Recently I’ve been reading Serre’s Faisceaux Algébriques Cohérents [Ser55]. The main point of the article is to develop the theory of sheaf cohomology and prove some properties of cohomology of coherent sheaves. Serre also computes the cohomology some of coherent sheaves on projective space. I don’t really like Serre’s treatment of varieties because it’s not based on the language of schemes. (Well I guess the language of schemes didn’t exist back then.) So I am going to mix bits of Serre [Ser55] with bits of Hartshorne [Har77]. In fact, Hartshorne actually contains most of the material. Following Hartshorne, I will define sheaf cohomology as derived functors, and then show that this is equivalent to Čech cohomology in nice cases. This is opposite to what Serre did, which is to define sheaf cohomology as Čech cohomology and show that it has the desired properties.
1. Sheaf cohomology#
Sheaf cohomology \( H^\ast(X, \mathscr{F}) \) is defined from the data of a base space \( X \) and a sheaf \( \mathscr{F} \) on it. We can work with a general topological space \( X \) with a sheaf \( \mathscr{F} \) of abelian groups on it, but we are more generally going to look at a ringed space \( (X, \mathscr{O} _ X) \) with a \( \mathscr{O} _ X \)-modules \( \mathscr{F} \).
Definition 1. A ringed space \( (X, \mathscr{O} _ X) \) is a topological space \( X \) with a sheaf of rings \( \mathscr{O} _ X \). An \( \mathscr{O} _ X \)-module is a sheaf of abelian groups \( \mathscr{F} \) on \( X \) along with the data of, for each open \( U \subseteq X \), an \( \mathscr{O} _ X(U) \)-module structure on \( \mathscr{F}(U) \) such that the restriction maps on \( \mathscr{F} \) and the restriction maps on \( \mathscr{O} _ X \) behave well with respect to the module structure.
Proposition 2. The category of \( \mathscr{O} _ X \)-modules is an abelian category, which we denote by \( \mathsf{Mod} _ {\mathscr{O} _ X} \).
For sheaves, everything is defined locally and there could be local-to-global issues. For instance, if \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) is a short exact sequence of \( \mathscr{O} _ X \)-modules, we only have a left exact sequence \[ \displaystyle 0 \rightarrow \Gamma(X, \mathscr{F}) \rightarrow \Gamma(X, \mathscr{G}) \rightarrow \Gamma(X, \mathscr{H}) \] of \( \Gamma(X, \mathscr{O} _ X) \)-modules. A natural thing to do is then to define the right derived functors, which is what sheaf cohomology is. This is similar to group cohomology \( H^i(G, A) \); in fact, I’m not sure but I think group cohomology is isomorphic to \( H^i(BG, EG \times _ G A) \) where \( EG \times _ G A \), which is an \( A \)-covering of \( BG \), is considered as a locally constant sheaf.
Proposition 3 (Hartshorne III.2.2). For every commutative ring \( R \), the category of \( R \)-modules have enough injectives. For every ringed space \( (X, \mathscr{O} _ X) \), the category of \( \mathscr{O} _ X \)-modules have enough injectives.
Proof. An \( R \)-module \( M \) is injective when every \( R \)-linear map \( I \rightarrow M \) extends to \( R \rightarrow M \) for every ideal \( I \subseteq R \). (Look up Baer’s crietrion.) So given arbitrary module \( M \), we consider the pushout
Now consider an arbitrary \( \mathscr{O} _ X \)-module \( \mathscr{F} \). For each \( x \in X \), find an injection \( \mathscr{F} _ x \hookrightarrow J _ x \) into an injective \( \mathscr{O} _ {X,x} \)-module, consider the skyscraper sheaves \( i _ {x,\ast} J _ x \) and take the product \( \mathscr{J} = \prod _ {x \in X}^{} i _ {x,\ast} J _ x \). Then \[ \displaystyle \mathrm{Hom} _ {\mathscr{O} _ X}(\mathscr{G}, \mathscr{J}) \cong \prod _ {x \in X}^{} \mathrm{Hom} _ {\mathscr{O} _ X}(\mathscr{G}, i _ {x,\ast} J _ x) \cong \prod _ {x \in X}^{} \mathrm{Hom} _ {\mathscr{O} _ {X,x}}(\mathscr{G} _ x, J _ x) \] shows that \( \mathscr{J} \) is an injective object. ▨
This allows us to construct injective resolution of \( \mathscr{O} _ X \)-modules, and thus define right derived functors of \( \Gamma(X, -) \).
Definition 4. For \( \mathscr{F} \) an \( \mathscr{O} _ X \)-module, its sheaf cohomology \( H^i(X, \mathscr{F}) \) is defined as the \( i \)-th right derived functors of \( \Gamma(X, -) : \mathsf{Mod} _ {\mathscr{O} _ X} \rightarrow \mathsf{Mod} _ {\Gamma(X, \mathscr{O} _ X)} \).
So, for instance, a short exact sequence \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) will induce a long exact sequence \[ \displaystyle 0 \rightarrow \Gamma(X, \mathscr{F}) \rightarrow \Gamma(X, \mathscr{G}) \rightarrow \Gamma(X, \mathscr{H}) \rightarrow H^1(X, \mathscr{F}) \rightarrow H^1(X, \mathscr{G}) \rightarrow \cdots. \]
2. Quasi-coherent algebraic sheaves#
Our goal is to compute cohomology of sheaves over schemes, in particular Noetherian schemes. If \( X \) is a scheme rather than an arbitrary ringed space, there is a special class of sheaves that are more natural to look at.
Definition 5. Let \( M \) be a module over a ring \( A \). Then we can construct a \( \mathscr{O} _ {\mathrm{Spec} A} \)-module \( \tilde{M} \), defined by \( \Gamma(\mathrm{Spec} A _ f, \tilde{M}) = M _ f \) for all \( f \in A \). This is indeed a sheaf with \( \Gamma(\mathrm{Spec} A, \tilde{M}) \cong M \).
If we restrict \( \tilde{M} \) to \( \mathrm{Spec} A _ f \), then the resulting sheaf is going to be \( \tilde{M} _ f \), where \( M _ f \) is the localization of \( M \) at \( f \).
Proposition 6 (Hartshorne II.5.4). Let \( A \) be a ring and suppose \( \bigcup _ {i} \mathrm{Spec} A _ {f _ i} = \mathrm{Spec} A \). Let \( \mathscr{F} \) be an \( \mathscr{O} _ {\mathrm{Spec} A} \)-module and assume that \( \mathscr{F} \vert _ {\mathrm{Spec} A _ {f _ i}} \cong \tilde{M _ i} \) for \( A _ {f _ i} \)-modules \( M _ i \).
Proof. We are going to set \( M = \Gamma(\mathrm{Spec} A, \mathscr{F}) \). Then we get maps from \( M _ {f _ i} = \Gamma(\mathrm{Spec} A, \mathscr{F}) _ {f _ i} \) to \( M _ i = \Gamma(\mathrm{Spec} A _ {f _ i}, \mathscr{F}) \) and need to show that they are isomorphisms. For injectivity, suppose \( s \in \Gamma(\mathrm{Spec} A, \mathscr{F}) \) restricts to \( 0 \) on \( \mathrm{Spec} A _ {f _ i} \). Then the isomorphisms \( (M _ i) _ {f _ j} \cong (M _ j) _ {f _ i} \) show that there is a sufficiently large \( N \) such that \( f _ i^N s \) restricted to \( f _ j \) is zero for all \( j \). This shows that \( f _ i^N s = 0 \in M \). For surjectivity, let \( s \in \Gamma(\mathrm{Spec} A _ {f _ i}, \mathscr{F}) \) be any section on \( \mathrm{Spec} A _ {f _ i} \). Then \( s \) restricts to a section in \( \Gamma(\mathrm{Spec} A _ {f _ i f _ j}, \mathscr{F}) \cong (M _ j) _ {f _ i} \) and so there exists a sufficiently large \( N \) such that \( f _ i^N s \) agrees \( s _ j \in M _ j \) on \( \mathrm{Spec} A _ {f _ i f _ j} \). To glue these sections, we still need to check that \( s _ j \) and \( s _ k \) agree on \( \mathrm{Spec} A _ {f _ j f _ k} \). This is not necessarily true, but they at least agree on \( \mathrm{Spec} A _ {f _ i f _ j f _ k} \) so there exists a sufficiently large \( M \) such that \( f _ i^M (s _ j - s _ k) = 0 \) on \( \mathrm{Spec} A _ {f _ j f _ k} \). So we can glue the sections \( f _ i^M s _ j \) to get a global section extending \( f _ i^{N+M} s \). ▨
This shows that the property of a sheaf being "isomorphic to some \( \tilde{M} \)" is affine-local. (See Vakil [Vak17] 5.3.2.)
Definition 7. Let \( X \) be a scheme. An \( \mathscr{O} _ X \)-module \( \mathscr{F} \) is called quasi-coherent if it satisfies the following equivalent properties:
- (a) For each affine open \( \mathrm{Spec} A \cong U \subseteq X \), there exists an \( A \)-module \( M \) with an isomorphism \( \mathscr{F} \vert _ U \cong \tilde{M} \).
- (b) There exists an affine open cover \( \{ \mathrm{Spec} A _ i \cong U _ i \} \) of \( X \) such that there exist \( A _ i \)-modules \( M _ i \) with isomorphisms \( \mathscr{F} \vert _ {U _ i} \cong \tilde{M} _ i \).
We denote the full subcategory of \( \mathsf{Mod} _ {\mathscr{O} _ X} \) consisting of quasi-coherent sheaves by \( \mathsf{QCoh} _ {\mathscr{O} _ X} \).
Corollary 8. Let \( X = \mathrm{Spec} A \). There is an equivalence of categories between \( \mathsf{QCoh} _ {\mathscr{O} _ X} \) and \( \mathsf{Mod} _ A \) given by \( \Gamma(X, -) \) and \( M \mapsto \tilde{M} \).
Proposition 9. Let \( X \) be a scheme.
- (a) The kernel, cokernel, and image of a map between quasi-coherent \( \mathscr{O} _ X \)-modules are quasi-coherent.
- (b) If \( \mathscr{F} \) and \( \mathscr{G} \) are quasi-coherent \( \mathscr{O} _ X \)-modules, then \( \mathscr{F} \oplus \mathscr{G} \) and \( \mathscr{F} \otimes _ {\mathscr{O} _ X} \mathscr{G} \) (but not necessarily \( \mathscr{H}om _ {\mathscr{O} _ X}(\mathscr{F}, \mathscr{G}) \)) are quasi-coherent.
Proof. Simply note that kernel, cokernel, image, direct sum, and tensor product all commute with \( M \mapsto \tilde{M} \). To show that tensoring commutes with \( M \mapsto \tilde{M} \), note that \( \mathrm{Hom} _ {\mathscr{O} _ {\mathrm{Spec} A}}(\tilde{M}, \mathscr{H}) \cong \mathrm{Hom} _ A(M, \Gamma(\mathrm{Spec} A, \mathscr{H})) \) for an arbitrary \( \mathscr{O} _ {\mathrm{Spec} A} \)-module \( \mathscr{H} \). ▨
Corollary 10. Quasi-coherent sheaves on \( X \) form an abelian category \( \mathsf{QCoh} _ {\mathscr{O} _ X} \).
Actually quasi-coherent sheaves can be defined on arbitrary ringed spaces.
Definition 11. Let \( (X, \mathscr{O} _ X) \) be a ringed space. An \( \mathscr{O} _ X \)-module is said to be quasi-coherent if it locally looks like the cokernel of some \( \mathscr{O} _ X \vert _ U^{\oplus J} \rightarrow \mathscr{O} _ X \vert _ U^{\oplus I} \) where \( I \) and \( J \) are indexing sets that are possibly infinite.
Proposition 12. The two definitions of quasi-coherence agree on an arbitrary scheme \( X \).
Proof. Every module is a cokernel of a map between free modules, so the first definition implies the second. Conversely, the second definition implies that the sheaf is locally a cokernel of a map of quasi-coherent modules, in the first sense, and hence quasi-coherent. Because we have shown that the first definition works locally, we see that the second implies the first. ▨
The reason I haven’t initially defined quasi-coherent sheaves on arbitrary ringed spaces is because I’m not sure if the category \( \mathsf{QCoh} _ {\mathscr{O} _ X} \) behaves well for a ringed space \( (X, \mathscr{O} _ X) \).
3. Comparison of cohomologies#
We want to compute cohomology of coherent sheaves, or more generally quasi-coherent sheaves. But the problem is that injective objects in \( \mathsf{Mod} _ {\mathscr{O} _ X} \) are too large to carry out computations. But if we work with quasi-coherent sheaves, there are fewer objects, and hence smaller injective objects. For instance, let’s try to compute the cohomology of a quasi-coherent sheave on an affine scheme. If \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{J}^\bullet \) is an injective resolution by quasi-coherent sheaves on \( X = \mathrm{Spec} A \), then it corresponds to an injective resolution of \( \Gamma(X, \mathscr{F}) \) by \( A \)-modules. So we would like to say that \( H^i(X, \mathscr{F}) = 0 \) for any quasi-coherent \( \mathscr{F} \). To make this kind of an argument, we should we have defined cohomology as the derived functors of \( \Gamma : \mathsf{Mod} _ {\mathscr{O} _ X} \rightarrow \mathsf{Mod} _ {\Gamma(X, \mathscr{O} _ X)} \). Even though \( \mathsf{QCoh} _ {\mathscr{O} _ X} \) is a full subcategory of \( \mathsf{Mod} _ {\mathscr{O} _ X} \), there is no guarantee that the derived functors will match up. But the derived functors actually do match up in the case of Noetherian schemes, and this is what we are trying to show.
Proposition 13. For \( X \) a scheme, the category \( \mathsf{QCoh} _ {\mathscr{O} _ X} \) has enough injectives, so that the right derived functors of \( \Gamma(X, -) \) makes sense.
Proof. I don’t know how to prove it, and I’m happy to accept this. Read Stacks 27.23 or Akhil Mathew’s post if you’re curious. ▨
So we have a lot of abelian categories of sheaves we could possibly work with. There is also the sheaves of abelian groups over \( X \), written as \( \mathsf{Ab} _ X \), which is the same as \( \underline{\mathbb{Z}} _ X \)-modules where \( \underline{\mathbb{Z}} _ X \) is the constant sheaf on \( X \). Then there are forgetful functors \[ \displaystyle \mathsf{QCoh} _ {\mathscr{O} _ X} \hookrightarrow \mathsf{Mod} _ {\mathscr{O} _ X} \rightarrow \mathsf{Ab} _ X. \] Each of them have enough injectives, and hence induce right derived functors of the global section functor. (\( \mathsf{Ab} _ X \) has enough injectives because it is an instance of \( \mathsf{Mod} \).) Let us call them \( H _ \mathsf{Q} \), \( H _ \mathsf{M} = H \), and \( H _ \mathsf{A} \) respectively.
Theorem 14 (Hartshorne III.2.6). Let \( (X, \mathscr{O} _ X) \) be a ringed space. Then \( H _ \mathsf{A}^i(X, \mathscr{F}) \cong H _ \mathsf{M}^i(X, \mathscr{F}) \) for every \( \mathscr{O} _ X \)-module \( \mathscr{F} \).
Proof. This is too long, but here is the sketch. A sheaf is called flabby or flasque if all the restriction maps are surjective. Flasque shaves have the following property: (Hartshorne Ex. II.1.16 and Lem. III.2.4)
- (a) If \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) is exact and \( \mathscr{F} \) is flasque, then \( 0 \rightarrow \Gamma(U, \mathscr{F}) \rightarrow \Gamma(U, \mathscr{G}) \rightarrow \Gamma(U, \mathscr{H}) \rightarrow 0 \) is exact for all open \( U \).
- (b) If \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{G} \rightarrow \mathscr{H} \rightarrow 0 \) is exact and \( \mathscr{F} \) and \( \mathscr{G} \) are flasque, then \( \mathscr{H} \) is flasque as well.
- (c) Any injective object in \( \mathsf{Mod} _ {\mathscr{O} _ X} \) is flasque, and hence setting \( \mathscr{O} _ X = \underline{\mathbb{Z}} _ X \) shows that any injective object in \( \mathsf{Ab} _ X \) is also flasque.
Using these properties, we see that flasque sheaves are \( \mathscr{H} _ \mathsf{A} \)-acyclic. To see this, take any flasque \( \mathscr{F} \in \mathsf{Ab} _ X \) and consider an embedding into an injective object in \( \mathsf{Ab} _ X \). If we write \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{J} \rightarrow \mathscr{G} \rightarrow 0 \), then we get \[ \displaystyle 0 \rightarrow \Gamma(X, \mathscr{F}) \rightarrow \Gamma(X, \mathscr{J}) \rightarrow \Gamma(X, \mathscr{G}) \rightarrow H _ \mathsf{A}^1(X, \mathscr{F}) \rightarrow 0 \] and \( H _ {\mathsf{A}}^{i}(X, \mathscr{G}) \cong H _ \mathsf{A}^{i+1}(X, \mathscr{F}) \) for all \( i \ge 1 \). But because \( \mathscr{F} \) is flasque, we get \( H _ \mathsf{A}^1(X, \mathscr{F}) = 0 \) by (a). By (c), \( \mathscr{J} \) is flasque and hence \( \mathscr{G} \) is flasque by (b). This inductively shows that \( H _ \mathsf{A}^i \) of any flasque sheaf is \( 0 \).
Now given any \( \mathscr{O} _ X \)-module \( \mathscr{F} \), we look at its injective resolution \( 0 \rightarrow \mathscr{F} \rightarrow \mathscr{J}^\bullet \). By (c), each \( \mathscr{J}^i \) is flasque, and hence \( H _ \mathsf{A} \)-acyclic. This means that we can compute \( H _ \mathsf{A}(X, \mathscr{F}) \) using this resolution. \( H _ \mathsf{M}(X, \mathscr{F}) \) is defined using this resolution, and hence the two agree. ▨
Actually Hartshorne defines sheaf cohomology as \( H _ \mathsf{A} \) instead of \( H _ \mathsf{M} \), but as we can see, it doesn’t matter.
Theorem 15 (Hartshorne Ex. III.3.6). Let \( X \) be a Noetherian scheme. Then \( H _ \mathsf{Q}^i(X, \mathscr{F}) \cong H _ \mathsf{A}^i(X, \mathscr{F}) \cong H _ \mathsf{M}^i(X, \mathscr{F}) \) for every quasi-coherent \( \mathscr{O} _ X \)-module \( \mathscr{F} \).
Proof. It suffices to prove that every injective object \( \mathscr{F} \) in \( \mathsf{QCoh} _ {\mathscr{O} _ X} \) is flasque, i.e., that \( \Gamma(V, \mathscr{F}) \rightarrow \Gamma(U, \mathscr{F}) \) is surjective for \( V \supseteq U \). It is sufficient to assume that \( V = X \), because surjectivity of the composition \( \Gamma(X, \mathscr{F}) \rightarrow \Gamma(V, \mathscr{F}) \rightarrow \Gamma(U, \mathscr{F}) \) implies what we want.
Set \( \mathscr{I} \subseteq \mathscr{O} _ X \) to be the quasi-coherent sheaf of ideals corresponding to the reduced closed subscheme for \( X \setminus U \). Then the embedding \( \mathscr{I} \hookrightarrow \mathscr{O} _ X \) is an isomorphism on \( U \), and moreover, \( \mathscr{I}^n \hookrightarrow \mathscr{O} _ X \) is also an isomorphism on \( U \). (Here, \( \mathscr{I}^n \) not \( \mathscr{I}^{\oplus n} \) but the sheaf that is locally \( I^n = I \cdots I \).) Then we have a restriction map \[ \displaystyle \varinjlim _ n \mathrm{Hom} _ {\mathscr{O} _ X} (\mathscr{I}^n, \mathscr{F}) \rightarrow \mathrm{Hom} _ {\mathscr{O} _ X \vert _ U} (\mathscr{O} _ X \vert _ U, \mathscr{F} \vert _ U) \cong \Gamma(U, \mathscr{F}). \]
My claim is that this map is an isomorphism if \( X \) is Noetherian. (See Stacks 29.10.4.) We first show that it is injective. For a map \( \mathscr{I}^n \rightarrow \mathscr{F} \) is \( 0 \) on \( U \), then affine-locally on \( \mathrm{Spec} A \) it looks like an \( A \)-linear map \( I^n \rightarrow M \) with any \( m \) in the image satisfying \( \mathrm{supp} m \subseteq V(I) \). But \( \mathrm{supp} m = V(\mathrm{Ann} m) \) and so \( I \subseteq \sqrt{\mathrm{Ann} m} \). The ideal is finitely generated because \( A \) is Noetherian, and then looking at the generators show that \( I^N \subseteq \mathrm{Ann} m \) for a sufficiently large \( N \). Then \( I^{n+N} \hookrightarrow I^n \rightarrow M \) is the zero map. Because \( X \) is quasi-compact we can do this for each affine cover and this shows that it is \( 0 \) in the colimit.
For surjectivity, let \( V = \mathrm{Spec} A \subseteq X \) with \( \mathscr{I} \vert _ V \cong \tilde{I} \) and \( \mathscr{F} \vert _ V \cong \tilde{M} \). Because \( A \) is Noetherian, \( I \) is finitely generated and so write \( I = (f _ 1, \ldots, f _ k) \). A section \( m \in \Gamma(U \cap V, \mathscr{F}) \) corresponds to a set of elements \( f _ i^{-\alpha} m _ i \in M _ {f _ i} \) such that \( (f _ i f _ j)^\beta (f _ j^\alpha m _ i - f _ i^\alpha m _ j) = 0 \in M \). We may replace \( M \) with the \( A \)-module generated by \( m _ i \), and hence assume that \( M \) is finitely generated. If we define \( M _ n \) as the submodule of \( M \) annihilated by \( I^n \), then \( M _ n \) is an ascending chain in \( M \) and hence converges to some submodule \( N \subseteq M \). By the Artin–Rees lemma, there exists an \( \gamma \) such that \( I^n M \cap N = I^{n-\gamma} (I^\gamma M \cap N) \subseteq I^{n-\gamma} N \) for all \( n \ge \gamma \). So for sufficienlty large \( \delta \), we get \( I^\delta M \cap N = 0 \), which means that anything annihilated by \( I^n \) is always \( 0 \). Increasing \( \alpha \) if necessary, we may supposed that \( m _ i \in I^\delta M \). Now we define a map \[ \displaystyle I^{k(\alpha+\beta)} \subseteq (f _ 1^{\alpha+\beta}, \ldots, f _ n^{\alpha+\beta}) \rightarrow I^\delta M; \quad \sum _ {i}^{} a _ i f _ i^{\alpha+\beta} \mapsto \sum _ {i}^{} a _ i f _ i^\beta m _ i. \] This is well-defined because if \( \sum _ {i}^{} a _ i f _ i^{\alpha+\beta} = 0 \) then \[ \displaystyle f _ j^{\alpha+\beta} \sum _ {i}^{} a _ i f _ i^\beta m _ i = \sum _ {i}^{} a _ i f _ i^{\alpha+\beta} f _ j^\beta m _ j = 0 \] and so \( \sum _ {i}^{} a _ i f _ i^{\alpha+\beta} = 0 \) by our condition that anything annihilated by \( I^n \) is \( 0 \). This map \( I^{k(\alpha+\beta}) \rightarrow I^\delta M \subseteq M \) corresponds to \( m \).
This is for one affine chart. Do this for every affine cover, but then the maps might not agree. But \( X \) Noetherian implies that it is quasi-separated, and so we only need to worry about a finite number of affine opens. By injectivity, we can make them agree by passing to a higher power of \( I \).
So a section of \( \mathscr{F} \) over \( U \) corresponds to a morphism \( \mathscr{I}^n \rightarrow \mathscr{F} \). Because \( \mathscr{F} \) is injective, there is an extension to \( \mathscr{O} _ X \rightarrow \mathscr{F} \). This is a global section that extends the original section on \( U \). ▨
Corollary 16. If \( X = \mathrm{Spec} A \) is affine with \( A \) Noetherian, and \( \mathscr{F} \) is a quasi-coherent sheaf, then \( H^i(X, \mathscr{F}) = 0 \).
Proof. Compute in the quasi-coherent category. The global sections functor is exact. ▨
References#
[Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1997, Graduate Texts in Mathematics, No. 52.
[Ser55] Jean-Pierre Serre, Faisceaux Algébriques Cohérents, Ann. of Math. (2) 61 (1955), 197–278.
[Vak17] Ravi Vakil, The rising sea—foundations of algebraic geometry, 2017.