The Stacks project

Proof. Part (1) is Lemma 31.5.9. Part (2) follows from (1) and Lemma 31.5.8.

Proof of part (3). To show the depth is $\geq 2$ it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{S, s}}(\kappa (s), (f_*\mathcal{F})_ s) = 0$ and $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{S, s}}(\kappa (s), (f_*\mathcal{F})_ s) = 0$, see Algebra, Lemma 10.72.5. Using the exact sequence $0 \to \mathfrak m_ s \to \mathcal{O}_{S, s} \to \kappa (s) \to 0$ it suffices to prove that the map

is an isomorphism. By flat base change (Cohomology of Schemes, Lemma 30.5.2) we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ and $X$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \times _ S X$. Denote $\mathfrak m \subset \mathcal{O}_ S$ the ideal sheaf of $s$. Then we see that

the first equality because $S$ is local with closed point $s$ and the second equality by adjunction for $f^*, f_*$ on quasi-coherent modules. However, since $s \not\in f(X)$ we see that $f^*\mathfrak m = \mathcal{O}_ X$. Working backwards through the arguments we get the desired equality.

For the proof of (4), (5), and (6) we use flat base change (Cohomology of Schemes, Lemma 30.5.2) to reduce to the case where $S$ is the spectrum of $\mathcal{O}_{S, s}$. Then a nonzerodivisor $a \in \mathcal{O}_{S, s}$ deterimines a short exact sequence

Since $\mathcal{F}$ is flat over $S$, we obtain an exact sequence

Pushing forward we obtain an exact sequence

This proves (4) and it shows that $f_*\mathcal{F}/ af_*\mathcal{F} \subset f_*(\mathcal{F}/a\mathcal{F})$. If $b$ is a nonzerodivisor on $\mathcal{O}_{S, s}/a\mathcal{O}_{S, s}$, then the exact same argument shows $b : \mathcal{F}/a\mathcal{F} \to \mathcal{F}/a\mathcal{F}$ is injective. Pushing forward we conclude

is injective and hence also $b : f_*\mathcal{F}/ af_*\mathcal{F} \to f_*\mathcal{F}/ af_*\mathcal{F}$ is injective. This proves (5). Part (6) follows from (4) and (5) and the definitions. $\square$