Theorem 69.22.5 (Theorem on formal functions). In Situation 69.22.1. Fix p \geq 0. The system of maps
H^ p(X, \mathcal{F})/I^ nH^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F}/I^ n\mathcal{F})
define an isomorphism of limits
H^ p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F})
where the left hand side is the completion of the A-module H^ p(X, \mathcal{F}) with respect to the ideal I, see Algebra, Section 10.96. Moreover, this is in fact a homeomorphism for the limit topologies.
Proof.
In fact, this follows immediately from Lemma 69.22.4. We spell out the details. Set M = H^ p(X, \mathcal{F}) and M_ n = H^ p(X, \mathcal{F}/I^ n\mathcal{F}). Denote N_ n = \mathop{\mathrm{Im}}(M \to M_ n). By the description of the limit in Homology, Section 12.31 we have
\mathop{\mathrm{lim}}\nolimits _ n M_ n = \{ (x_ n) \in \prod M_ n \mid \varphi _ i(x_ n) = x_{n - 1}, \ n = 2, 3, \ldots \}
Pick an element x = (x_ n) \in \mathop{\mathrm{lim}}\nolimits _ n M_ n. By Lemma 69.22.4 part (3) we have x_ n \in N_ n for all n since by definition x_ n is the image of some x_{n + m} \in M_{n + m} for all m. By Lemma 69.22.4 part (1) we see that there exists a factorization
M \to N_ n \to M/I^{n - c_1}M
of the reduction map. Denote y_ n \in M/I^{n - c_1}M the image of x_ n for n \geq c_1. Since for n' \geq n the composition M \to M_{n'} \to M_ n is the given map M \to M_ n we see that y_{n'} maps to y_ n under the canonical map M/I^{n' - c_1}M \to M/I^{n - c_1}M. Hence y = (y_{n + c_1}) defines an element of \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM. We omit the verification that y maps to x under the map
M^\wedge = \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n M_ n
of the lemma. We also omit the verification on topologies.
\square
Comments (0)