The Stacks project

58.17 Restriction to a closed subscheme

In this section we prove some results about the restriction functor

\[ \textit{FÉt}_ X \longrightarrow \textit{FÉt}_ Y,\quad U \longmapsto V = U \times _ X Y \]

where $X$ is a scheme and $Y$ is a closed subscheme. Using the topological invariance of the fundamental group, we can relate the study of this functor to the completion functor on finite locally free modules.

In the following lemmas we use the concept of coherent formal modules defined in Cohomology of Schemes, Section 30.23. Given a Noetherian scheme and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ we will say an object $(\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ is finite locally free if each $\mathcal{F}_ n$ is a finite locally free $\mathcal{O}_ X/\mathcal{I}^ n$-module.

Lemma 58.17.1. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Assume the completion functor

\[ \textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is fully faithful.

Proof. Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $U$ finite étale over $X$ and a morphism $t : Y \to U$ over $X$ there exists a unique section $s : X \to U$ such that $t = s|_ Y$. Picture

\[ \xymatrix{ & U \ar[d]^ f \\ Y \ar[r] \ar[ru] & X \ar@{..>}@/^1em/[u] } \]

Finding the dotted arrow $s$ is the same thing as finding an $\mathcal{O}_ X$-algebra map

\[ s^\sharp : f_*\mathcal{O}_ U \longrightarrow \mathcal{O}_ X \]

which reduces modulo the ideal sheaf of $Y$ to the given algebra map $t^\sharp : f_*\mathcal{O}_ U \to \mathcal{O}_ Y$. By Lemma 58.8.3 we can lift $t$ uniquely to a compatible system of maps $t_ n : Y_ n \to U$ and hence a map

\[ \mathop{\mathrm{lim}}\nolimits t_ n^\sharp : f_*\mathcal{O}_ U \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{Y_ n} \]

of sheaves of algebras on $X$. Since $f_*\mathcal{O}_ U$ is a finite locally free $\mathcal{O}_ X$-module, we conclude that we get a unique $\mathcal{O}_ X$-module map $\sigma : f_*\mathcal{O}_ U \to \mathcal{O}_ X$ whose completion is $\mathop{\mathrm{lim}}\nolimits t_ n^\sharp $. To see that $\sigma $ is an algebra homomorphism, we need to check that the diagram

\[ \xymatrix{ f_*\mathcal{O}_ U \otimes _{\mathcal{O}_ X} f_*\mathcal{O}_ U \ar[r] \ar[d]_{\sigma \otimes \sigma } & f_*\mathcal{O}_ U \ar[d]^\sigma \\ \mathcal{O}_ X \otimes _{\mathcal{O}_ X} \mathcal{O}_ X \ar[r] & \mathcal{O}_ X } \]

commutes. For every $n$ we know this diagram commutes after restricting to $Y_ n$, i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we conclude. $\square$

Lemma 58.17.2. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Assume the completion functor

\[ \textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is an equivalence.

Proof. The restriction functor is fully faithful by Lemma 58.17.1.

Let $U_1 \to Y$ be a finite étale morphism. To finish the proof we will show that $U_1$ is in the essential image of the restriction functor.

For $n \geq 1$ let $Y_ n$ be the $n$th infinitesimal neighbourhood of $Y$. By Lemma 58.8.3 there is a unique finite étale morphism $\pi _ n : U_ n \to Y_ n$ whose base change to $Y = Y_1$ recovers $U_1 \to Y_1$. Consider the sheaves $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$. We may and do view $\mathcal{F}_ n$ as an $\mathcal{O}_ X$-module on $X$ wich is locally isomorphic to $(\mathcal{O}_ X/f^{n + 1}\mathcal{O}_ X)^{\oplus r}$. This $(\mathcal{F}_ n)$ is a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. By assumption there exists a finite locally free $\mathcal{O}_ X$-module $\mathcal{F}$ and a compatible system of isomorphisms

\[ \mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n \]

of $\mathcal{O}_ X$-modules.

To construct an algebra structure on $\mathcal{F}$ consider the multiplication maps $\mathcal{F}_ n \otimes _{\mathcal{O}_ X} \mathcal{F}_ n \to \mathcal{F}_ n$ coming from the fact that $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$ are sheaves of algebras. These define a map

\[ (\mathcal{F}\otimes _{\mathcal{O}_ X} \mathcal{F})^\wedge \longrightarrow \mathcal{F}^\wedge \]

in the category $\textit{Coh}(X, \mathcal{I})$. Hence by assumption we may assume there is a map $\mu : \mathcal{F}\otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}$ whose restriction to $Y_ n$ gives the multiplication maps above. After possibly shrinking further we may assume $\mu $ defines a commutative $\mathcal{O}_ X$-algebra structure on $\mathcal{F}$ compatible with the given algebra structures on $\mathcal{F}_ n$. Setting

\[ U = \underline{\mathop{\mathrm{Spec}}}_ X((\mathcal{F}, \mu )) \]

we obtain a finite locally free scheme $\pi : U \to X$ whose restriction to $Y$ is isomorphic to $U_1$. The the discriminant of $\pi $ is the zero set of the section

\[ \det (Q_\pi ) : \mathcal{O}_ X \longrightarrow \wedge ^{top}(\pi _*\mathcal{O}_ U)^{\otimes -2} \]

constructed in Discriminants, Section 49.3. Since the restriction of this to $Y_ n$ is an isomorphism for all $n$ by Discriminants, Lemma 49.3.1 we conclude that it is an isomorphism. Thus $\pi $ is étale by Discriminants, Lemma 49.3.1. $\square$

Lemma 58.17.3. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

defines is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor $\mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{FÉt}_ V \to \textit{FÉt}_ Y$ is fully faithful.

Proof. Observe that $\mathcal{V}$ is a directed set, so the colimits are as in Categories, Section 4.19. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 58.17.1; we urge the reader to skip it.

Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $U$ finite étale over $V \in \mathcal{V}$ and a morphism $t : Y \to U$ over $V$ there exists a $V' \geq V$ and a morphism $s : V' \to U$ over $V$ such that $t = s|_ Y$. Picture

\[ \xymatrix{ & & U \ar[d]^ f \\ Y \ar[r] \ar[rru] & V' \ar@{..>}[ru] \ar[r] & V } \]

Finding the dotted arrow $s$ is the same thing as finding an $\mathcal{O}_{V'}$-algebra map

\[ s^\sharp : f_*\mathcal{O}_ U|_{V'} \longrightarrow \mathcal{O}_{V'} \]

which reduces modulo the ideal sheaf of $Y$ to the given algebra map $t^\sharp : f_*\mathcal{O}_ U \to \mathcal{O}_ Y$. By Lemma 58.8.3 we can lift $t$ uniquely to a compatible system of maps $t_ n : Y_ n \to U$ and hence a map

\[ \mathop{\mathrm{lim}}\nolimits t_ n^\sharp : f_*\mathcal{O}_ U \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{Y_ n} \]

of sheaves of algebras on $V$. Observe that $f_*\mathcal{O}_ U$ is a finite locally free $\mathcal{O}_ V$-module. Hence we get a $V' \geq V$ a map $\sigma : f_*\mathcal{O}_ U|_{V'} \to \mathcal{O}_{V'}$ whose completion is $\mathop{\mathrm{lim}}\nolimits t_ n^\sharp $. To see that $\sigma $ is an algebra homomorphism, we need to check that the diagram

\[ \xymatrix{ (f_*\mathcal{O}_ U \otimes _{\mathcal{O}_ V} f_*\mathcal{O}_ U)|_{V'} \ar[r] \ar[d]_{\sigma \otimes \sigma } & f_*\mathcal{O}_ U|_{V'} \ar[d]^\sigma \\ \mathcal{O}_{V'} \otimes _{\mathcal{O}_{V'}} \mathcal{O}_{V'} \ar[r] & \mathcal{O}_{V'} } \]

commutes. For every $n$ we know this diagram commutes after restricting to $Y_ n$, i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we deduce that there exists a $V'' \geq V'$ such that $\sigma |_{V''}$ is an algebra homomorphism as desired. $\square$

Lemma 58.17.4. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{FÉt}_ V \to \textit{FÉt}_ Y \]

is an equivalence.

Proof. Observe that $\mathcal{V}$ is a directed set, so the colimits are as in Categories, Section 4.19. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 58.17.2; we urge the reader to skip it.

The restriction functor is fully faithful by Lemma 58.17.3.

Let $U_1 \to Y$ be a finite étale morphism. To finish the proof we will show that $U_1$ is in the essential image of the restriction functor.

For $n \geq 1$ let $Y_ n$ be the $n$th infinitesimal neighbourhood of $Y$. By Lemma 58.8.3 there is a unique finite étale morphism $\pi _ n : U_ n \to Y_ n$ whose base change to $Y = Y_1$ recovers $U_1 \to Y_1$. Consider the sheaves $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$. We may and do view $\mathcal{F}_ n$ as an $\mathcal{O}_ X$-module on $X$ wich is locally isomorphic to $(\mathcal{O}_ X/f^{n + 1}\mathcal{O}_ X)^{\oplus r}$. This $(\mathcal{F}_ n)$ is a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. By assumption there exists a $V \in \mathcal{V}$ and a finite locally free $\mathcal{O}_ V$-module $\mathcal{F}$ and a compatible system of isomorphisms

\[ \mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n \]

of $\mathcal{O}_ V$-modules.

To construct an algebra structure on $\mathcal{F}$ consider the multiplication maps $\mathcal{F}_ n \otimes _{\mathcal{O}_ V} \mathcal{F}_ n \to \mathcal{F}_ n$ coming from the fact that $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$ are sheaves of algebras. These define a map

\[ (\mathcal{F}\otimes _{\mathcal{O}_ V} \mathcal{F})^\wedge \longrightarrow \mathcal{F}^\wedge \]

in the category $\textit{Coh}(X, \mathcal{I})$. Hence by assumption after shrinking $V$ we may assume there is a map $\mu : \mathcal{F}\otimes _{\mathcal{O}_ V} \mathcal{F} \to \mathcal{F}$ whose restriction to $Y_ n$ gives the multiplication maps above. After possibly shrinking further we may assume $\mu $ defines a commutative $\mathcal{O}_ V$-algebra structure on $\mathcal{F}$ compatible with the given algebra structures on $\mathcal{F}_ n$. Setting

\[ U = \underline{\mathop{\mathrm{Spec}}}_ V((\mathcal{F}, \mu )) \]

we obtain a finite locally free scheme over $V$ whose restriction to $Y$ is isomorphic to $U_1$. It follows that $U \to V$ is étale at all points lying over $Y$, see More on Morphisms, Lemma 37.12.3. Thus after shrinking $V$ once more we may assume $U \to V$ is finite étale. This finishes the proof. $\square$

Lemma 58.17.5. Let $X$ be a scheme and let $Y \subset X$ be a closed subscheme. If every connected component of $X$ meets $Y$, then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is faithful.

Proof. Let $a, b : U \to U'$ be two morphisms of schemes finite étale over $X$ whose restriction to $Y$ are the same. The image of a connected component of $U$ is an connected component of $X$; this follows from Topology, Lemma 5.7.7 applied to the restriction of $U \to X$ to a connected component of $X$. Hence the image of every connected component of $U$ meets $Y$ by assumption. We conclude that $a = b$ after restriction to each connected component of $U$ by Étale Morphisms, Proposition 41.6.3. Since the equalizer of $a$ and $b$ is an open subscheme of $U$ (as the diagonal of $U'$ over $X$ is open) we conclude. $\square$

Lemma 58.17.6. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_ n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Assume one of the following holds

  1. $X$ is quasi-affine and $\Gamma (X, \mathcal{O}_ X) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{O}_{Y_ n})$ is an isomorphism, or

  2. $X$ has an ample invertible module $\mathcal{L}$ and $\Gamma (X, \mathcal{L}^{\otimes m}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n})$ is an isomorphism for all $m \gg 0$, or

  3. for every finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ the map $\Gamma (X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{E}|_{Y_ n})$ is an isomorphism.

Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is fully faithful.

Proof. This lemma follows formally from Lemma 58.17.1 and Algebraic and Formal Geometry, Lemma 52.15.1. $\square$

Lemma 58.17.7. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme. Let $Y_ n \subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume one of the following holds

  1. $X$ is quasi-affine and

    \[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{O}_ V) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{O}_{Y_ n}) \]

    is an isomorphism, or

  2. $X$ has an ample invertible module $\mathcal{L}$ and

    \[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n}) \]

    is an isomorphism for all $m \gg 0$, or

  3. for every $V \in \mathcal{V}$ and every finite locally free $\mathcal{O}_ V$-module $\mathcal{E}$ the map

    \[ \mathop{\mathrm{colim}}\nolimits _{V' \geq V} \Gamma (V', \mathcal{E}|_{V'}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{E}|_{Y_ n}) \]

    is an isomorphism.

Then the functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{FÉt}_ V \to \textit{FÉt}_ Y \]

is fully faithful.

Proof. This lemma follows formally from Lemma 58.17.3 and Algebraic and Formal Geometry, Lemma 52.15.2. $\square$


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