schematically dense in $H$ is called the Zariski closure of $S$ in $G$.
*p.26. Proposition 1.65. In the proof of (a), one should choose $U$ before passing to the algebraic closure.

Chapter 2.

*p.48 The group $G$ has height $\leq n$ if $Ker(F_G^n)=G$, i.e., $F_G^n$ is trivial (rather than 0).
*p.51. Replace $X$ with $G$ thrice in the first paragraph of Section 2g.
*p.60. Section 2j: Define torsors over nonaffine bases, because such torsors occur later in the book.

Chapter 3.

*p.80, 3.52. Two inner forms $(G,f)$ and $(G',f')$ are said to be equivalent if there exists an isomorphism $\varphi\colon G\to G'$ such that $f'=\varphi_K\circ f$ up to an inner automorphism of $G_{0K}$, i.e., such that $f'^{-1}\circ\varphi_K\circ f$ is inner.
*p.85. $G_W$ is the stabilizer of $W$ in $G$ (not $V$).

Chapter 4.

*p.95. In the proof of 4.29, $V^{\otimes m}\otimes W$ should be $V^{\otimes m}\otimes V'$ (twice).

Chapter 5.

*p.123. Delete Coker$(g')$ from the diagram and sequence in Exercise 5-7 (the homomorphism $g'$ need not be normal).

Chapter 6.

*p.136. In the example 6.48, take $p=2$, otherwise the multiplication doesn't preserve the defining relation. Also, the kernel $N$ of $\varphi$ is $X^p-tY^p=1$ (not 0).

Chapter 8.

*p.150. The proof of Proposition 8.9 requires X to be reduced. Corollary 8.10 is correct, but the proof requires an addition step. This suffices for the applications.
*p.157. The line before Proposition 8.37 should read: central extension if $G$ is connected.

Chapter 9.

*p.173, 9.25. $\lambda_X$ should lie in $\text{End}(\omega(X))...$, not $\text{End}(X)...$
*p.181. In the proof of 9.44 requires correction.

Chapter 11.

*p.223. Proposition 11.36 should read $\ldots \alpha=\text{Lie}(\varphi)$ not $\ldots \alpha=\text{Lie}(\varphi)\circ\text{Lie}(\varphi)$.
*p.224. Throughout the section on the Verschiebung morphism, $G$ is commutative (as in the first paragraph). In definition 11.39 the second arrow is reversed.

Chapter 12.

*p.244. In 12.41, $N$ should be a central subgroup of $H$. This suffices for the applications.
*p.249. The proof of 12.57 is missing a step: first show that the Reynolds operator is $A^G$-linear. Also $a_i$ should be $r_i$.

Chapter 13.

*p.255. In the proof of 13.4, replace $=\subset$ with $\simeq$.
*p.256. In the second line of the proof of 13.5, replace $\supset$ with $\subset$.
*p.262. In (86), set $Z=X$.

Chapter 14.

*p.273. In Theorem 13.47(c) (and corollaries), it is necessary to require $X$ to be proper, otherwise there may be no fixed points.

Chapter 15.

*p.317. Delete 15.29 and 15.30. The proofs in Section 15h have been rewritten.

Chapter 16.

*p.324. In the last sentence of first paragraph, replace "diagonalizable" with "trigonalizable".
*p.330. Near the end of 16.20, the field $k'$ is $k[c^{1/p}]$.
*p.349. In 16.65, the subgroup $H$ should be connected (obviously).

Chapter 17.

*p.360. The statement before 17.28 should say that $\text{SL}_2$ (not $\text{SL}_3$) has dimension $3$ (obviously).
*p.364. In 17.42, take $I=\{1,2,\ldots,n\}$ (or rewrite the last sentence).
*p.386. Delete Exercise 17-7. It is true that Steinberg 1968, 7.2, proves that every surjective endomorphism of an algebraic group over an algebraically closed field fixes a Borel subgroup, but you should read Steinberg's proof rather than trying to prove it yourself.

Chapter 18.

*p.388 and p.396 In the last sentence of 18.4 and in 18-1, $G$ should be reductive.
*p.389. In the statement of 18.8, the target of $\alpha$ is $G'$, not $G$. This is correct in the diagram.

Chapter 19.

*p.399 Definition 19.7 should read: An algebraic group $G$ over $k$ is almost-simple if it is semisimple and non-commutative and its only smooth connected normal subgroups are $G$ and $e$. It is simple if in addition its centre is trivial. Similarly change in 19.8.

Chapter 20.

*p.417. In the proof of 20.27, interchange second and third.
*p.419. In the statement of 20.33, add "$G$ of semisimple rank 1" after "reductive group". In the final sentence of the proof, (first,second,third)$\to$(second,third,first).
*p.420. The reference to "Section 2k" should be to "Section 3k".

Chapter 21.

*p.440 The proof of Theorem 21.51 has been rewritten.
*p.445. The necessity in Theorem 21.68(d) fails in characteristics $2,3$ (and is not "obvious" in characteristics $\neq 2,3$).
*p.447. The proof of Proposition 21.71 only shows that $B_ue_w\subset Y(w)$. To prove equality, show as in the proof of 21.75 that (T3) holds, and deduce that $G=\bigcup_w B_uwB$.
*p.449. In the proof of Proposition 21.11, replace the appeal to 21.67 with a direct argument.
*p.456--60. In several places in Example ($A_n$), I write $n$ instead of $n+1$. In 21.96, replace $\text{SL}_n$ with $\text{SL}_{n+1}$ and $\mathbb{Z}/n\mathbb{Z}$ with $\mathbb{Z}/(n+1)\mathbb{Z}$. Set braces are missing in the lines displaying $\Phi$ and $\Delta$ in 21.97, 21.98, 21.99, and an equality symbol is missing for $\Delta$ in 21.98. In Example $C_n$ there is an errant $\text{SO}_{2n+1}$. In Example ($D_n$), $\text{SO}_n$ should be $\text{SO}_{2n}$ and, two lines later, $\text{SO}_{2n+1}$ should be $\text{SO}_{2n}$.
*p.460 middle, definition of $\phi$, the last $y_{2n}$ should be $y_n$.
*p.460 last paragraph, $2n+1\times 2n+1$ should be $2n\times 2n$.
*p.461. In Exercise 21-2 one needs to assume that the unipotent group is smooth, otherwise there are the infamous unipotent isogenies in characteristic 2, as in 18.3 and as discussed in Section 2 of Prasad and Yu 2006.

Chapter 22.

*p.468. Redefined $E(\lambda)$ to be induced from the opposite Borel subgroup $B^{\prime}$ so as to agree with the literature. Now it is nonzero if and only if $\lambda$ is dominant etc.
*p.469. Expanded the proof of the crucial Lemma 22.24.
*p.473. Example 22.35 mysteriously repeats example 22.34 instead of giving the fundamental weights of $\text{PGL}_n$.
*p.480. In Theorem 22.52, either assume $k$ has characteristic zero or replace $V$ with $H^0(G/B',L(\chi))$.

Chapter 23.

*p.488. End of second sentence of last paragraph of the proof of 23.11, change $U_{\alpha_2}$ to $U_{\alpha_2}(k)$.
*p.494. In the statement of Corollary, 23.32, change "an algebraic group" to "a reductive algebraic group" (so $(G',T)$ is a split reductive group, and having a root datum makes sense).
*p.497 23.42. Any inner automorphism of a pinned reductive group is trivial.
*p.498.The statement of Corollary 23.48 is incorrect. When $G$ is split, the map $H^1(k,\text{Inn}(G))\to H^1(k,\text{Aut}(G))$ is injective as a map of pointed sets, but not necessarily as a map of sets. See p.523 below. This is not used, but it means that it is necessary to distinguish the two notions of inner form, even for split groups.
*p.498. In the proof of Corollary 23.47, the first $G$ should be Aut$(G)$.
*p.498. Corollary 23.48 should only say that an inner form $(G,f)$ is trivial if and only if the inner form $G$ is trivial.
p.499. In 23.50, $\text{Hom}(G,H)$ is not affine unless $\text{Out}(G)$ is finite (because $(\mathbb{Z})_k$ is not affine).
*p.500 In 23.52, $G$ is a reductive group.
*p.500. Here is a more precise statement of Corollary 23.54: Let $G$ be a reductive group over $k$. There exists an inner form $(H,f)$ of $G$ such that $H$ is quasi-split, and any two such inner forms are equivalent. In particular, the class of $(H,f)$ in $H^{1}(k,G^{\text{ad}})$ is uniquely determined.
*p.501. In the statement of Corollary 23.56, the objects of the second category are reduced root data.
*p.504. Line 4, should read $[X_i,Y_i]=H_i\ldots$, not $[X_i,X_i]\ldots$.
*p.504. In the last displayed equation of the "First proof", $J_j$ should be $Y_j$.
*p.504. Proposition 23.64 has been deleted and a remark added.
*p.505. The last displayed equation before Lemma 23.65 should read $h_i(v_{\alpha})=\langle \alpha,\alpha_i^{\vee}\rangle v_{\alpha}$ (subscript $i$ is missing).
*p.506. In the condition (c), replace $\beta_R$ with $\beta$. In the next line, replace $\mathfrak{g}_R$ with $\mathfrak{g}_\mathcal{R}$.
*p.506. In the proof of Lemma 23.68, p.475 should be p.476.
*p.507. In Theorem 23.70, (d) and (e), $G$ should be $G(\mathfrak{g})$.

Chapter 24.

*p.513 In line 3, replace "isogeny" with "central isogeny". In 24.3, "the field of definition of $G_{i}$ as a subgroup of $G$" is the fixed field of the subgroup of $\mathrm{Gal}(k^s/k)$ fixing $G_{i}$.
*p.515, Caution: the second exact sequence in the displayed pair is only exact as a sequence of pointed sets. See p.498 above.
*p.516. Line 2: replace "subset" with "image of".
*p.516. Line 4: Homs modulo conjugation in the case $D_4$.
*p.518. Simple algebras over $k$ are required to be nonzero.
*p.518. In fact, 24.20 is not proved in Jacobson 1989, Section 4.6, with the "separable" condition.
*p.524. Throughout this section, $n>2$.
*p.531. At the end of the first paragraph of Section 24i, replace $C(V,q)$ with $C_0(V,q)$.
*p.540. Near the bottom, $F_2$ should be $F_4$ --- there is no $F_2$.

Chapter 25.

*p.545 et seq. In 25.6(b), in the final sentence of 25.24, and in 25.27, the parabolic subgroup ($P$ or $Q$) should be minimal, and in 25.27, $S$ should be maximal split in $P$.
*p.546. The proof of 25.7 is only a brief sketch. The "nontrivial" on the first line of the proof of 25.10 should be "noncentral".
p.549. The statement in 25.16, that the relative root system is a root system, is certainly correct but (as explained to me by Brian Conrad) the proofs of this in the classical literature (Borel and Tits 1965, Borel 1991, and Springer 1998) are incomplete. For a complete proof, see CGP, C.2.15.
*p.549. In line 1 of 25.19, add "torus" to "maximal split".
*p.563. In 25.66 and 25.68, the direct products should be direct sums. In 25.67 and 25.69, the $G^{(v)}$ should be required to be quasi-split for all but finitely many $v$.

Appendix A.

*p.570. The proof in footnote 3 requires correction.
*p.582. In A.70, delete the word "faithfully".