The Stacks project

Lemma 51.8.4. Let $X$ be an integral locally Noetherian scheme. Let $j : U \to X$ be the inclusion of a nonempty open subscheme with complement $Z$. Assume that for all $z \in Z$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge $ we have $\dim (\mathcal{O}_{X, z}^\wedge /\mathfrak p) \geq 2$. Then $j_*\mathcal{O}_ U$ is coherent.

Proof. We may assume $X$ is affine. Using Lemmas 51.7.2 and 51.8.2 we reduce to $X = \mathop{\mathrm{Spec}}(A)$ where $(A, \mathfrak m)$ is a Noetherian local domain and $\mathfrak m \in Z$. Then we can use induction on $d = \dim (A)$. (The base case is $d = 0, 1$ which do not happen by our assumption on the local rings.) Set $V = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $. Observe that the local rings of $V$ have dimension strictly smaller than $d$. Repeating the arguments for $j' : U \to V$ we and using induction we conclude that $j'_*\mathcal{O}_ U$ is a coherent $\mathcal{O}_ V$-module. Pick a nonzero $f \in A$ which vanishes on $Z$. Since $D(f) \cap V \subset U$ we find an $n$ such that multiplication by $f^ n$ on $U$ extends to a map $f^ n : j'_*\mathcal{O}_ U \to \mathcal{O}_ V$ over $V$ (for example by Cohomology of Schemes, Lemma 30.10.4). This map is injective hence there is an injective map

\[ j_*\mathcal{O}_ U = j''_* j'_* \mathcal{O}_ U \to j''_*\mathcal{O}_ V \]

on $X$ where $j'' : V \to X$ is the inclusion morphism. Hence it suffices to show that $j''_*\mathcal{O}_ V$ is coherent. In other words, we may assume that $X$ is the spectrum of a local Noetherian domain and that $Z$ consists of the closed point.

Assume $X = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m)$ local and $Z = \{ \mathfrak m\} $. Let $A^\wedge $ be the completion of $A$. Set $X^\wedge = \mathop{\mathrm{Spec}}(A^\wedge )$, $Z^\wedge = \{ \mathfrak m^\wedge \} $, $U^\wedge = X^\wedge \setminus Z^\wedge $, and $\mathcal{F}^\wedge = \mathcal{O}_{U^\wedge }$. The ring $A^\wedge $ is universally catenary and Nagata (Algebra, Remark 10.158.9 and Lemma 10.160.8). Moreover, condition (3) of Lemma 51.8.3 for $X^\wedge , Z^\wedge , U^\wedge , \mathcal{F}^\wedge $ holds by assumption! Thus we see that $(U^\wedge \to X^\wedge )_*\mathcal{O}_{U^\wedge }$ is coherent. Since the morphism $c : X^\wedge \to X$ is flat we conclude that the pullback of $j_*\mathcal{O}_ U$ is $(U^\wedge \to X^\wedge )_*\mathcal{O}_{U^\wedge }$ (Cohomology of Schemes, Lemma 30.5.2). Finally, since $c$ is faithfully flat we conclude that $j_*\mathcal{O}_ U$ is coherent by Descent, Lemma 35.7.1. $\square$


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