Errata/addenda for Algebraic Groups
Please send corrections to me for a possible corrected reprinting.

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

Erratum

p.19 Corrected Theorem 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 $H$.
Proof: Let $H$ be the (unique) reduced closed subscheme of $G$ such that $|H|=\bar{S}$. Then $S=G(k)\cap\bar{S}=G(k)\cap|H|=H(k)$. As $H(k)$ is dense in $|H|$ and $H$ is reduced, $H(k)$ is schematically dense in $H$ (see 1.10), $H(k)$ is dense in $H(k^{\mathrm{a}})$ (see 1.11), and $H$ is geometrically reduced (1.12). Hence $H(k^{\mathrm{a}})$ is a subgroup of $G(k^{\mathrm{a}})$ (by 1.40) and $H\times H$ is reduced (A43), which implies that $m_{G}\colon H\times H\rightarrow G$ factors through $H$. Similarly, $\text{inv}_{G}$ restricts to a morphism $H\rightarrow H$ and $\ast\rightarrow G$ factors through $H$. Therefore $H$ is an algebraic subgroup of $G$ such that $H(k)$ is schematically dense in $H$.
Conversely, let $H$ be an algebraic subgroup of $G$. Then $H(k)=G(k)\cap|H|$ and so $H(k)$ is closed in $G(k)$. If $H(k)$ is schematically dense in $H$, then the above construction starting with $S=H(k)$ gives back $H$.
p.20 Corrected Corollary 1.46: Let $G$ be an algebraic group over $k$ and $S$ a subgroup of $G(k)$. There is a unique algebraic subgroup $H$ of $G$ such that $S$ is schematically dense in $H$.
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 $G_K$, i.e., such that $f'^{-1}\circ\varphi_K\circ f$ is inner.
p.123. Delete Coker$(g')$ from the diagram in Exercise 5-6 (the homomorphism need not be normal).
p.136. In the example 6.48, take $p=2$, otherwise the multiplication doesn't preserve the defining relation. See 25.38 for more general examples.
p.150.The proof of Proposition 8.9 only works if X is reduced. This is implicitly used at the end of the argument "$X^H$ contains an open neigbourhood of $P$ and is closed in $X$. Since X is irreducible, $X^H = X$." I do not know how to prove this proposition as stated but with an additional argument, one gets Corollary 8.10: see SGA3, Expose VI B, Corollaire 6.2.6. This should suffice for most applications in the book. (M. Brion)
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.
p.273 In Theorem 13.47(c), it is necessary to require $X$ to be proper, otherwise $\mathbb{G}_m$ acting on itself by translation provides a counterexample.
p.399 Definition 19.7: A simple algebraic group should be defined to be an almost-simple group with trivial centre. [Need to rethink the definitions 19.7 and 19.8.]
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.461. Exercise 21-2 needs an additional hypothesis to exclude the examples in 18.5.
p.498. Replace $(X,\Phi,\Delta)$ with $(X,\Phi,\Phi^{\vee},\Delta)$ twice. 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.531. In 24.58 and 24.59, assume $(V,q)$ is regular. In 24.58(b), exclude the case $n=1,\,q=0$. At the end of the first paragraph of Section 24i, replace $C(V,q)$ with $C_0(V,q)$.
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.547. In 25.12, the field $k$ has characteristic zero.
p.582. In A.70, delete the word "faithfully".

Bjorn Poonen has posted Comments on the book, which themselves contains errors, for example, his items 1, 2, 3, 6, 8 (Feb 18 version) are incorrect.

Contributors include: Jarod Alper; Dylon Chow; Brian Conrad; Ofer Gabber, Bjorn Poonen, Zev Rosengarten, Thierry Stulemeijer, Christian Voigt, Qijun Yan.

Misprints, minor errors, complements, …

p.3. Readers concerned that I ignore nonclosed points of algebraic schemes should see 1.19 of my notes Reductive Groups (RG).
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. Here's a correct proof of 1.40 (from the draft version): For $a\in G(k)$, the map $x\mapsto ax\colon G(k)\rightarrow G(k)$ is a homeomorphism because its inverse is of the same form. For $a\in S$, we have $aS\subset S\subset\bar{S}$, and so $a\bar{S}=(aS)^{-}\subset\bar{S}$. Thus, for $a\in\bar{S}$, we have $Sa\subset\bar{S}$, and so $\bar{S}a=(Sa)^{-}% \subset\bar{S}$. Hence $\bar{S}\bar{S}\subset\bar{S}$. The map $x\mapsto x^{-1}\colon G(k)\rightarrow G(k)$ is a homeomorphism, and so $(\bar{S}% )^{-1}=(S^{-1})^{-}=\bar{S}$.
p.26. Proposition 1.65. In the proof of (a), one should choose $U$ before passing to the algebraic closure.
p.51. Replace $X$ with $G$ twice in the first paragraph of Section 2g.
p.85. $G_W$ is the stabilizer of $W$ in $G$ (not $V$).
p.95. In the proof of 4.29, $V^{\otimes m}\otimes W$ should be $V^{\otimes m}\otimes V'$.
p.98. Chapter 5. Instead of the ad hoc arguments in this chapter, one can make use of the following statement. In a category $C$ satisfying the following four axioms, the two Noether isomorphism theorems and the Zassenhaus (butterfly) lemma hold (hence also the Jordan-Hölder and Schreier refinement theorems):
Z0. $C$ has zero objects.
Z1. $C$ has inverse images and cokernels of kernels.
Z2. If a morphism in $C$ can be written as the composite $e\circ m$ of a kernel $m$ and a cokernel $e$, then it can be written $m'\circ e'$ as the composite of a cokernel $e'$ and a kernel $m'$.
Z3. If a morphism in $C$ can be written as the composite of a monomorphism and a cokernel, then it can be written as the composite of a cokernel and a monomorphism.
See Wyler, The Zassenhaus lemma for categories, Archiv der Math. 1971.
p.129, 6.19. In (c,d,e), $G$ should be assumed to be smooth (to be safe). See DG, II, Sect. 5, 4.8, p.247, for a proof of (c) in the smooth case.
p.134, 6.32 We show that the $H$ in the proof (with greatest dimension) is maximal. Suppose that $H'$ is a smooth connected subgroup of $G$ containing $H$ as a normal subgroup and having the same dimension as $H$. Then $H'/H$ is smooth connected and finite, hence étale and connected, hence trivial (2.17).
p.166, footnote. It would be better to define the eigenvalues of an endomorphism to be the multiset (rather than family) of roots...
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, the pair $(\mathrm{Comod(C)}\times \mathrm{Comod(C)},\omega\otimes\omega)$ doesn't satisfy the conditions of 9.37, but the equality $\varinjlim\ldots$ does hold. Now a functor $\phi\colon \mathrm{Comod}(C)\times\mathrm{Comod}(C)\rightarrow\mathrm{Comod}(C)$ such that $\phi(V,W)=V\otimes_{k}W$ as a $k$-vector space defines a homomorphism $\mathrm{End}(\omega)\rightarrow\mathrm{End}(\omega\otimes\omega)$, and hence a homomorphism $C\otimes_k C\rightarrow C$. This gives an inverse to $u\mapsto \phi^{u}$.
p.223. Proposition 11.36 should read $\ldots \alpha=\text{Lie}(\varphi)$ not $\ldots \alpha=\text{Lie}(\varphi)\circ\text{Lie}(\varphi)$.
p.300, 14.74. Here's a proof that a noncommutative wound unipotent group $U$ over $k$ cannot be a normal subgroup of a split unipotent group. We may suppose that $k$ is separably closed because separable extensions preserve woundness. Let $W$ be a split unipotent group containing $U$ as a normal subgroup. For all $u\in U(k)$, the map $w\mapsto wuw^{-1}\colon W\rightarrow U$ of schemes is constant because $W$ is isomorphic to $\mathbb{A}^{\dim(W)}$ (14.66), and so $U(k)$ is central in $W(k)$. This implies that $U$ is commutative (by smoothness), contradicting the hypothesis (Gabber, via Zev Rosengarten).
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).
p.352 M. Brion has pointed out to me that, under the assumptions of Theorem 17.1, it is possible to prove a much stronger statement, namely, that $G/H$ is affine. For this, we may assume $k$ algebraically closed. Choose an equivariant immersion of $G/H$ in $\mathbb{P}(V)$ for some $G$-module $V$ and consider the closure $X$ of $G/H$, with boundary $Y$. Let $R$ be the homogeneous coordinate ring of $X$, and $I$ the ideal of $Y$ in $R$. Applying the Lie-Kolchin theorem 16.30 to the homogeneous components of $I$ yields a homogeneous $f$ in $I$ which is an eigenvector of $G$. Then the set of zeroes of $f$ is $Y$, and hence $G/H = X - Y$ is affine.
p.360. The statement before 17.28 should say that $\text{SL}_2$, not $\text{SL}_3$, has dimension $3$.
p.364. In 17.42, take $I=\{1,2,\ldots,n\}$ (or rewrite the last sentence).
p.389. In the statement of 18.8, the target of $\alpha$ is $G'$, not $G$. This is correct in the diagram.
p.417. In the proof of 20.27, interchange second and third.
p.420. The reference to "Section 2k" should be to "Section 3k".
p.441. Here are some additional details for the proof of 21.51. First, a smooth connected normal subgroup $N$ of a semisimple group $G$ is semisimple. In proving this, we may suppose that $k$ is algebraically closed. Then it suffices to show that the radical $R$ of $N$ is normal in $G$, and for this it suffices to show that $R$ is stable under $\text{inn}(g)$ for all $g\in G(k)$ (see 1.85), but this is obvious. In particular, the $G_i$ in the proof of 22.51 are semisimple, and hence almost-simple.
The rest of the proof is valid when $k$ is algebraically closed. When $k$ is arbitrary, replace the last paragraph with: It remains to show that $H=G$. For this it suffices to show that, if not, then the centralizer of $H$ in $G$ contains a connected subgroup variety of dimension $\geq 1$. This follows from the case $k=k^{a}$.
p.442. The note 21.57 repeats the definition of "almost pseudo-simple group" from 19.8, which, as noted in the text follows Tits rather than CGP. To conform to CGP, replace "proper normal" with "smooth connected proper normal".
p.443.In the proof of Corollary 21.63, "generated" should be "generate".
p.456. In the first paragraph, "determines the group up to isogeny (23.9 below)" is correct, but "determines the group up to a central isogeny (23.62 below)" would be better.
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.473. Example 22.35 mysteriously repeats example 22.34 instead of giving the fundamental weights of $\text{PGL}_n$. It needs to be rewritten.
p.480. Just after Remark 22.49, replace 18.23 with 18.24.
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.496. It is worth noting that 23.37(b) forces the isogeny $\varphi$ in 23.40 to be central.
p.498. In the proof of Corollary 23.47, the first $G$ should be Aut$(G)$.
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. 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.
To deduce this from the preceding results, it remains to show that, if $H$ is quasi-split, then the map $\text{Aut}_{k^{s}}(H)^{\Gamma}% \rightarrow\text{Out}_{k^{s}}(H)^{\Gamma}$ is surjective. We may suppose that $H$ is obtained by twisting a split group $G$ by a one-cocycle $a$ of $P=\text{Aut}_{k^{s}}(G,T,\Delta,(e_{\sigma}))$. The group $P$ acts on the terms of the sequence \[ 1\rightarrow\text{Inn}_{k^{s}}(G)\rightarrow\text{Aut}_{k^{s} }(G)\rightarrow\text{Out}_{k^{s}}(G)\rightarrow1 \] by conjugation. When twisted by $a$ this sequence becomes \[ 1\rightarrow\text{Inn}_{k^{s}}(H)\rightarrow\text{Aut}_{k^{s} }(H)\rightarrow\text{Out}_{k^{s}}(H)\rightarrow1. \] As $\text{Aut}_{k^{s}}(H)=\text{Aut}_{k^{s}}(G)_{a}$, it contains a subgroup $P_{a}$, which maps by a $\Gamma$-equivariant isomorphism onto $\text{Out}_{k^{s}}(H)$.
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. In the statement of Proposition 23.64, $G$ should be $\mathfrak{g}$.
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.505. In the proof of Theorem 23.67, it would have been better to refer to Proposition 23.64 rather than "the proposition" since there have been a few lemmas in between.
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 $G(\mathfrak{g})$.
p.509. "Since the Tannakian theory works ... category representations." An "of" is missing.
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". Line 4: Homs modulo conjugation in the case $D_4$.
p.518. Algebras $A$ over $k$ are required to be nonzero. Line 12 "... which are proved ... in Jacobson 1989", or maybe not --- see Conrad's comments below.
p.523. Here is a more precise statement of Theorem 24.43. Let $A$ be a central simple algebra $A$ of degree $n^2$ over $k$. The choice of an isomorphism $M_n(k^s)\to A\otimes k^s$ determines an isomorphism $\text{SL}_{nk^s}\to\text{SL}_1(A)_{k^s}$ and the pair $(\text{SL}_1(A),f)$ is an inner form of $\text{SL}_n$. Every inner form of $\text{SL}_n$ arises in this way, and the inner forms $(\text{SL}_1(A),f)$ and $(\text{SL}_1(A'),f')$ are isomorphic if and only if $A$ and $A'$ are isomorphic. Caution: $\text{SL}_1(A)$ and $\text{SL}_1(A^{\text{opp}})$ are isomorphic as algebraic groups because $\text{SL}_1(A)\simeq\text{SU}(A\times A^{\text{opp}},*)\simeq\text{SL}_1(A^{\text{opp}})$ where $*(a,b)=(b,a)$.
p.540. Near the bottom, $F_2$ should be $F_4$ --- there is no $F_2$.
p.544. In the second part of the proof of 25.1, I use that a parabolic subgroup $P$ of a connected group variety $G$ contains a maximal torus of $G$. Here's the proof. When $k$ is algebraically closed, every Borel subgroup (hence every parabolic subgroup) contains a maximal torus of $G$ (bottom of p.355); moreover, every maximal torus of $P$ is maximal in $G$ (17.10). Now apply 17.83 to $P$.
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.547. A theorem of Deligne (Saavedra 1972, p.229) says that a tensor filtration on $\mathrm{Rep}(G)$ is splittable if $k$ is of characteristic zero. When $G$ is reductive, this follows from 25.1 with no hypothesis on $k$. In 25.12, there is no hypothesis on $k$.
p.549. The statement in 25.16, that the relative root system is a root system, is certainly correct, but the standard proofs in the old literature have been questioned; see Conrad's comments below.
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 $v$.
p.568. In line 1, "Spm($A$)" should be "spm($A$)".
p.570. The proof in footnote 3 is incorrect. Instead, let $T=Z(f_{1},\ldots,f_{r})$ be a closed subset of $\mathrm{spm}(A_{k^{\mathrm{a}}})$. If $f_{1},\ldots,f_{r}\in A$, then $\pi(T)=Z(f_{1},\ldots,f_{r})$ in $\mathrm{spm}(A)$. In general, $f_{1},\ldots,f_{r}\in A_{k^{\prime}}$ for some finite extension $k^{\prime}$ of $k$ in $k^{\mathrm{a}}$, and we can use that the map $\mathrm{spm}(A_{k^{\prime}})\rightarrow \mathrm{spm}(A)$ is closed (because $A_{k^{\prime}}$ is a finite $A$-algebra).
p.582 A70. Optionally , one can relax "integral" to "reduced": if the algebraic scheme $X$ is reduced, then it contains a disjoint union of integral schemes as a dense open subscheme.
p.632. "McNinch, G." should be "McNinch, G. J.", so it prints in the correct order.

Conrad's comments After the book was published, Brian Conrad kindly provided me with seven pages of comments and corrections. I haven't posted his list here because it also contains errors (e.g., the statements 1.65, 1.70, 1.71, 7.4, 7.5, 7.6, 7.12, 17.64, 23.54,... in my book are in fact correct). Most of his corrections have been included in the above list. Here are some of his comments.