04.06.01 · algebraic-geometry / coherent

Quasi-coherent sheaf

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II; Vakil; Grothendieck-Dieudonné EGA I; Serre FAC 1955

Intuition [Beginner]

A quasi-coherent sheaf on a scheme $X$ is a sheaf that, locally on every affine open $U = Spec (R) \subseteq X$ , looks like a module over the ring $R$ . It is the natural notion of a sheaf adapted to the algebraic structure of a scheme — sheaves whose sections behave like modules.

The slogan: on an affine scheme $Spec (R)$ , quasi-coherent sheaves correspond bijectively to $R$ -modules. Module $M$ goes to the sheaf $M$ , characterised by $M (D (f)) = M [f^{- 1}]$ . Conversely, the global sections of $M$ recover $M$ .

The class of quasi-coherent sheaves is exactly the right class of sheaves on a scheme. Coherent sheaves (a stricter subclass) are well-behaved analogues of finitely-generated modules. Vector bundles, line bundles, and structure sheaves are all quasi-coherent. The theory of cohomology and many key constructions (pullback, pushforward, tensor product) work cleanly only for quasi-coherent sheaves.

Visual [Beginner]

A scheme covered by affine pieces; on each piece, the quasi-coherent sheaf restricts to a module over the local ring.

Worked example [Beginner]

The structure sheaf $O_{X}$ is the canonical quasi-coherent sheaf. On an affine $U = Spec (R)$ , $O_{X} ∣_{U} = R$ — the sheaf associated to $R$ as an $R$ -module over itself.

Another example: for a homogeneous ideal $I \subseteq k [x_{0}, \dots, x_{n}]$ , the ideal sheaf $I_{X} \subseteq O_{P^{n}}$ of the closed subscheme $X = V_{+} (I)$ is quasi-coherent. On affine charts, it is the sheaf associated to the ideal in the local ring.

A non-example: the constant sheaf $\underline{Z}$ on a scheme. This is a sheaf, and well-defined topologically, but does not arise from any module structure on the affine charts (since modules over different rings on different affines don't naturally agree on overlaps). It is sheaf-cohomology-meaningful but not a quasi-coherent sheaf.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a scheme. A sheaf $F$ of $O_{X}$ -modules is quasi-coherent if every point $x \in X$ has an affine open neighbourhood $U = Spec (R) ∋ x$ such that $F ∣_{U} ≅ M$ for some $R$ -module $M$ , where $M$ denotes the sheaf associated to $M$ :

M (D (f)) := M [f^{- 1}] = M \otimes_{R} R [f^{- 1}], f \in R .

Equivalently: $F$ is quasi-coherent iff for every affine open $U = Spec (R) \subseteq X$ , $F ∣_{U} ≅ F (U)$ .

Equivalence of categories on affine schemes. The functors

Mod (R) (\cdot) Γ QCoh (Spec (R))

are mutually inverse equivalences. The forward functor sheafifies an $R$ -module; the backward functor takes global sections.

Properties.

Closure under standard operations: kernel, cokernel, image, direct sum, tensor product, $Hom$ (for finite presentation in the second slot).
Pullback: for a morphism $f : X \to Y$ and quasi-coherent $G$ on $Y$ , $f^{*} G$ is quasi-coherent on $X$ .
Pushforward: for quasi-compact and quasi-separated $f : X \to Y$ and quasi-coherent $F$ on $X$ , $f_{*} F$ is quasi-coherent on $Y$ .
Cohomology: on an affine scheme, quasi-coherent sheaves have zero higher cohomology (Serre's vanishing). On projective schemes, the higher cohomology is governed by the Serre formula and Hilbert polynomial.

Affine criterion. A scheme $X$ is affine iff for every quasi-coherent ideal sheaf $I \subseteq O_{X}$ , $H^{1} (X; I) = 0$ . This characterises affine schemes cohomologically.

Examples.

Structure sheaf. $O_{X}$ is quasi-coherent.
Ideal sheaves. For a closed subscheme $Y ↪ X$ , the ideal sheaf $I_{Y} \subset O_{X}$ is quasi-coherent.
Twisting sheaves on $P^{n}$ . $O (d)$ for $d \in Z$ — quasi-coherent (and even coherent).
Module sheafification. For any $R$ -module $M$ , $M$ is quasi-coherent on $Spec (R)$ .
Direct images. For a morphism $f : X \to Y$ with mild hypotheses, $f_{*} O_{X}$ is quasi-coherent.

On Proj. A graded $S$ -module $M$ has an associated quasi-coherent sheaf $M$ on $Proj (S)$ :

M (D (f)) := (M [f^{- 1}])_{0} for homogeneous f \in S_{+} .

The functor $(\cdot)$ from graded $S$ -modules to quasi-coherent sheaves on $Proj (S)$ is not an equivalence — graded modules differing only in low degrees give the same sheaf. The kernel is the torsion category of modules supported in low degrees. Quotienting out gives the equivalence

grMod (S) / Tor \sim QCoh (Proj (S)) .

Key theorem with proof [Intermediate+]

Theorem (equivalence of $R$ -modules and quasi-coherent sheaves on $Spec (R)$ ). The functor $(\cdot) : Mod (R) \to QCoh (Spec (R))$ , $M \mapsto M$ , is an equivalence of abelian categories. Its inverse is the global sections functor $Γ$ .

Proof sketch.

Step 1. $(\cdot)$ produces a quasi-coherent sheaf. By construction, $M ∣_{D (f)} = M [f^{- 1}]$ is the sheafification of the $R [f^{- 1}]$ -module $M [f^{- 1}]$ , which is the affine-cover quasi-coherent property.

Step 2. $Γ \circ (\cdot) = id$ on modules. For an $R$ -module $M$ , $Γ (Spec (R), M) = M (D (1)) = M [1^{- 1}] = M$ .

Step 3. $(\cdot) \circ Γ = id$ on quasi-coherent sheaves. For a quasi-coherent sheaf $F$ on $Spec (R)$ , set $M = F (Spec (R))$ . The natural map $M \to F$ is constructed by sending $m / f^{n} \in M [f^{- 1}]$ to the corresponding section in $F (D (f))$ (using the localisation property of quasi-coherent sheaves). It is an isomorphism on every $D (f)$ by the affine quasi-coherent property, hence on $Spec (R)$ globally.

Step 4. Exactness. Both functors preserve exact sequences (sheafification is exact on the affine cover; global sections is exact for quasi-coherent sheaves on affine schemes by Serre's vanishing). $□$

This equivalence is the fundamental dictionary of algebraic geometry: commutative algebra (modules over rings) on one side, sheaf theory (quasi-coherent sheaves on schemes) on the other. Most algebraic-geometric constructions are translations between the two languages.

Bridge. The construction here builds toward 04.06.02 (coherent sheaf), where the same data is upgraded, and the symmetry side is taken up in 04.05.03 (line bundle on a scheme). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 5 (medium, conceptual).

Why isn't the constant sheaf $\underline{Z}$ quasi-coherent on a scheme of positive dimension?

Hint

The constant sheaf does not arise from a single module on each affine.

Answer

On $Spec (R)$ for $R$ Noetherian and connected, the constant sheaf $\underline{Z}$ has all sections equal to $Z$ (one copy per connected component — by definition of constant sheaf). So $Γ (Spec (R); \underline{Z}) = Z$ .

If $\underline{Z}$ were quasi-coherent, it would equal $Z$ as a sheaf of $Z$ -modules. But $Z (D (f)) = Z [f^{- 1}]$ for $f \in R$ , whereas the constant sheaf gives $Z$ on every connected open.

For example on $Spec (Z)$ : $D (2) = Spec (Z) ∖ {(2)}$ is connected. $Z (D (2)) = Z [2^{- 1}]$ , but $\underline{Z} (D (2)) = Z$ . So $\underline{Z} \neq = Z$ , and $\underline{Z}$ is not quasi-coherent.

The constant sheaf is well-defined as a sheaf of abelian groups but doesn't have the "module-over-the-ring" property that quasi-coherence requires.

Exercise 6 (hard, proof).

State Serre's vanishing theorem on affine schemes for quasi-coherent sheaves.

Hint

Higher sheaf cohomology vanishes on affines.

Answer

Serre's vanishing theorem. Let $X = Spec (R)$ be an affine scheme and $F$ a quasi-coherent sheaf on $X$ . Then $H^{i} (X; F) = 0$ for $i \geq 1$ .

This is the foundational vanishing theorem for affine schemes. The proof uses that $M$ has zero higher cohomology because $Spec (R)$ has a basis of distinguished opens with vanishing higher Čech cohomology.

This implies: a short exact sequence of quasi-coherent sheaves on an affine scheme remains short exact in global sections. The $H^{1}$ obstruction (Mittag-Leffler problem in topological terms) vanishes.

The dual statement (cohomology of projective schemes via twists) involves Serre's projective vanishing: for coherent $F$ on $P^{n}$ , $H^{i} (P^{n}; F (d)) = 0$ for $i \geq 1$ and $d$ large.

Exercise 7 (hard, conceptual).

Why is the equivalence "graded modules ↔ quasi-coherent sheaves on $Proj (S)$ " only modulo torsion?

Hint

Modules supported in low degrees vanish on Proj.

Answer

The Proj construction excludes the irrelevant ideal $S_{+}$ . So a graded $S$ -module $M$ supported only in low degrees (every element annihilated by some power of $S_{+}$ ) sheafifies to zero on $Proj (S)$ :

M (D (f)) = (M [f^{- 1}])_{0} = 0 for f \in S_{+} .

This is because any element $m \in M$ has $f^{n} \cdot m = 0$ for some $n$ (since $M$ is torsion at $S_{+}$ ), hence $m / f^{k} = 0$ in $M [f^{- 1}]$ .

So torsion modules in the graded sense — those supported in low degrees, equivalently $H_{S_{+}}^{0} (M) = M$ — go to zero in $QCoh (Proj (S))$ . The equivalence is

grMod (S) / Tor_{S_{+}} \sim QCoh (Proj (S)) .

This phenomenon — that low-degree graded data is forgotten by Proj — is one of the fundamental subtleties of projective algebraic geometry. It is the algebraic shadow of "the irrelevant ideal corresponds to the origin we removed before projectivising." Castelnuovo-Mumford regularity quantifies how much of the low-degree data can be safely forgotten.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has AlgebraicGeometry.QuasiCoherentSheaf and the equivalence with $R$ -modules on affines.

[object Promise]

Advanced results [Master]

Quasi-coherent sheaves and module categories. $QCoh (X)$ is a Grothendieck abelian category — has all colimits, filtered colimits exact, generators. On a quasi-compact and quasi-separated scheme, $QCoh$ has enough injectives.

Pushforward exact sequences. For $f : X \to Y$ a morphism, the pushforward $f_{*}$ on quasi-coherent sheaves preserves limits but not in general colimits. The right-derived $R^{i} f_{*}$ are quasi-coherent under quasi-compact-quasi-separated hypotheses; this is the foundation of Grothendieck's six-functor formalism.

Quasi-coherent sheaves with extra structure. Sheaves of $O_{X}$ -algebras (sheaves of rings extending $O_{X}$ ); sheaves of Lie algebroids; D-modules. All are quasi-coherent with extra structure.

Spec / QCoh duality. For a Noetherian scheme $X$ with sufficient finiteness conditions, the spectrum of $QCoh (X)$ as a tensor-triangulated category recovers $X$ — a reconstruction theorem (Bondal-van den Bergh, Balmer). $QCoh$ thus contains all geometric information.

Tannakian formalism. Quasi-coherent sheaves with extra structure (e.g., flat connection) form a tannakian category whose Tannaka dual is a (group) scheme. This extends the Spec-construction one categorical level up.

Geometric Langlands. $D$ -modules on $Bun_{G}$ (the moduli of $G$ -bundles on a curve) are conjecturally dual to quasi-coherent sheaves on $Loc_{G^{\lor}}$ (the moduli of $G^{\lor}$ -local systems). This is the categorical Langlands correspondence, with quasi-coherent sheaves at its heart.

Synthesis. This construction generalises the pattern fixed in 04.01.01 (sheaf), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs of: Serre's vanishing on affines, the equivalence with $R$ -modules, the Proj-side equivalence modulo torsion, and the affine criterion (Serre's cohomological criterion for affine-ness) are deferred to companion units. The proof of the basic equivalence is sketched in the formal-definition section.

Connections [Master]

Sheaf 04.01.01 — quasi-coherent sheaves are a special class of sheaves of $O_{X}$ -modules.
Affine scheme 04.02.02 — equivalence $Mod (R) ≅ QCoh (Spec (R))$ .
Projective scheme 04.02.03 — quasi-coherent sheaves on $Proj (S)$ correspond to graded modules modulo torsion.
Coherent sheaf 04.06.02 — coherent sheaves are quasi-coherent + finiteness conditions.
Sheaf cohomology 04.03.01 — Serre's vanishing on affine schemes is the foundational theorem for quasi-coherent cohomology.
Line bundle / invertible sheaf 04.05.03 — line bundles are rank-1 locally free quasi-coherent sheaves.
Picard group 04.05.02 — classifies invertible quasi-coherent sheaves up to isomorphism.

Historical & philosophical context [Master]

The notion of coherent sheaf was introduced by Henri Cartan and Jean-Pierre Serre in the early 1950s, in the context of complex-analytic geometry. Cartan's seminars (1951–54) developed coherent analytic sheaves and the Cartan-Serre-Grauert coherence theorems for proper holomorphic maps.

The algebraic version — quasi-coherent and coherent sheaves on schemes — was formalised by Serre's foundational 1955 paper Faisceaux Algébriques Cohérents (FAC). FAC introduced sheaf-theoretic methods to algebraic geometry, including:

Quasi-coherent and coherent sheaves on a variety.
Cohomology of coherent sheaves (later understood as sheaf cohomology in the Grothendieck framework).
Serre's vanishing theorem on affine varieties.
The Serre-twist sheaves $O (d)$ and the cohomology of $O (d)$ on $P^{n}$ .

FAC was a foundational achievement, providing the framework that Grothendieck's EGA (Éléments de Géométrie Algébrique, 1960–67) extended to schemes. EGA Chapter I §1 systematically develops quasi-coherent modules in scheme-theoretic generality.

The conceptual revolution: algebraic geometry is fundamentally the study of quasi-coherent sheaves on schemes. Every important geometric object — varieties, line bundles, vector bundles, ideal sheaves, moduli spaces — is built from quasi-coherent data. The category $QCoh (X)$ contains essentially all the algebraic-geometric information about $X$ .

Today, quasi-coherent sheaves are ubiquitous: in derived algebraic geometry (Lurie's $\infty$ -categorical refinement), in geometric Langlands (D-modules and quasi-coherent sheaves on moduli stacks), in derived categories of coherent sheaves (Bondal-Orlov reconstruction, Bridgeland stability conditions), and in arithmetic geometry (étale sheaves, perverse sheaves, motives). The legacy of Cartan-Serre-Grothendieck's foundational work permeates modern mathematics.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §II.5 is the standard introduction.
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §13 covers quasi-coherent sheaves systematically.
Grothendieck-Dieudonné, Éléments de Géométrie Algébrique I (EGA I, 1960) — Ch. I §1 is the foundational reference.
Serre, Faisceaux Algébriques Cohérents (FAC, 1955) — the seminal paper introducing coherent sheaf theory to algebraic geometry.
Eisenbud-Harris, The Geometry of Schemes — Ch. III, geometric perspective.
Liu, Algebraic Geometry and Arithmetic Curves — strong Chapter 5 on quasi-coherent sheaves.
Görtz & Wedhorn, Algebraic Geometry I — comprehensive modern treatment.