﻿ Errata -- J.S. Milne
Errata for Algebraic Groups (J.S. Milne)
p.26. Proposition 1.65 (= Demazure and Gabriel 1970, II, \S 5, 3.1). The proof passes from $k$ to its algebraic closure, and so "reduced" should be replaced by "geometrically reduced" in (a) and (c). Consequently, "reduced" should be replaced by "geometrically reduced" in 1.70, 1.71, 7.4, 7.5, and 7.12 (and in the corresponding statements in Demazure and Gabriel).
p.51. Replace $X$ with $G$ twice in the first paragraph of Section 2g.
p.80, 3.52, Two inner forms $(G,f)$ and $(G',f')$ are said to be equivalent if there exists an isomorphism $\varphi:G\to G'$ such that $f'=\varphi_K\circ f$ up to conjugation.
p.85. $G_W$ is the stabilizer of $W$ in $G$ (not $V$).
p.129. 6.18 and 6.19 (also Waterhouse 1979, 10.1) should be treated with caution in the nonsmooth case (see Conrad's comments below).
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.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.223. Proposition 11.36 should read $\ldots \alpha=\text{Lie}(\varphi)$ not $\ldots \alpha=\text{Lie}(\varphi)\circ\text{Lie}(\varphi)$.
p.224. The definition 11.39 of the Verschiebung requires correction.
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.360. The statement before 17.28 should say that $\text{SL}_2$, not $\text{SL}_3$, has dimension $3$.
p.373. As its proof uses 7.5, Proposition 17.64 should be treated with caution when $k$ is not perfect.
p.389. In the statement of 18.8, the target of $\alpha$ is $G'$, not $G$. This is correct in the diagram.
p.399/442. In definition 19.8 and note 21.57, replace "proper normal" with "smooth connected proper normal".
p.456 et seq. I sometimes write "isogeny" when I should write "central isogeny", e.g., in the first paragraph on p.456. It is the central isogenies that preserve the Dynkin diagram and, in general, behave as isogenies in characteristic zero. See the discussion pp.493--494. I'll try to track down more examples.
p.457. In 21.96, replace $\text{SL}_n$ and $\mathbb{Z}/n\mathbb{Z}$ with $\text{SL}_{n+1}$ and $\mathbb{Z}/(n+1)\mathbb{Z}$.
p.459. 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.
p.460. In the 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.461. Exercise 21-2 needs an additional hypothesis to exclude the examples in 18.5.
p.480. Just after Remark 23.49, replace 18.23 with 18.24.
p.499. In 23.50, $\underline{\text{Hom}}(G,H)$ will not be affine when $\underline{\text{Out}}(G)$ is not finite (because $(\mathbb{Z})_k$ is not affine).
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.518. In Section 24d, the algebras $A$ over $k$ are required to be nonzero.
p.518. "... which are proved in Jacobson 1989", or maybe not --- see Conrad's comments (the statements are true!).
p.527 et seq. I should have incorporated the Appendix to Conrad 2014 into the exposition of orthogonal groups.
p.531. In 24.58 and 24.59, assume $(V,q)$ is regular; in 24.58(b), exclude the case $n=1,\,q=0$.
p.531. 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, it is necessary to take the parabolic subgroup ($P$ or $Q$) to be minimal, and in 25.27, it is necessary to take $S$ to be maximal split in $P$.
p.546. The proofs of 25.7 and 25.10 are incomplete. Also, "nontrivial" on the first line of the proof of 25.10 should be "noncentral".
p.549. The assertion 25.16(a), that the relative root system really is a root system, is true, but the standard proofs of this in the old literature are either false or incomprehensible. See Conrad's comments.
p.549. In line 1 of 25.19, add "torus" to "maximal split".

Seven pages of comments and corrections from Brian Conrad here (most corrections have been included above).