Errata — Tome III

Misprints and errors in the new edition of SGA 3, Tome 3.

13 October 2024.

1. SGA 3, Tome 3

p. 15, line 4: after "take $T$ split", add: "and use the proof of Th. 2.5".
p. 23, line −4 of 5.13: $t \neq = 0$ should be $b \neq = 0$ .
p. 24, at the level of page 33 of the original: add "of $r$ copies" before "of $n G =$ ". And, 6 lines below, replace $n G_{m, S^{'}}$ by $(n G_{m, S^{'}})^{r}$ .

p. 29, 2 lines below original page 39: $x$ should be $X$ .
p. 37, 3 lines above B), $G (0) = 1$ should be $E (0) = 1$ .
p. 38, line above C), $r$ should be $α$ .
p. 38, 2 lines below C), $N (S^{'})$ should be $N^{'} (S^{'})$ .
p. 40, 1st formula in D): $E (x)$ should be $E (x)^{- 1}$ . (This formula was not written in the original.)
p. 40, paragraph E): in the 3 first formulas with $α (H)$ on the left, replace on the right $(1 + b x)$ by $(1 + a x)$ . The proof is unchanged.
p. 45, Th. 3.1 (v): the 1st formula is incorrect and $Y$ is undefined. (This error appears in the original.) One should understand that $Y = z X$ , with $z \in G_{m} (S^{'})$ ; then by (iv) one has $w_{α} (Y) = w_{α} (X) α^{\lor} (z^{- 1})$ , hence $w_{α} (X)^{2} = α^{\lor} (- 1)$ gives
```
w_α(X) w_α(Y) = α^∨(−z⁻¹).
```
Finally, $X^{- 1}$ and $Y^{- 1}$ are defined in Corollary 2.6: one has $X X^{- 1} = 1$ and $Y^{- 1} = z^{- 1} X^{- 1}$ , hence indeed $- z^{- 1} = - X Y^{- 1}$ .
p. 51, Prop. 3.12: replace $Q (S)$ by $N^{\times} (S)$ .