The Stacks project

Lemma 50.24.2. In Situation 50.24.1 the pushforward $f_*\mathcal{O}_ X$ is a finite étale $\mathcal{O}_ S$-algebra and locally on $S$ we have $Rf_*\mathcal{O}_ X = f_*\mathcal{O}_ X \oplus P$ in $D(\mathcal{O}_ S)$ with $P$ perfect of tor amplitude in $[1, \infty )$. The map $\text{d} : f_*\mathcal{O}_ X \to f_*\Omega _{X/S}$ is zero.

Proof. The first part of the statement follows from Derived Categories of Schemes, Lemma 36.32.8. Setting $S' = \underline{\mathop{\mathrm{Spec}}}_ S(f_*\mathcal{O}_ X)$ we get a factorization $X \to S' \to S$ (this is the Stein factorization, see More on Morphisms, Section 37.53, although we don't need this) and we see that $\Omega _{X/S} = \Omega _{X/S'}$ for example by Morphisms, Lemma 29.32.9 and 29.36.15. This of course implies that $\text{d} : f_*\mathcal{O}_ X \to f_*\Omega _{X/S}$ is zero. $\square$

Comments (2)

Comment #8670 by Sveta M on

Typo: "In Situation 0G8G the psuhforward" should be "... pushforward".

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  • 2 comment(s) on Section 50.24: Relative Poincaré duality

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