52.19 Algebraization of coherent formal modules, III
We continue the discussion started in Sections 52.16 and 52.17. We will use the distance function of Section 52.18 to formulate a some natural conditions on coherent formal modules in Situation 52.16.1.
In Situation 52.16.1 given a point $y \in U \cap Y$ we can consider the $I$-adic completion
\[ \mathcal{O}_{X, y}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{X, y}/I^ n\mathcal{O}_{X, y} \]
This is a Noetherian local ring complete with respect to $I\mathcal{O}_{X, y}^\wedge $ with maximal ideal $\mathfrak m_ y^\wedge $, see Algebra, Section 10.97. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let us define the “stalk” of $(\mathcal{F}_ n)$ at $y$ by the formula
\[ \mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y} \]
This is a finite module over $\mathcal{O}_{X, y}^\wedge $. See Algebra, Lemmas 10.98.2 and 10.96.12.
Definition 52.19.1. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $a, b$ be integers. Let $\delta ^ Y_ Z$ be as in (52.18.0.1). We say $(\mathcal{F}_ n)$ satisfies the $(a, b)$-inequalities if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$
if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) \not= \{ \mathfrak m_ y^\wedge \} $, then
\[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \delta ^ Y_ Z(y) \geq a \quad \text{or}\quad \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > b \]
if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) = \{ \mathfrak m_ y^\wedge \} $, then
\[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \delta ^ Y_ Z(y) > a \]
We say $(\mathcal{F}_ n)$ satisfies the strict $(a, b)$-inequalities if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$ we have
\[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \delta ^ Y_ Z(y) > a \quad \text{or}\quad \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > b \]
Here are some elementary observations.
Lemma 52.19.2. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $a, b$ be integers.
If $(\mathcal{F}_ n)$ is annihilated by a power of $I$, then $(\mathcal{F}_ n)$ satisfies the $(a, b)$-inequalities for any $a, b$.
If $(\mathcal{F}_ n)$ satisfies the $(a + 1, b)$-inequalities, then $(\mathcal{F}_ n)$ satisfies the strict $(a, b)$-inequalities.
If $\text{cd}(A, I) \leq d$ and $A$ has a dualizing complex, then
$(\mathcal{F}_ n)$ satisfies the $(s, s + d)$-inequalities if and only if for all $y \in U \cap Y$ the tuple $\mathcal{O}_{X, y}^\wedge , I\mathcal{O}_{X, y}^\wedge , \{ \mathfrak m_ y^\wedge \} , \mathcal{F}_ y^\wedge , s - \delta ^ Y_ Z(y), d$ is as in Situation 52.10.1.
If $(\mathcal{F}_ n)$ satisfies the strict $(s, s + d)$-inequalities, then $(\mathcal{F}_ n)$ satisfies the $(s, s + d)$-inequalities.
Proof.
Immediate except for part (4) which is a consequence of Lemma 52.10.5 and the translation in (3).
$\square$
Lemma 52.19.3. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. If $\text{cd}(A, I) = 1$, then $\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if
\[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > 3 \]
for all $y \in U \cap Y$ and $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$.
Proof.
Observe that for a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $, $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$ we have $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) = \{ \mathfrak m_ y^\wedge \} \Leftrightarrow \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = 1$ as $\text{cd}(A, I) = 1$. See Local Cohomology, Lemmas 51.4.5 and 51.4.10. OK, consider the three numbers $\alpha = \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) \geq 0$, $\beta = \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) \geq 1$, and $\gamma = \delta ^ Y_ Z(y) \geq 1$. Then we see Definition 52.19.1 requires
if $\beta > 1$, then $\alpha + \gamma \geq 2$ or $\alpha + \beta + \gamma > 3$, and
if $\beta = 1$, then $\alpha + \gamma > 2$.
It is trivial to see that this is equivalent to $\alpha + \beta + \gamma > 3$.
$\square$
In the rest of this section, which we suggest the reader skip on a first reading, we will show that, when $A$ is $I$-adically complete, the category of $(\mathcal{F}_ n)$ of $\textit{Coh}(U, I\mathcal{O}_ U)$ which extend to $X$ and satisfy the strict $(1, 1 + \text{cd}(A, I))$-inequalities is equivalent to a full subcategory of the category of coherent $\mathcal{O}_ U$-modules.
Lemma 52.19.4. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume
$A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,
the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities.
Let $x \in X$ be a point. Let $W = \overline{\{ x\} }$. If $W \cap Y$ has an irreducible component contained in $Z$ and one which is not, then $\text{depth}(\mathcal{F}_ x) \geq 1$.
Proof.
Let $W \cap Y = W_1 \cup \ldots \cup W_ n$ be the decomposition into irreducible components. By assumption, after renumbering, we can find $0 < m < n$ such that $W_1, \ldots , W_ m \subset Z$ and $W_{m + 1}, \ldots , W_ n \not\subset Z$. We conclude that
\[ W \cap Y \setminus \left((W_1 \cup \ldots \cup W_ m) \cap (W_{m + 1} \cup \ldots \cup W_ n)\right) \]
is disconnected. By Lemma 52.14.2 we can find $1 \leq i \leq m < j \leq n$ and $z \in W_ i \cap W_ j$ such that $\dim (\mathcal{O}_{W, z}) \leq d + 1$. Choose an immediate specialization $y \leadsto z$ with $y \in W_ j$, $y \not\in Z$; existence of $y$ follows from Properties, Lemma 28.6.4. Observe that $\delta ^ Y_ Z(y) = 1$ and $\dim (\mathcal{O}_{W, y}) \leq d$. Let $\mathfrak p \subset \mathcal{O}_{X, y}$ be the prime corresponding to $x$. Let $\mathfrak p' \subset \mathcal{O}_{X, y}^\wedge $ be a minimal prime over $\mathfrak p\mathcal{O}_{X, y}^\wedge $. Then we have
\[ \text{depth}(\mathcal{F}_ x) = \text{depth}((\mathcal{F}^\wedge _ y)_{\mathfrak p'}) \quad \text{and}\quad \dim (\mathcal{O}_{W, y}) = \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p') \]
See Algebra, Lemma 10.163.1 and Local Cohomology, Lemma 51.11.3. Now we read off the conclusion from the inequalities given to us.
$\square$
Lemma 52.19.5. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume
$A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,
the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1+ d)$-inequalities, and
for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$.
Then $H^0(U, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F})$ is an isomorphism.
Proof.
We will prove this by showing that Lemma 52.12.4 applies. Thus we let $x \in \text{Ass}(\mathcal{F})$ with $x \not\in Y$. Set $W = \overline{\{ x\} }$. By condition (3) we see that $W \cap Y \not\subset Z$. By Lemma 52.19.4 we see that no irreducible component of $W \cap Y$ is contained in $Z$. Thus if $z \in W \cap Z$, then there is an immediate specialization $y \leadsto z$, $y \in W \cap Y$, $y \not\in Z$. For existence of $y$ use Properties, Lemma 28.6.4. Then $\delta ^ Y_ Z(y) = 1$ and the assumption implies that $\dim (\mathcal{O}_{W, y}) > d$. Hence $\dim (\mathcal{O}_{W, z}) > 1 + d$ and we win.
$\square$
Lemma 52.19.6. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume
$A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,
the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities, and
for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$.
Then the map
\[ \mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)}(\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]
is bijective for every coherent $\mathcal{O}_ U$-module $\mathcal{G}$.
Proof.
Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathcal{F})$. Using Cohomology of Schemes, Lemma 30.11.2 or More on Algebra, Lemma 15.23.10 we see that the completion of $\mathcal{H}$ satisfies the strict $(1, 1 + d)$-inequalities and that for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{H}_ x) \geq 2$. Details omitted. Thus by Lemma 52.19.5 we have
\[ \mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) = H^0(U, \mathcal{H}) = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]
See Cohomology of Schemes, Lemma 30.23.5 for the final equality.
$\square$
Lemma 52.19.7. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$ and $d \geq 1$. Assume
$A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,
$(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module,
$(\mathcal{F}_ n)$ satisfies the strict $(1, 1 + d)$-inequalities.
Then there exists a unique coherent $\mathcal{O}_ U$-module $\mathcal{F}$ whose completion is $(\mathcal{F}_ n)$ such that for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$.
Proof.
Choose a coherent $\mathcal{O}_ U$-module $\mathcal{F}$ whose completion is $(\mathcal{F}_ n)$. Let $T = \{ x \in U \mid \overline{\{ x\} } \cap Y \subset Z\} $. We will construct $\mathcal{F}$ by applying Local Cohomology, Lemma 51.15.4 with $\mathcal{F}$ and $T$. Then uniqueness will follow from the mapping property of Lemma 52.19.6.
Since $T$ is stable under specialization in $U$ the only thing to check is the following. If $x' \leadsto x$ is an immediate specialization of points of $U$ with $x \in T$ and $x' \not\in T$, then $\text{depth}(\mathcal{F}_{x'}) \geq 1$. Set $W = \overline{\{ x\} }$ and $W' = \overline{\{ x'\} }$. Since $x' \not\in T$ we see that $W' \cap Y$ is not contained in $Z$. If $W' \cap Y$ contains an irreducible component contained in $Z$, then we are done by Lemma 52.19.4. If not, we choose an irreducible component $W_1$ of $W \cap Y$ and an irreducible component $W'_1$ of $W' \cap Y$ with $W_1 \subset W'_1$. Let $z \in W_1$ be the generic point. Let $y \leadsto z$, $y \in W'_1$ be an immediate specialization with $y \not\in Z$; existence of $y$ follows from $W'_1 \not\subset Z$ (see above) and Properties, Lemma 28.6.4. Then we have the following $z \in Z$, $x \leadsto z$, $x' \leadsto y \leadsto z$, $y \in Y \setminus Z$, and $\delta ^ Y_ Z(y) = 1$. By Local Cohomology, Lemma 51.4.10 and the fact that $z$ is a generic point of $W \cap Y$ we have $\dim (\mathcal{O}_{W, z}) \leq d$. Since $x' \leadsto x$ is an immediate specialization we have $\dim (\mathcal{O}_{W', z}) \leq d + 1$. Since $y \not= z$ we conclude $\dim (\mathcal{O}_{W', y}) \leq d$. If $\text{depth}(\mathcal{F}_{x'}) = 0$ then we would get a contradiction with assumption (3); details about passage from $\mathcal{O}_{X, y}$ to its completion omitted. This finishes the proof.
$\square$
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