Page: -1 [1998-07-18] * p. -1, Page-numbers i,ii,...: were shifted by 2 in the final version. This is bad: There are now wrong references in the index Perhaps one should start the regular page numbers 1,2,... already with the Conventions and Notations section. Nevertheless, there are references to the introduction (pagnumbers in bibliography), so one has to be careful. Page: -1 [1998-07-18] * p.-1, Library of Congress Catalogue: ___pp.__cm Numbers are missing. Page: -1 [1998-07-20] * p.-1, Library of Congress Catalogue: The semicolons are preceeded by spaces (2 times). Page: -13 [2005-05-27] * p. -13, Introduction (page xiii): Tony Springer is Tonny. Page: -4 [1998-07-20] * p.-4, Summary (page iv in print): ... cohomology, or Jordan algebras. (colon missing) Page: 9 [1998-07-27] * p.9, Prop. 1.5, line 4: $U\subset V$ of dimension $k/\ind A$ (instead of $r/\ind A$). Page: 17 [1999-08-13] * p. 17, line -10, (2.7): ... such that \sigma(u)=\pm u instead of ``u=\pm u'' Page: 22 [1999-01-07] * p. 22 line 7 from below: Symd(B,\tau) ---> Sym(B,\tau) Page: 27 [1999-03-23] * On p. 27: $\Lie(A)$ was used for $A$ viewed as a Lie algebra. In (42.6) (and some other places) $\Lie(A, \sigma)$ is used for $\Skew(A,\sigma)$. I wonder if it would be not better to keep with the notation $\Skew(A,\sigma)$. Page: 41 [2006-03-06] * p. 41, Remark 3.20: one has to assume that E_1 x E_2 has constant rank over F x F (noticed by R. Loetscher). Page: 57 [1999-08-13] * p. 57, line +20: \Sym(A,\sigma_b) instead of `` \Sym(B,\sigma_b)'' Page: 77 [1999-02-27] * page 77, line 3: should be $(D,\theta)$ instead of $(D,\sigma)$. Page: 81 [1998-11-30] * page 81: Proposition (7.3.2): \disc has to be replaced by \det. Page: 93 [1999-03-23] * p.93, two lines above "Examples.": a period is missing at the end of a sentence. Page: 93 [1999-08-30] * p. 93: I found the following mistake in the BOI: on page 93, in the middle, a sentence should read: "We then simply write $C(A,\sigma)$ for $C(A,\sigma,f)$." (The full stop in that sentence is also missing.) JP: Another typo. (I had noticed the absence of the period, but not the interchange of C(A,\sigma) and C(A, \sigma,f) ! Page: 99 [1998-04-30] * p.99: 2 displays mid-page have equal signs at end of line and again at beginning of next line (remove first =, leaving only those aligning below) (MR:Doubtful) Page: 106 [1998-07-27] * p.106, line 11: insert "isotropic": "...suppose $I\subset A$ is an isotropic right ideal..." Page: 113-114 [1999-08-30] * pp 113-114: substitute B(A,\sigma,f) for B(A,\sigma) in the displayed equations and on the first line of p. 114. Page: 118 [1999-03-31] * p.118: there is a pair of extra brackets in equation (10.8) Page: 150 [1998-07-27] * p.150: in (a) and (b): the $C_\pm$ should be $C^\pm$ (four times). Page: 155 [1999-08-30] * p. 155, line -4: b_q instead of b in \Sim(V,b) Page: 156 [1999-08-30] * p.156, lines +15 through -9: the base field should be called F instead of K Page: 168 [1999-03-23] * p.168, line 7 from below: $v' \in V'$ should be replaced by $v \in V$. Page: 202 [1998-04-30] * p.202: first display on page has errant period. (MR:OK) Page: 202 [1999-03-23] * p.202, Ex.13.: $(A,\sigma)_+$ should be replaced by $\Sym(A,\sigma)$. Page: 237 [1999-08-30] * p.237, in (16.9) and the line below: substitute \Symd for \Sym Page: 276 [1998-11-30] * page 276: Correct the proof of (17.30) or remove (17.30). Page: 281 [2000-12-19] * p. 281, line 20: the reference should be to Proposition 7, p.~V.40 instead of Proposition 1, p.~V.47. (The reference to Proposition 3, p.~V.36 is OK.) Page: 284 [1998-05-25] * p.284: Prop. (18.8) 3. S_{L/F} instead of L/F Page: 286 [2002-01-18] * p. 286, line 2, right side of the displayed equation: the exponent of $x$ in the second term should be $n-2$ instead of $n-1$. Page: 289 [2000-12-19] * p. 289, line 12: substitute \Ind^G_H M for L. Page: 298 [2002-01-18] * p. 298, second line of the proof of Corollary (18.28): $\Theta\bigl((1-e)(1\otimes\ell)\bigr)$ instead of $\Theta(1\otimes\ell)$. Page: 299 [2002-01-18] * p. 299, line -8, right side of the displayed equation: $S_{L/F}$ instead of $S_{L/K}$. Page: 327 [1999-09-29] * page 327, line -11: replace $F[t]$ by $F[t,t \inv]$ twice. Page: 331 [1999-02-01] * Page 331, Example(20.13), second line: delete extra "is". Page: 331 [2001-05-31] * p. 331, Example (20.13) (3): Yes, one should add to Example (20.13) (3) : if $n$ is prime to $char(F)$. Page: 334 [1998-09-12] * Page 334.: In the text \subsection{The Lie algebra of a group scheme} Let $G$ be an algebraic group scheme over~$F$ and let $A = F [G]$. delete "algebraic". Page: 349 [1999-09-23] * page 349, line 4: replace last $q$ by $g$. Page: 356 [1998-09-12] * Page 356.: In the text Consider the adjoint representation (see Example \Eqref{gp.adjoint}{gp.adjoint.2}): \[ \ad\col G \to \gGL \bigl(\LIE(G)\bigr). \] By the theory of representations of diagonalizable groups (see~\eqref{gp.repdiag}) applied to the restriction of~$\ad$ to~$T$, we get a decomposition $\ad$ should be replaced by $\Ad$ twice. Page: 357 [1998-09-12] * Page 357.: In The character group~$T^*$ then is identified with $\LZ^n/ (e_1 + e_2 + \dots + e_{n+1})\LZ$ by $\bar{e}_i \leftrightarrow \chi_i$. replace $\LZ^n$ by $\LZ^{n+1}$. Page: 358 [2013-07-31] * p.358, Theorem (25.10): the condition on the form q is that it should be regular of Witt index n. (The condition that the form b_q be hyperbolic on V/rad(b_q) is automatic if charac F = 2, and it is nonsense if charac F \neq 2 since in that case rad(b_q)=0 so V/rad(b_q) has odd dimension.) Page: 358 [2013-07-31] * p.358, line 5: "b_q(v_i,v_j) = 0 otherwise" should be: "b_q(v_i,v_j) = 0 if i, j \geq 1 and i\neq j\pm n" (because b_q(v_0,v_0) \neq 0 when charac F \neq 2); Page: 369 [1999-02-01] * page 369, Remark (26.13), second line: delete extra "to". Page: 370 [1998-04-28] * p.370: at the bottom of the page, there is a period at the end of the second to last display. (MR:It should be a colon) Page: 379 [2000-12-19] * p. 379, line 2: C^* should be C^*_\sep Page: 380 [1998-09-12] * Page 380. Exercise 5: should be deleted since it is contined in Prop. 21.7. Page: 388 [2002-01-11] * p. 388, pf. of (28.12), line 1: x_\sigma \to \alpha_\sigma, not \in Same comment for y_\sigma \in \gamma_\sigma Page: 390 [2006-01-13] * p. 390: By the way, let me point out another mistake in the BoI, which was pointed out to me by Melanie Raczek (a student of mine): on p. 390 we talk about the induced \Gamma-group \Ind^\Gamma_\Gamma_0A_0 for \Gamma_0 any closed subgroup of \Gamma. As far as I can see, the action of \Gamma on this group is not necessarily continuous if the index of \Gamma_0 in \Gamma is infinite. Page: 394 [1999-05-10] * page 394, line 18: misspelled "exercise". Objection: this is in a reference to Serre's "Cohomologie galoisienne" book, which is in French. In French, "exercise" is spelled "exercice". JP Page: 402 [1999-05-10] * page 402, line -7: delete extra "correspondence". Page: 411 [1998-04-30] * p.411: text in paragraph above display (29.36) beginning "and define..." should align on left margin (MR:OK) Page: 425 [2002-01-11] * p. 425, displayed equation below (31.3): When Aut(Dyn(G)) is nonabelian (G of type D_4), Hom is not the same as H^1. Page: 444 [1998-12-22] * p. 444, in exercise 10: - on line 5 of this exercise, there is a subscript "n" missing (to \mu); - the suggestion for the definition of the isomorphism in part (a) is wrong: the isomorphism should take x F^{\times 2} to [x,x]. (Markus: this is the inverse of [x,y] maps to x N_{Z/F}(z) F^{\times 2}) Page: 453 [1999-01-07] * p. 453 first line of the proof of (32.3): d_{b,c}(a) ---> d_{a,c}(b) Page: 455 [1999-01-07] * p. 455 line 3 from below: Corollary (33.19) does not exist. The proof has to be rewritten.... Page: 461 [1999-10-15] * page 461, Korollar (33.21): "extends to an isomorphism or an anti-isomorphism". This is correct but seems strange since \Phi always extends to an isomorphism. ??? Page: 461 [2000-10-13] * p. 461, Korollar (33.21): Lieber Herr Rost, ich w"are dankbar, wenn Sie mir die Einzelheiten zur Bemerkung (33.22) in Ihrem Buch klar machen k"onnten. (Mail von Richard M. Weiss) Page: 463 [1998-07-20] * p.463, Chap.VII, (33.28): 1, 2, 4, or 8 (colon missing) Page: 472 [1998-04-30] * p.472: first line on page should align on left margin (MR:OK) Page: 474 [1999-01-07] * p. 474, in (34.24) second line: ... map f:A \to F such x \star x = .. has to be rewritten as: ... map f:A \to F such that f(0)=0 and x \star x = .. Page: 478 [1999-01-07] * p. 478 line -14: given by z \mapsto (T_C(z), \xi \otimes z + ...) (add (T_C(z), ..) Page: 478 [2006-03-06] * p. 478 in 34.33: one should read n(x) = 1/6T_B(x^2) Page: 481 [1999-01-07] * p. 481: The proof of (35.2) is given assuming char F not 2. This can be changed. (I have a copy of a nice proof due to Elduque; I shall type it soon) (Who are you? Max?) Page: 482 [1999-01-07] * p. 482 line 19: b_1(b_q(x)u_1,u_1) --- > b_1(q(x)u_1,u_1) Page: 482 [1999-01-07] * p. 482 line 8: new: \lambda_x \in \Hom_F(U_1,U_2) and \rho_x \in \Hom_F(U_2,U_1) (Switch U_1 and U_2) Page: 485 [1999-01-07] * p. 485: (35.7) can be proven without assuming char F not 2 (I have a copy of a nice proof due to Elduque; I shall type it soon) (Who are you? Max?) last line of (35.7) \chi^{pm} --- > \chi^{\pm} Page: 489 [1999-01-07] * p. 489, line 4: b_{q_1}( ..., v_2) ---> b_{q_1}( ..., v_1) Page: 493 [1999-10-29] * p. 493, l. -5: "with fixed L" can be misleading. It should be pointed out that all automorphisms keep L pointwise fixed. Page: 494 [1999-10-29] * p. 494, last diagram: Is it correct that N_{L/F}^1 carries the exponent? It does not appear in the surrounding text. Page: 494 [1999-10-29] * p. 494, line 2,3 after (36.8): Replace C by \tilde C. ? Page: 494 [2006-03-06] * p. 494 in 36.B (4): Q(z) instead of Q(x). Page: 495 [1999-01-07] * p. 495, line -6: b_Q(... , ...) = \rho(...) \rho^2(Q(x)) ---> b_Q(... , ...) = \rho(...) \rho^2(Q(z)) Page: 495 [1999-10-29] * p. 495, line 2: Replace K by \Delta. ? Page: 497 [2002-01-11] * p. 497: This is not an error, but there is something missing on this page. The question of whether or not p is surjective is not addressed. Petersson pointed out to me that p is typically not surjective, as follows from Thm. 9 of Section 9 of (*) A.A. Albert and N.Jacobson, On reduced exceptional simple Jordan algebras, Ann. of Math. 66 (3) (1957), 400 - 417 (see also N. Jacobson, Collected Mathematical Papers, Vol. II, pp. 310 - 327) Page: 505 [2001-06-21] * p. 505, line 2: There is a mess with \star\tens 1 (in boi-8.tex line 5084). Page: 519 [1999-10-29] * p. 519, l. -2: In T(x^#,y) and T(y^#,x) the x,y have to exchanged. Page: 519 [2002-01-11] * p. 519 displayed equation at bottom of page: The powers \lambda^2 and \lambda^1 are swapped. Page: 543 [2002-01-11] * p. 543, paragraph 1, line -5: "Observe that this Lie algebra [F_4] is not simple in characteristic 3." Probably this is just a typo: According to the literature, the Lie algebra F_4 is simple if and only if char =/= 2. I have not checked the proofs, but I have these references: (*) Gerhard Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen, Arch. Math. _42_ (1984), 408-416 This one gives that F_4 / Z(F_4) = F_4 is simple if and only if char =/= 2. (*) James F. Hurley, Centers of Chevalley algebras, J. Math. Soc. Japan, _34_ #2 (1982), 219-222 This gives that the center of the Chevalley algebra of type F_4 is 0 in all characteristics. Page: 544 [2007-01-25] * p. 544: Tits Resume: "kind kind" -> "kind" (terrible misprint!) Page: 551 [1998-12-22] * p. 551, 6 line from below: Lemma 42.9 ---> Proposition 42.9. Page: 553 [1998-12-22] * p. 553, 8 line from below: In the formula for T a copy of F is needed. Page: 554 [1998-12-22] * p.554, line before Proposition 43.6: Symvol $N_{L_i/F}[A_i]$ is missing. Page: 555 [1999-08-10] * page 555, Prop. 43.9 (2): Replace N_L(a) with N_{L/F}(a). Page: 560-561 [1998-12-22] * pp. 560-561: In $PGO^+$ the plus sign in lower case in several places. Page: 566 [1999-03-23] * p. 566 -567 : replace $C(\End(V),\sigma_q)$ by $C_0(V,q)$ (twice) Page: 567 [1999-03-23] * p.567, In (45.4) : $\Lie\bigl(C_0(q),\tau\bigr)$ should be replaced by $\Lie\bigl(C_0(q),\tau_0\bigr)$ (upto remark 1)!) Page: 569 [1999-03-23] * On p. 569: we associate a Lie algebra $\Lie{T}$ to a trialitarian algebra $T$. This notation is not compatible with the notation $\Lie(A)$ p. 27 (see 1)) but I do not know a better Page: 586 [1999-01-07] * p. 586: Inner form ---> inner form (to change p. 425) Page: ?? [1998-07-18] * p. ??, boi-10.tex: \index{A}{tau'@{$\tau'$ (second involution)}}% Change \index{A} -> \index{B} Page: ?? [1998-08-13] * p.??, boi-10.tex, line 73: \sigma_{A)}) Page: ?? [1999-01-20] * p.??: Brauer-equivalent versus Brauer equivalence ?? Page: ?? [1999-01-29] * p.??, boi-10.tex, mismatching paranthesis: l.73, l.1127. Page: ?? [1999-01-29] * p.??, boi-2.tex, mismatching paranthesis: l. 6976 Page: ?? [1999-01-29] * p.??, boi-6.tex, mismatching paranthesis: l. 1447, l. 3193: Page: ?? [1999-01-29] * p.??, boi-7.tex, mismatching paranthesis: l. 728, Page: ?? [1999-01-29] * p.??, boi-8.tex, mismatching paranthesis: l. 2742, l. 3226, l. 3966. Page: ?? [1999-01-29] * p.??, mismatching paranthesis: 3 or 4 of them are harmless like ($A)$. Page: ??? [1998-04-??] * boi.bib: Van Drooge (thesis, Utrecht, 1967) is cited in 2.tex. Can someone produce a bib-entry? Page: ??? [1998-07-27] * in the references: [68] has now appeared: Commun. Algebra 26 (5) (1998) 1589--1612. Page: ??? [1998-09-01] * Chap. VII, Notes, last page: relatively --> comparatively Page: ??? [1998-11-23] * Notation Index: Nrp, Trp are missing (pfaffian norm and trace) Page: ??? [1998-11-27] * Notation Index: Replace entry "discriminant 291" by "discriminant of an \'etale algebra". Page: ??? [1998-11-30] * bibliography: preprint [148] did appear meanwhile. Page: ??? [2001-05-31] * bibliography: Dan Shapiro's book "Compositions of Quadratic Forms" appeared. Might be cited. See boi-add.bib Page: ??? [2002-09-18] * The discriminant of an algebra of even degree with an involution of second kind can also be found in the paper Representation theory and the notion of the discriminant by Tamagawa, which appeared in the proc. Algebraic Number Theory, Papers contributed for the International Symposium, Kyoto 1976; S. Iyanaga (Ed.), Japan Society for the Promotion of Science, Tokyo, 1977. We did not refer to that paper. Did somebody knew it?