Lemma 20.37.5. Let (X, \mathcal{O}_ X) be a ringed space. Let (K_ n) be an inverse system in D(\mathcal{O}_ X). Let x \in X and m \in \mathbf{Z}. Assume there exist an integer n(x) and a fundamental system \mathfrak {U}_ x of open neighbourhoods of x such that for U \in \mathfrak {U}_ x
R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) = 0, and
H^ m(U, K_ n) \to H^ m(U, K_{n(x)}) is injective for n \geq n(x).
Then the map on stalks H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)_ x \to H^ m(K_{n(x)})_ x is injective.
Proof.
Let \gamma be an element of H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)_ x which maps to zero in H^ m(K_{n(x)})_ x. Since H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n) is the sheafification of U \mapsto H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) (by Lemma 20.32.3) we can choose U \in \mathfrak {U}_ x and an element \tilde\gamma \in H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) mapping to \gamma . Then \tilde\gamma maps to \tilde\gamma _{n(x)} \in H^ m(U, K_{n(x)}). Using that H^ m(K_{n(x)}) is the sheafification of U \mapsto H^ m(U, K_{n(x)}) (by Lemma 20.32.3 again) we see that after shrinking U we may assume that \tilde\gamma _{n(x)} = 0. For this U we consider the short exact sequence
0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) \to H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(U, K_ n) \to 0
of Lemma 20.37.1. By assumption (1) the group on the left is zero and by assumption (2) the group on the right maps injectively into H^ m(U, K_{n(x)}). We conclude \tilde\gamma = 0 and hence \gamma = 0 as desired.
\square
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