The Stacks project

52.4 Formal sections, III

In this section we generalize some of the results of Section 52.3 to the case of an ideal $I \subset A$ of cohomological dimension $1$.

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.

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 30.10.4 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

\[ \Phi : \bigoplus \nolimits _{n \geq 0} I^{nc} \to A[T_1, \ldots , T_ r] \]

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 $\bigoplus _{n \geq 0} I^{nc} \to \bigoplus _{n \geq 0} I^ n$.

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

\[ \Phi _ N : \bigoplus \nolimits _{n \geq 0} I^{nc}N \to N[T_1, \ldots , T_ r] \]

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 $\bigoplus _{n \geq 0} I^{nc}N \to \bigoplus _{n \geq 0} I^ nN$.

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 30.10.4. (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

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

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

\[ \Phi _\mathcal {F} : \bigoplus \nolimits _{n \geq 0} I^{nc}\mathcal{F} \longrightarrow \mathcal{F}[T_1, \ldots , T_ r] \]

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 $\bigoplus _{n \geq 0} I^{nc}\mathcal{F} \to \bigoplus _{n \geq 0} I^ n\mathcal{F}$.

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

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

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.

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 52.2.4 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

\[ g = \frac{t}{x} = \frac{s}{y} \]

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.


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