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The Stacks project

18.42 Constant sheaves

Let E be a set and let \mathcal{C} be a site. We will denote \underline{E} the constant sheaf with value E on \mathcal{C}. If E is an abelian group, ring, module, etc, then \underline{E} is a sheaf of abelian groups, rings, modules, etc.

Lemma 18.42.1. Let \mathcal{C} be a site. If 0 \to A \to B \to C \to 0 is a short exact sequence of abelian groups, then 0 \to \underline{A} \to \underline{B} \to \underline{C} \to 0 is an exact sequence of abelian sheaves and in fact it is even exact as a sequence of abelian presheaves.

Proof. Since sheafification is exact it is clear that 0 \to \underline{A} \to \underline{B} \to \underline{C} \to 0 is an exact sequence of abelian sheaves. Thus 0 \to \underline{A} \to \underline{B} \to \underline{C} is an exact sequence of abelian presheaves. To see that \underline{B} \to \underline{C} is surjective, pick a set theoretical section s : C \to B. This induces a section \underline{s} : \underline{C} \to \underline{B} of sheaves of sets left inverse to the surjection \underline{B} \to \underline{C}. \square

Lemma 18.42.2. Let \mathcal{C} be a site. Let \Lambda be a ring and let M and Q be \Lambda -modules. If Q is a finitely presented \Lambda -module, then we have \underline{M \otimes _\Lambda Q}(U) = \underline{M}(U) \otimes _\Lambda Q for all U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}).

Proof. Choose a presentation \Lambda ^{\oplus m} \to \Lambda ^{\oplus n} \to Q \to 0. This gives an exact sequence M^{\oplus m} \to M^{\oplus n} \to M \otimes Q \to 0. By Lemma 18.42.1 we obtain an exact sequence

\underline{M}(U)^{\oplus m} \to \underline{M}(U)^{\oplus n} \to \underline{M \otimes Q}(U) \to 0

which proves the lemma. (Note that taking sections over U always commutes with finite direct sums, but not arbitrary direct sums.) \square

Lemma 18.42.3. Let \mathcal{C} be a site. Let \Lambda be a coherent ring. Let M be a flat \Lambda -module. For U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the module \underline{M}(U) is a flat \Lambda -module.

Proof. Let I \subset \Lambda be a finitely generated ideal. By Algebra, Lemma 10.39.5 it suffices to show that \underline{M}(U) \otimes _\Lambda I \to \underline{M}(U) is injective. As \Lambda is coherent I is finitely presented as a \Lambda -module. By Lemma 18.42.2 we see that \underline{M}(U) \otimes I = \underline{M \otimes I}. Since M is flat the map M \otimes I \to M is injective, whence \underline{M \otimes I} \to \underline{M} is injective. \square

Lemma 18.42.4. Let \mathcal{C} be a site. Let \Lambda be a Noetherian ring. Let I \subset \Lambda be an ideal. The sheaf \underline{\Lambda }^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{\Lambda /I^ n} is a flat \underline{\Lambda }-algebra. Moreover we have canonical identifications

\underline{\Lambda }/I\underline{\Lambda } = \underline{\Lambda }/\underline{I} = \underline{\Lambda }^\wedge /I\underline{\Lambda }^\wedge = \underline{\Lambda }^\wedge /\underline{I} \cdot \underline{\Lambda }^\wedge = \underline{\Lambda }^\wedge /\underline{I}^\wedge = \underline{\Lambda /I}

where \underline{I}^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{I/I^ n}.

Proof. To prove \underline{\Lambda }^\wedge is flat, it suffices to show that \underline{\Lambda }^\wedge (U) is flat as a \Lambda -module for each U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), see Lemmas 18.28.2 and 18.28.3. By Lemma 18.42.3 we see that

\underline{\Lambda }^\wedge (U) = \mathop{\mathrm{lim}}\nolimits \underline{\Lambda /I^ n}(U)

is a limit of a system of flat \Lambda /I^ n-modules. By Lemma 18.42.1 we see that the transition maps are surjective. We conclude by More on Algebra, Lemma 15.27.4.

To see the equalities, note that \underline{\Lambda }(U)/I\underline{\Lambda }(U) = \underline{\Lambda /I}(U) by Lemma 18.42.2. It follows that \underline{\Lambda }/I\underline{\Lambda } = \underline{\Lambda }/\underline{I} = \underline{\Lambda /I}. The system of short exact sequences

0 \to \underline{I/I^ n}(U) \to \underline{\Lambda /I^ n}(U) \to \underline{\Lambda /I}(U) \to 0

has surjective transition maps, hence gives a short exact sequence

0 \to \mathop{\mathrm{lim}}\nolimits \underline{I/I^ n}(U) \to \mathop{\mathrm{lim}}\nolimits \underline{\Lambda /I^ n}(U) \to \mathop{\mathrm{lim}}\nolimits \underline{\Lambda /I}(U) \to 0

see Homology, Lemma 12.31.3. Thus we see that \underline{\Lambda }^\wedge /\underline{I}^\wedge = \underline{\Lambda /I}. Since

I \underline{\Lambda }^\wedge \subset \underline{I} \cdot \underline{\Lambda }^\wedge \subset \underline{I}^\wedge

it suffices to show that I \underline{\Lambda }^\wedge (U) = \underline{I}^\wedge (U) for all U. Choose generators I = (f_1, \ldots , f_ r). For every n we obtain a short exact sequence

0 \to K_ n/(I^ n)^{\oplus r} \to (\Lambda /I^ n)^{\oplus r} \xrightarrow {(f_1, \ldots , f_ r)} I/I^{n + 1} \to 0

where K_ n = \{ (x_1, \ldots , x_ r) \in \Lambda ^{\oplus r} \mid \sum x_ i f_ i \in I^{n + 1}\} . We obtain short exact sequences

0 \to \underline{K_ n/(I^ n)^{\oplus r}}(U) \to \underline{(\Lambda /I^ n)^{\oplus r}}(U) \to \underline{I/I^{n + 1}}(U) \to 0

A calculation shows K_ n = K + (I^ n)^{\oplus r}, hence the transition maps K_{n + 1}/(I^{n + 1})^{\oplus r} \to K_ n/(I^ n)^{\oplus r} are surjective. Hence the system of modules on the left hand side has surjective transition maps and a fortiori has ML. Thus we see that (f_1, \ldots , f_ r) : (\underline{\Lambda }^\wedge )^{\oplus r}(U) \to \underline{I}^\wedge (U) is surjective by Homology, Lemma 12.31.3 which is what we wanted to show. \square

Lemma 18.42.5. Let \mathcal{C} be a site. Let \Lambda be a ring and let M be a \Lambda -module. Assume \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) is not the empty topos. Then

  1. \underline{M} is a finite type sheaf of \underline{\Lambda }-modules if and only if M is a finite \Lambda -module, and

  2. \underline{M} is a finitely presented sheaf of \underline{\Lambda }-modules if and only if M is a finitely presented \Lambda -module.

Proof. Proof of (1). If M is generated by x_1, \ldots , x_ r then x_1, \ldots , x_ r define global sections of \underline{M} which generate it, hence \underline{M} is of finite type. Conversely, assume \underline{M} is of finite type. Let U \in \mathcal{C} be an object which is not sheaf theoretically empty (Sites, Definition 7.42.1). Such an object exists as we assumed \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) is not the empty topos. Then there exists a covering \{ U_ i \to U\} and finitely many sections s_{ij} \in \underline{M}(U_ i) generating \underline{M}|_{U_ i}. After refining the covering we may assume that s_{ij} come from elements x_{ij} of M. Then x_{ij} define global sections of \underline{M} whose restriction to U generate \underline{M}.

Assume there exist elements x_1, \ldots , x_ r of M which define global sections of \underline{M} generating \underline{M} as a sheaf of \underline{\Lambda }-modules. We will show that x_1, \ldots , x_ r generate M as a \Lambda -module. Let x \in M. We can find a covering \{ U_ i \to U\} _{i \in I} and f_{i, j} \in \underline{\Lambda }(U_ i) such that x|_{U_ i} = \sum f_{i, j} x_ j|_{U_ i}. After refining the covering we may assume f_{i, j} \in \Lambda . Since U is not sheaf theoretically empty, there is at least one i \in I such that U_ i is not sheaf theoretically empty. Then the map M \to \underline{M}(U_ i) is injective (details omitted). We conclude that x = \sum f_{i, j}x_ j in M as desired.

Proof of (2). Assume \underline{M} is a \underline{\Lambda }-module of finite presentation. By (1) we see that M is of finite type. Choose generators x_1, \ldots , x_ r of M as a \Lambda -module. This determines a short exact sequence 0 \to K \to \Lambda ^{\oplus r} \to M \to 0 which turns into a short exact sequence

0 \to \underline{K} \to \underline{\Lambda }^{\oplus r} \to \underline{M} \to 0

by Lemma 18.42.1. By Lemma 18.24.1 we see that \underline{K} is of finite type. Hence K is a finite \Lambda -module by (1). Thus M is a \Lambda -module of finite presentation. \square


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