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Tag 02OC

Chapter 29: Cohomology of Schemes > Section 29.20: The theorem on formal functions

Theorem 29.20.5 (Theorem on formal functions). Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\rm Spec}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. 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 $$ H^p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\rm lim}\nolimits_n H^p(X, \mathcal{F}/I^n\mathcal{F}) $$ 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 10.95. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof. In fact, this follows immediately from Lemma 29.20.4. 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 $N_n = \mathop{\rm Im}(M \to M_n)$. By the description of the limit in Homology, Section 12.27 we have $$ \mathop{\rm lim}\nolimits_n M_n = \{(x_n) \in \prod M_n \mid \varphi_i(x_n) = x_{n - 1}, ~n = 2, 3, \ldots\} $$ Pick an element $x = (x_n) \in \mathop{\rm lim}\nolimits_n M_n$. By Lemma 29.20.4 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 29.20.4 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 $\mathop{\rm lim}\nolimits_n M/I^nM$. We omit the verification that $y$ maps to $x$ under the map $$ M^\wedge = \mathop{\rm lim}\nolimits_n M/I^nM \longrightarrow \mathop{\rm lim}\nolimits_n M_n $$ of the lemma. We also omit the verification on topologies. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 5204–5227 (see updates for more information).

    \begin{theorem}[Theorem on formal functions]
    \label{theorem-formal-functions}
    Let $A$ be a Noetherian ring.
    Let $I \subset A$ be an ideal.
    Let $f : X \to \Spec(A)$ be a proper morphism.
    Let $\mathcal{F}$ be a coherent sheaf on $X$.
    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
    $$
    H^p(X, \mathcal{F})^\wedge
    \longrightarrow
    \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})
    $$
    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 \ref{algebra-section-completion}.
    Moreover, this is in fact a homeomorphism for the limit topologies.
    \end{theorem}
    
    \begin{proof}
    In fact, this follows immediately from
    Lemma \ref{lemma-ML-cohomology-powers-ideal}. 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 $N_n = \Im(M \to M_n)$.
    By the description of the limit in Homology, Section
    \ref{homology-section-inverse-systems} we have
    $$
    \lim_n M_n
    =
    \{(x_n) \in \prod M_n \mid \varphi_i(x_n) = x_{n - 1}, \ n = 2, 3, \ldots\}
    $$
    Pick an element $x = (x_n) \in \lim_n M_n$.
    By Lemma \ref{lemma-ML-cohomology-powers-ideal} 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 \ref{lemma-ML-cohomology-powers-ideal} 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 $\lim_n M/I^nM$.
    We omit the verification that $y$ maps to $x$ under the
    map
    $$
    M^\wedge = \lim_n M/I^nM \longrightarrow \lim_n M_n
    $$
    of the lemma. We also omit the verification on topologies.
    \end{proof}

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