# stacks/stacks-project

Much better depth >= 2 of pushforward thing

Finally! Not sure why this was missed earlier...
 @@ -924,7 +924,7 @@ \section{Morphisms and weakly associated points} then $s$ is not associated to $f_*\mathcal{F}$. \item If $s \not \in f(X)$, $(f_*\mathcal{F})_s$ is a finite $\mathcal{O}_{S, s}$-module, and $\mathcal{O}_{S, s}$ is Noetherian, then $\text{depth}(f_*\mathcal{O}_{S, s}) \geq 1$. is Noetherian, then $\text{depth}(f_*\mathcal{O}_{S, s}) \geq 2$. \item If $\mathcal{F}$ is flat over $S$ and $a \in \mathfrak m_s$ is a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\mathcal{F})_s$. \item If $\mathcal{F}$ is flat over $S$ and $a, b \in \mathfrak m_s$ @@ -940,9 +940,36 @@ \section{Morphisms and weakly associated points} \begin{proof} Part (1) is Lemma \ref{lemma-weakass-pushforward}. Part (2) follows from (1) and Lemma \ref{lemma-ass-weakly-ass}. Part (3) follows from (2) and Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors} or Algebra, Lemma \ref{algebra-lemma-ideal-nonzerodivisor}. \medskip\noindent Proof of part (3). To show the depth is $\geq w$ it suffices to show that $\Hom_{\mathcal{O}_{S, s}}(\kappa(s), (f_*\mathcal{F})_s) = 0$ and $\Ext^1_{\mathcal{O}_{S, s}}(\kappa(s), (f_*\mathcal{F})_s) = 0$, see Algebra, Lemma \ref{algebra-lemma-depth-ext}. 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 $$\Hom_{\mathcal{O}_{S, s}}(\mathcal{O}_{S, s}, (f_*\mathcal{F})_s) \to \Hom_{\mathcal{O}_{S, s}}(\mathfrak m_s, (f_*\mathcal{F})_s)$$ is an isomorphism. By flat base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) we may replace $S$ by $\Spec(\mathcal{O}_{S, s})$ and $X$ by $\Spec(\mathcal{O}_{S, s}) \times_S X$. Denote $\mathfrak m \subset \mathcal{O}_S$ the ideal sheaf of $s$. Then we see that $$\Hom_{\mathcal{O}_{S, s}}(\mathfrak m_s, (f_*\mathcal{F})_s) = \Hom_{\mathcal{O}_S}(\mathfrak m, f_*\mathcal{F}) = \Hom_{\mathcal{O}_X}(f^*\mathfrak m, \mathcal{F})$$ 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. \medskip\noindent For the proof of (4), (5), and (6) we use flat base change