Lemma 52.17.2 (0EHF)—The Stacks project

Lemma 52.17.2. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. If the inverse system $H^0(U, \mathcal{F}_ n)$ has Mittag-Leffler, then the canonical maps

\[ \widetilde{M/I^ nM}|_ U \to \mathcal{F}_ n \]

are surjective for all $n$ where $M$ is as in (52.16.6.1).

Proof. Surjectivity may be checked on the stalk at some point $y \in Y \setminus Z$. If $y$ corresponds to the prime $\mathfrak q \subset A$, then we can choose $f \in \mathfrak a$, $f \not\in \mathfrak q$. Then it suffices to show

\[ M_ f \longrightarrow H^0(U, \mathcal{F}_ n)_ f = H^0(D(f), \mathcal{F}_ n) \]

is surjective as $D(f)$ is affine (equality holds by Properties, Lemma 28.17.1). Since we have the Mittag-Leffler property, we find that

\[ \mathop{\mathrm{Im}}(M \to H^0(U, \mathcal{F}_ n)) = \mathop{\mathrm{Im}}(H^0(U, \mathcal{F}_ m) \to H^0(U, \mathcal{F}_ n)) \]

for some $m \geq n$. Using the long exact sequence of cohomology we see that

\[ \mathop{\mathrm{Coker}}(H^0(U, \mathcal{F}_ m) \to H^0(U, \mathcal{F}_ n)) \subset H^1(U, \mathop{\mathrm{Ker}}(\mathcal{F}_ m \to \mathcal{F}_ n)) \]

Since $U = X \setminus V(\mathfrak a)$ this $H^1$ is $\mathfrak a$-power torsion. Hence after inverting $f$ the cokernel becomes zero. $\square$

The Stacks project

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