Lemma 52.7.3. Let A be a Noetherian ring and let I \subset A be an ideal. Let X be a Noetherian scheme over A. Let \mathcal{F} be a coherent \mathcal{O}_ X-module. Assume that H^ p(X, \mathcal{F}) is a finite A-module for all p. Then there are short exact sequences
0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathcal{F})^\wedge \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0
of A-modules where H^ p(X, \mathcal{F})^\wedge is the usual I-adic completion. If f is proper, then the R^1\mathop{\mathrm{lim}}\nolimits term is zero.
Proof.
Consider the two spectral sequences of Lemma 52.6.20. The first degenerates by More on Algebra, Lemma 15.94.4. We obtain H^ p(X, \mathcal{F})^\wedge in degree p. This is where we use the assumption that H^ p(X, \mathcal{F}) is a finite A-module. The second degenerates because
\mathcal{F}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}
is a sheaf by Lemma 52.7.2. We obtain H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) in degree p. Since R\Gamma (X, -) commutes with derived limits (Injectives, Lemma 19.13.6) we also get
R\Gamma (X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F}/I^ n\mathcal{F})
By More on Algebra, Remark 15.87.6 we obtain exact sequences
0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0
of A-modules. Combining the above we get the first statement of the lemma. The vanishing of the R^1\mathop{\mathrm{lim}}\nolimits term follows from Cohomology of Schemes, Lemma 30.20.4.
\square
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