It’s quite awkward to be introducing coherent sheaves at this point, when the title of the paper is "Coherent algebraic sheaves". So far, we’ve mostly gotten away with quasi-coherent sheaves. Serre introduces the notion of coherent sheaves, which contain some idea of finite generation in addition to being an quasi-coherent sheaf. But we want the class of sheaves to be form an abelian category. Finitely generated modules over a ring generally don’t form an abelian category, so we need an alternative notion.
1. Coherent sheaves#
Definition 1. An \( R \)-module is \( M \) called finitely generated if there exists an integer \( n \) with a surjection \( R^n \rightarrow M \). An \( R \)-module \( M \) is called coherent if it is finitely generated, and the cokernel of every (not necessarily surjective) map \( R^n \rightarrow M \) is finitely generated.
This is equivalent to the seemingly weaker statement that every finitely generated submodule is finitely presented. If \( R \) is Noetherian, a module is coherent if and only if it is finitely generated.
Proposition 2. If \( 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \) is a short exact sequence of \( R \)-modules, and two of them is coherent, the last one is also coherent.
Proof. Suppose \( A \) and \( B \) are coherent. Because \( B \) is finitely generated, \( C \) is too. For an arbitrary map \( R^n \rightarrow C \), we can lift it to \( R^n \rightarrow B \). Also, there is a surjection \( R^k \rightarrow A \) which induces \( R^k \rightarrow B \). Put them together to get a map \( R^n \oplus R^k \rightarrow B \), and continue it to an exact sequence \( R^m \rightarrow R^{n+k} \rightarrow B \) using that \( B \) is coherent. Then \( R^m \rightarrow R^n \rightarrow C \) is exact after diagram chasing.
Suppose that \( A \) and \( C \) are coherent. Then the generators of \( A \) and any inverse images of generators of \( C \) together will be generators for \( B \), and hence \( B \) is finitely generated. For the other condition, consider an arbitrary \( R^n \rightarrow B \), which induces a map \( R^n \rightarrow C \). Extend it to an exact \( R^k \rightarrow R^n \rightarrow C \) and we get \( R^k \rightarrow A \) because \( A \) is the kernel of \( B \rightarrow C \). Then extend to \( R^m \rightarrow R^k \rightarrow A \) and then \( R^m \rightarrow R^n \rightarrow B \) will be exact.
Suppose that \( B \) and \( C \) are coherent. If we look at a surjection \( R^n \twoheadrightarrow B \), induce to \( R^n \twoheadrightarrow C \) and extend to \( R^m \rightarrow R^n \rightarrow C \), then the induced map \( R^m \rightarrow A \) will be surjective. For an arbitrary map \( R^n \rightarrow A \), the kernel of \( R^n \rightarrow A \) is equal to \( R^n \rightarrow B \), and hence finitely generated. This shows that \( A \) is coherent. ▨
Proposition 3. The kernel, cokernel, and image of a map between coherent \( R \)-modules are coherent.
Proof. Let \( \varphi : A \rightarrow B \) be the map. Because \( A \) is finitely generated, \( \mathrm{im} f \) is finitely generated. Because \( \mathrm{im} f \hookrightarrow B \) is injective, the kernel of \( R^n \rightarrow \mathrm{im} f \) is the same as the kernel of \( R^n \rightarrow B \). This shows that \( \mathrm{im} f \) is coherent. Then note that \( 0 \rightarrow \ker f \rightarrow A \rightarrow \mathrm{im} f \rightarrow 0 \) and \( 0 \rightarrow \mathrm{im} f \rightarrow B \rightarrow \mathrm{coker} f \rightarrow 0 \) are short exact. ▨
Proposition 4. If \( A \) and \( B \) are coherent \( R \)-modules, then \( A \oplus B \), \( A \otimes B \), and \( \mathrm{Hom}(A, B) \) are also coherent.
Proof. For the direct sum, just note that \( 0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0 \) is short exact. For the tensor product, note that \( A \) is the cokernel of some \( R^m \rightarrow R^n \). Tensoring with \( B \) gives \( A \otimes B \) as the cokernel of some \( B^m \rightarrow B^n \), where \( B^n \) and \( B^m \) are both coherent. For \( \mathrm{Hom} \), similarly note that taking \( \mathrm{Hom}(-, B) \) to \( R^m \rightarrow R^n \rightarrow A \rightarrow 0 \) gives \( \mathrm{Hom}(A, B) \) as a kernel of \( B^n \rightarrow B^m \). ▨
Now coherent sheaves are supposed to be locally modeled on coherent modules.
Definition 5. An \( \mathscr{O} _ X \)-module \( \mathscr{F} \) is called finitely generated if for each \( x \in X \), there is a neighborhood \( U \), and integer \( n \) with a surjective map \( \mathscr{O} \vert _ U^n \rightarrow \mathscr{F} \vert _ U \). An \( \mathscr{O} _ X \)-module \( \mathscr{F} \) is called coherent if it is finitely generated, and for every open \( U \), the cokernel of every (not necessarily surjective) map \( \mathscr{O} \vert _ U^n \rightarrow \mathscr{F} \) is finitely generated.
Then we have the analogues of all the properties we have shown above.
Proposition 6 (Serre n13,14). Let \( (X, \mathscr{O} _ X) \) be a ringed space.
- (a) For a short exact sequence \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) of \( \mathscr{O} _ X \)-modules, if two of them are coherent, then the last one is coherent as well.
- (b) The kernel, cokernel, and image of a map between coherent \( \mathscr{O} _ X \)-modules are coherent.
- (c) If \( \mathscr{F} \) and \( \mathscr{G} \) are coherent \( \mathscr{O} _ X \)-modules, then \( \mathscr{F} \oplus \mathscr{G} \), \( \mathscr{F} \otimes _ {\mathscr{O} _ X} \mathscr{G} \), \( \mathscr{H}\mathit{om} _ {\mathscr{O} _ X}(\mathscr{F}, \mathscr{G}) \) are also coherent.
Proof. Proceed similarly but locally. ▨
Corollary 7. The category of coherent \( \mathscr{O} _ X \)-sheaves form an abelian category, which we denote by \( \mathsf{Coh} _ {\mathscr{O} _ X} \).
2. Coherent algebraic sheaves#
As the name suggests, coherent sheaves over schemes are quasi-coherent. Indeed, every coherent sheaf \( \mathscr{F} \) is locally a cokernel of a map \( \mathscr{O} \vert _ U^m \rightarrow \mathscr{O} \vert _ U^n \). Because both \( \mathscr{O} \vert _ U^m \) and \( \mathscr{O} \vert _ U^n \) are clearly quasi-coherent, \( \mathscr{F} \vert _ U \) is quasi-coherent, and because quasi-coherence is local we conclude that \( \mathscr{F} \) is quasi-coherent. But there is more to coherence than quasi-coherence.
Proposition 8. If \( M \) is a coherent \( A \)-module, then \( M _ f \) is a coherent \( A _ f \)-module for each \( f \in A \). Conversely, if \( (f _ 1, \ldots, f _ n) = (1) \) and each \( M _ {f _ i} \) is a coherent \( A _ {f _ i} \)-module, then \( M \) is a coherent \( A \)-module.
Proof. First suppose that \( M \) is a coherent \( A \)-module. The generators of \( M \) as an \( A \)-module are generators of \( M _ f \) as an \( A _ f \)-module, so \( M _ f \) is finitely generated. For an \( A _ f \)-linear map \( \varphi : A _ f^n \rightarrow M _ f \), there exists an large \( N \) such that \( f^N \varphi \) comes from an \( A \)-linear map \( \psi : A^n \rightarrow M \). Because \( M \) is coherent, we can extend to an exact \( A^m \rightarrow A^n \xrightarrow{\psi} M \), and localizing gives \( A _ f^m \rightarrow A _ f^n \rightarrow M _ f \). This map \( A _ f^n \rightarrow M _ f \) is actually \( \varphi \) times \( f^N \), but \( f^N \) is invertible so \( \ker \varphi \) and \( \ker (f^N \varphi) \) are equal.
Now suppose that each \( M _ {f _ i} \) is a coherent \( A _ {f _ i} \)-module. Pick generators \( m _ {i,\alpha} \) of \( M _ {f _ i} \) as an \( A _ {f _ i} \)-module such that \( m _ {i,\alpha} \in M \). This is possible because \( f _ i \) is a unit. Then \( m _ {i,\alpha} \) generate \( f _ i^{N _ i} M \) as \( A \)-modules, where \( N _ i \) is sufficiently large. Since \( (f _ 1, \ldots, f _ n) = (1) \), the union \( { m _ {i,\alpha} } \) generate \( M \) as an \( A \)-module. Now consider an arbitrary \( \varphi : A^n \rightarrow M \). We then have \( (\ker \varphi) _ {f _ i} = \ker (\varphi _ {f _ i}) \) and because \( M _ {f _ i} \) are coherent, \( (\ker \varphi) _ {f _ i} \) are finitely generated. By the previous argument, \( \ker \varphi \) is finitely generated. ▨
Proposition 9. Let \( X = \mathrm{Spec} A \). A quasi-coherent \( \mathscr{O} _ X \)-module \( \tilde{M} \) is coherent if and only if \( M \) is a coherent \( A \)-module.
Proof. First suppose that \( M \) is coherent. It is clear that \( \tilde{M} \) is finitely generated because a surjection \( A^n \rightarrow M \) induces a surjection \( \mathscr{O} _ X^n \rightarrow \tilde{M} \). Now consider an arbitrary \( \mathscr{O} \vert _ U^n \rightarrow \tilde{M} \vert _ U \). Because finite generation is a local notion, we may take a smaller neighborhood \( V = \mathrm{Spec} A _ f \subseteq U \) and show that the kernel of \( \mathscr{O} \vert _ V^n \rightarrow \tilde{M} \vert _ V \cong \tilde{M} _ f \) is finitely generated. This is true because \( M _ f \) is coherent over \( A _ f \).
Now suppose that \( \tilde{M} \) is coherent. Because it is locally finitely generated, we see that there are \( f _ 1, \ldots, f _ n \) with \( (f _ 1, \ldots, f _ n) = (1) \) and \( M _ {f _ i} \) finitely generated over \( A _ {f _ i} \). But the proof of the previous proposition shows that \( M _ {f _ i} \) finitely generated implies \( M \) finitely generated. To show that \( M \) is coherent, consider any \( A^n \rightarrow M \), which induces a map \( \mathscr{O}^n \rightarrow \tilde{M} \). The kernel is finitely generated (as a sheaf) and quasi-coherent, and thus by the previous argument we see that it is \( \tilde{N} \) for some finitely generated \( A \)-module \( N \). Here \( N \) is the kernel of \( A^n \rightarrow M \). ▨
Corollary 10. If \( X \) is a scheme and \( \mathscr{F} \) is an \( \mathscr{O} _ X \)-module, the following conditions are equivalent:
- (a) \( \mathscr{F} \) is coherent.
- (b) For each affine open \( \mathrm{Spec} A \cong U \subseteq X \), the sheaf \( \mathscr{F} \vert _ U \) is isomorphic to \( \tilde{M} \) where \( M \) is a coherent \( A \)-module.
- (c) There exists an affine open cover \( { \mathrm{Spec} A _ i \cong U _ i } \) of \( X \) such that each \( \mathscr{F} \vert _ {U _ i} \) is isomorphic to \( \tilde{M _ i} \) where \( M \) is a coherent \( A _ i \)-module.
This is why we have defined coherent sheaves in an ad-hoc way before, on \( \mathbb{P} _ k^n \).
Corollary 11. If \( X \) is a locally Noetherian scheme and \( \mathscr{F} \) is an \( \mathscr{O} _ X \)-module, the following conditions are equivalent:
- (a) \( \mathscr{F} \) is coherent.
- (b) For each affine open \( \mathrm{Spec} A \cong U \subseteq X \), the sheaf \( \mathscr{F} \vert _ U \) is isomorphic to \( \tilde{M} \) where \( M \) is a finitely generated \( A \)-module.
- (c) There exists an affine open cover \( { \mathrm{Spec} A _ i \cong U _ i } \) of \( X \) such that each \( \mathscr{F} \vert _ {U _ i} \) is isomorphic to \( \tilde{M _ i} \) where \( M \) is a finitely generated \( A _ i \)-module.
References#
[Ser55] Jean-Pierre Serre, Faisceaux Algébriques Cohérents, Ann. of Math. (2) 61 (1955), 197–278.