Lemma 33.16.6 (0B2F)—The Stacks project

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Lemma 33.16.6. Let $f : X \to Y$ be a morphism of schemes over a base scheme $S$ . Let $x \in X$ be a point. Set $y = f(x)$ . If $\kappa (y) = \kappa (x)$ , then $f$ induces a natural linear map

$\text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}$

which is dual to the linear map $\Omega _{Y/S, y} \otimes \kappa (y) \to \Omega _{X/S, x}$ via the identifications of Lemma 33.16.4.

Proof. Omitted. $\square$

Comments (3)

Comment #3268 by Dario Weißmann on April 23, 2018 at 10:59

Typo: after the map df there is a point but the sentence is not yet finished

Comment #3271 by Dario Weißmann on April 25, 2018 at 20:29

Also $\Omega_{X/S,\kappa(x)}$ should be replaced by $\Omega_{X/S,x}$ .

Comment #3363 by Johan on May 19, 2018 at 17:06

Thanks, fixed here.

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