Lemma 52.4.1. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$. If $\text{cd}(A, I) = 1$, then there exist $c \geq 1$ and maps $\varphi _ j : I^ c \to A$ such that $\sum f_ j \varphi _ j : I^ c \to I$ is the inclusion map.
52.4 Formal sections, III
In this section we prove Lemma 52.4.5 which (in the setting of Noetherian schemes and coherent modules) is the analogue of Cohomology, Lemma 20.36.2 in case the ideal $I$ is not assumed principal but has the property that $\text{cd}(A, I) = 1$.
Proof. Since $\text{cd}(A, I) = 1$ the complement $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$ is affine (Local Cohomology, Lemma 51.4.8). Say $U = \mathop{\mathrm{Spec}}(B)$. Then $IB = B$ and we can write $1 = \sum _{j = 1, \ldots , r} f_ j b_ j$ for some $b_ j \in B$. By Cohomology of Schemes, Lemma 30.10.5 we can represent $b_ j$ by maps $\varphi _ j : I^ c \to A$ for some $c \geq 0$. Then $\sum f_ j \varphi _ j : I^ c \to I \subset A$ is the canonical embedding, after possibly replacing $c$ by a larger integer, by the same lemma. $\square$
Lemma 52.4.2. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi _ j : I^ c \to A$, $j = 1, \ldots , r$ be as in Lemma 52.4.1. Then there is a unique graded $A$-algebra map with $\Phi (g) = \sum \varphi _ j(g) T_ j$ for $g \in I^ c$. Moreover, the composition of $\Phi $ with the map $A[T_1, \ldots , T_ r] \to \bigoplus _{n \geq 0} I^ n$, $T_ j \mapsto f_ j$ is the inclusion map $\bigoplus _{n \geq 0} I^{nc} \to \bigoplus _{n \geq 0} I^ n$.
\[ \Phi : \bigoplus \nolimits _{n \geq 0} I^{nc} \to A[T_1, \ldots , T_ r] \]
Proof. For each $j$ and $m \geq c$ the restriction of $\varphi _ j$ to $I^ m$ is a map $\varphi _ j : I^ m \to I^{m - c}$. Given $j_1, \ldots , j_ n \in \{ 1, \ldots , r\} $ we claim that the composition
\[ \varphi _{j_1} \ldots \varphi _{j_ n} : I^{nc} \to I^{(n - 1)c} \to \ldots \to I^ c \to A \]
is independent of the order of the indices $j_1, \ldots , j_ n$. Namely, if $g = g_1 \ldots g_ n$ with $g_ i \in I^ c$, then we see that
\[ (\varphi _{j_1} \ldots \varphi _{j_ n})(g) = \varphi _{j_1}(g_1) \ldots \varphi _{j_ n}(g_ n) \]
is independent of the ordering as multiplication in $A$ is commutative. Thus we can define $\Phi $ by sending $g \in I^{nc}$ to
\[ \Phi (g) = \sum \nolimits _{e_1 + \ldots + e_ r = n} (\varphi _1^{e_1} \circ \ldots \circ \varphi _ r^{e_ r})(g) T_1^{e_1} \ldots T_ r^{e_ r} \]
It is straightforward to prove that this is a graded $A$-algebra homomorphism with the desired property. Uniqueness is immediate as is the final property. This proves the lemma. $\square$
Lemma 52.4.3. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi _ j : I^ c \to A$, $j = 1, \ldots , r$ be as in Lemma 52.4.1. Let $A \to B$ be a ring map with $B$ Noetherian and let $N$ be a finite $B$-module. Then, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, there is a unique unique graded $B$-module map with $\Phi _ N(g x) = \Phi (g) x$ for $g \in I^{nc}$ and $x \in N$ where $\Phi $ is as in Lemma 52.4.2. The composition of $\Phi _ N$ with the map $N[T_1, \ldots , T_ r] \to \bigoplus _{n \geq 0} I^ nN$, $T_ j \mapsto f_ j$ is the inclusion map $\bigoplus _{n \geq 0} I^{nc}N \to \bigoplus _{n \geq 0} I^ nN$.
\[ \Phi _ N : \bigoplus \nolimits _{n \geq 0} I^{nc}N \to N[T_1, \ldots , T_ r] \]
Proof. The uniqueness is clear from the formula and the uniqueness of $\Phi $ in Lemma 52.4.2. Consider the Noetherian $A$-algebra $B' = B \oplus N$ where $N$ is an ideal of square zero. To show the existence of $\Phi _ N$ it is enough (via Lemma 52.4.1) to show that $\varphi _ j$ extends to a map $\varphi '_ j : I^ cB' \to B'$ after possibly increasing $c$ to some $c'$ (and replacing $\varphi _ j$ by the composition of the inclusion $I^{c'} \to I^ c$ with $\varphi _ j$). Recall that $\varphi _ j$ corresponds to a section
\[ h_ j \in \Gamma (\mathop{\mathrm{Spec}}(A) \setminus V(I), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}) \]
see Cohomology of Schemes, Lemma 30.10.5. (This is in fact how we chose our $\varphi _ j$ in the proof of Lemma 52.4.1.) Let us use the same lemma to represent the pullback
\[ h'_ j \in \Gamma (\mathop{\mathrm{Spec}}(B') \setminus V(IB'), \mathcal{O}_{\mathop{\mathrm{Spec}}(B')}) \]
of $h_ j$ by a $B'$-linear map $\varphi '_ j : I^{c'}B' \to B'$ for some $c' \geq c$. The agreement with $\varphi _ j$ will hold for $c'$ sufficiently large by a further application of the lemma: namely we can test agreement on a finite list of generators of $I^{c'}$. Small detail omitted. $\square$
Lemma 52.4.4. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$ with $\text{cd}(A, I) = 1$. Let $c \geq 1$ and $\varphi _ j : I^ c \to A$, $j = 1, \ldots , r$ be as in Lemma 52.4.1. Let $X$ be a Noetherian scheme over $\mathop{\mathrm{Spec}}(A)$. Let be an inverse system of coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. Then, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, there exists a unique graded $\mathcal{O}_ X$-module map with $\Phi _\mathcal {F}(g s) = \Phi (g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$ where $\Phi $ is as in Lemma 52.4.2. The composition of $\Phi _\mathcal {F}$ with the map $\mathcal{F}[T_1, \ldots , T_ r] \to \bigoplus _{n \geq 0} I^ n\mathcal{F}$, $T_ j \mapsto f_ j$ is the canonical inclusion $\bigoplus _{n \geq 0} I^{nc}\mathcal{F} \to \bigoplus _{n \geq 0} I^ n\mathcal{F}$.
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
\[ \Phi _\mathcal {F} : \bigoplus \nolimits _{n \geq 0} I^{nc}\mathcal{F} \longrightarrow \mathcal{F}[T_1, \ldots , T_ r] \]
Proof. The uniqueness is immediate from the $\mathcal{O}_ X$-linearity and the requirement that $\Phi _\mathcal {F}(g s) = \Phi (g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$. Thus we may assume $X = \mathop{\mathrm{Spec}}(B)$ is affine. Observe that $(\mathcal{F}_ n)$ is an object of the category $\textit{Coh}(X, I\mathcal{O}_ X)$ introduced in Cohomology of Schemes, Section 30.23. Let $B' = B^\wedge $ be the $I$-adic completion of $B$. By Cohomology of Schemes, Lemma 30.23.1 the object $(\mathcal{F}_ n)$ corresponds to a finite $B'$-module $N$ in the sense that $\mathcal{F}_ n$ is the coherent module associated to the finite $B$-module $N/I^ n N$. Applying Lemma 52.4.3 to $I \subset A \to B'$ and $N$ we see that, after possibly increasing $c$ and adjusting $\varphi _ j$ accordingly, we get unique maps
\[ \Phi _ N : \bigoplus \nolimits _{n \geq 0} I^{nc}N \to N[T_1, \ldots , T_ r] \]
with the corresponding properties. Note that in degree $n$ we obtain an inverse system of maps $N/I^ mN \to \bigoplus _{e_1 + \ldots + e_ r = n} N/I^{m - nc}N \cdot T_1^{e_1} \ldots T_ r^{e_ r}$ for $m \geq nc$. Translating back into coherent sheaves we see that $\Phi _ N$ corresponds to a system of maps
\[ \Phi ^ n_ m : I^{nc}\mathcal{F}_ m \longrightarrow \bigoplus \nolimits _{e_1 + \ldots + e_ r = n} \mathcal{F}_{m - nc} \cdot T_1^{e_1} \ldots T_ r^{e_ r} \]
for varying $m \geq nc$ and $n \geq 1$. Taking the inverse limit of these maps over $m$ we obtain $\Phi _\mathcal {F} = \bigoplus _ n \mathop{\mathrm{lim}}\nolimits _ m \Phi ^ n_ m$. Note that $\mathop{\mathrm{lim}}\nolimits _ m I^ t\mathcal{F}_ m = I^ t \mathcal{F}$ as can be seen by evaluating on affines for example, but in fact we don't need this because it is clear there is a map $I^ t\mathcal{F} \to \mathop{\mathrm{lim}}\nolimits _ m I^ t\mathcal{F}_ m$. $\square$
Lemma 52.4.5. Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian scheme over $\mathop{\mathrm{Spec}}(A)$. Let be an inverse system of coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. If $\text{cd}(A, I) = 1$, then for all $p \in \mathbf{Z}$ the limit topology on $\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ is $I$-adic.
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
Proof. First it is clear that $I^ t \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ maps to zero in $H^ p(X, \mathcal{F}_ t)$. Thus the $I$-adic topology is finer than the limit topology. For the converse we set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$, we pick generators $f_1, \ldots , f_ r$ of $I$, we pick $c \geq 1$, and we choose $\Phi _\mathcal {F}$ as in Lemma 52.4.4. We will use the results of Lemma 52.2.1 without further mention. In particular we have a short exact sequence
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}_ n) \to H^ p(X, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n) \to 0 \]
Thus we can lift any element $\xi $ of $\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ to an element $\xi ' \in H^ p(X, \mathcal{F})$. Suppose $\xi $ maps to zero in $H^ p(X, \mathcal{F}_{nc})$ for some $n$, in other words, suppose $\xi $ is “small” in the limit topology. We have a short exact sequence
\[ 0 \to I^{nc}\mathcal{F} \to \mathcal{F} \to \mathcal{F}_{nc} \to 0 \]
and hence the assumption means we can lift $\xi '$ to an element $\xi '' \in H^ p(X, I^{nc}\mathcal{F})$. Applying $\Phi _\mathcal {F}$ we get
\[ \Phi _\mathcal {F}(\xi '') = \sum \nolimits _{e_1 + \ldots + e_ r = n} \xi '_{e_1, \ldots , e_ r} \cdot T_1^{e_1} \ldots T_ r^{e_ r} \]
for some $\xi '_{e_1, \ldots , e_ r} \in H^ p(X, \mathcal{F})$. Letting $\xi _{e_1, \ldots , e_ r} \in \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ be the images and using the final assertion of Lemma 52.4.4 we conclude that
\[ \xi = \sum f_1^{e_1} \ldots f_ r^{e_ r} \xi _{e_1, \ldots , e_ r} \]
is in $I^ n \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ as desired. $\square$
Example 52.4.6. Let $k$ be a field. Let $A = k[x, y][[s, t]]/(xs - yt)$. Let $I = (s, t)$ and $\mathfrak a = (x, y, s, t)$. Let $X = \mathop{\mathrm{Spec}}(A) - V(\mathfrak a)$ and $\mathcal{F}_ n = \mathcal{O}_ X/I^ n\mathcal{O}_ X$. Observe that the rational function is regular in an open neighbourhood $V \subset X$ of $V(I\mathcal{O}_ X)$. Hence every power $g^ e$ determines a section $g^ e \in M = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{F}_ n)$. Observe that $g^ e \to 0$ as $e \to \infty $ in the limit topology on $M$ since $g^ e$ maps to zero in $\mathcal{F}_ e$. On the other hand, $g^ e \not\in IM$ for any $e$ as the reader can see by computing $H^0(U, \mathcal{F}_ n)$; computation omitted. Observe that $\text{cd}(A, I) = 2$. Thus the result of Lemma 52.4.5 is sharp.
\[ g = \frac{t}{x} = \frac{s}{y} \]
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