8.3 Quasi-coherent sheaves

Quasi-coherent sheaves are a key concept in algebraic geometry, bridging local and global perspectives on geometric objects. They generalize coherent sheaves and provide a way to associate algebraic data with open sets of a scheme.

These sheaves are crucial for understanding scheme structure and cohomology. They include important examples like the structure sheaf and differential forms, and play a vital role in various geometric constructions and theorems in algebraic geometry.

Quasi-coherent sheaves

• Quasi-coherent sheaves are a fundamental concept in algebraic geometry that generalizes the notion of coherent sheaves
• They provide a way to study geometric objects locally and globally by associating algebraic data to each open set of a scheme
• Quasi-coherent sheaves play a crucial role in understanding the structure and properties of schemes and their cohomology

Definition of quasi-coherent sheaves

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• A quasi-coherent sheaf $\mathcal{F}$ on a scheme $X$ is an $\mathcal{O}_X$-module such that for every open affine subset $U \subset X$, there exists an $\mathcal{O}_X(U)$-module $M$ with $\mathcal{F}|_U \cong \widetilde{M}$
  • $\mathcal{O}_X$ denotes the structure sheaf of the scheme $X$
  • $\widetilde{M}$ is the sheaf associated to the module $M$
• Intuitively, a quasi-coherent sheaf locally looks like the sheaf associated to a module over the ring of regular functions
• Examples of quasi-coherent sheaves include the structure sheaf $\mathcal{O}_X$ and the sheaf of differential forms $\Omega_X$

Locally free sheaves

• A locally free sheaf $\mathcal{E}$ on a scheme $X$ is a quasi-coherent sheaf such that for every point $x \in X$, there exists an open neighborhood $U$ of $x$ with $\mathcal{E}|_U \cong \mathcal{O}_U^{\oplus n}$ for some integer $n$
  • $\mathcal{O}_U$ is the restriction of the structure sheaf to the open set $U$
• Locally free sheaves are the algebraic analogue of vector bundles in differential geometry
• The rank of a locally free sheaf is the integer $n$ appearing in the local trivialization
• Examples of locally free sheaves include the tangent sheaf $\mathcal{T}_X$ and the cotangent sheaf $\Omega_X^1$

Coherent vs quasi-coherent sheaves

• Coherent sheaves are a special case of quasi-coherent sheaves with an additional finiteness condition
• A coherent sheaf $\mathcal{F}$ on a scheme $X$ is a quasi-coherent sheaf such that for every open affine subset $U \subset X$, there exists an exact sequence $\mathcal{O}_U^{\oplus m} \to \mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0$
  • This means that locally, $\mathcal{F}$ can be described as the cokernel of a morphism between free $\mathcal{O}_U$-modules of finite rank
• Every coherent sheaf is quasi-coherent, but the converse is not true in general
• Coherent sheaves are more restrictive but have better finiteness properties compared to quasi-coherent sheaves

Characterization of quasi-coherence

• A sheaf $\mathcal{F}$ on a scheme $X$ is quasi-coherent if and only if for every open affine subset $U \subset X$ and every morphism of schemes $f: V \to U$ with $V$ affine, the natural map $\mathcal{O}_X(U) \otimes_{\mathcal{O}_X(V)} \mathcal{F}(V) \to \mathcal{F}(U)$ is an isomorphism
  • This characterization provides a way to check quasi-coherence using affine open coverings and tensor products
• Alternatively, a sheaf $\mathcal{F}$ is quasi-coherent if and only if there exists an affine open covering $\{U_i\}$ of $X$ and $\mathcal{O}_X(U_i)$-modules $M_i$ such that $\mathcal{F}|_{U_i} \cong \widetilde{M_i}$ for each $i$, and the restriction maps $\mathcal{F}(U_i) \to \mathcal{F}(U_i \cap U_j)$ correspond to the natural maps $M_i \to M_i \otimes_{\mathcal{O}_X(U_i)} \mathcal{O}_X(U_i \cap U_j)$

Quasi-coherent subsheaves

• A subsheaf $\mathcal{G}$ of a quasi-coherent sheaf $\mathcal{F}$ on a scheme $X$ is quasi-coherent if for every open affine subset $U \subset X$, the restriction $\mathcal{G}|_U$ is the sheaf associated to a submodule of $\mathcal{F}(U)$
• The kernel and image of a morphism between quasi-coherent sheaves are quasi-coherent subsheaves
• The intersection and sum of two quasi-coherent subsheaves are again quasi-coherent

Quasi-coherent quotient sheaves

• A quotient sheaf $\mathcal{F}/\mathcal{G}$ of a quasi-coherent sheaf $\mathcal{F}$ by a quasi-coherent subsheaf $\mathcal{G}$ is quasi-coherent
• For every open affine subset $U \subset X$, the restriction $(\mathcal{F}/\mathcal{G})|_U$ is the sheaf associated to the quotient module $\mathcal{F}(U)/\mathcal{G}(U)$
• The cokernel of a morphism between quasi-coherent sheaves is a quasi-coherent quotient sheaf

Tensor products of quasi-coherent sheaves

• The tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ of two quasi-coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on a scheme $X$ is a quasi-coherent sheaf
• For every open affine subset $U \subset X$, the restriction $(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})|_U$ is the sheaf associated to the tensor product of modules $\mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$
• The tensor product operation is associative, commutative, and distributes over direct sums

Pullbacks of quasi-coherent sheaves

• Let $f: X \to Y$ be a morphism of schemes. If $\mathcal{F}$ is a quasi-coherent sheaf on $Y$, then the pullback $f^*\mathcal{F}$ is a quasi-coherent sheaf on $X$
• For every open affine subset $U \subset X$ with $f(U)$ contained in an open affine subset $V \subset Y$, the restriction $(f^*\mathcal{F})|_U$ is the sheaf associated to the module $\mathcal{F}(V) \otimes_{\mathcal{O}_Y(V)} \mathcal{O}_X(U)$
• Pullbacks are functorial: $(f \circ g)^* = g^* \circ f^*$ and $\mathrm{id}_X^* = \mathrm{id}$

Pushforwards of quasi-coherent sheaves

• Let $f: X \to Y$ be a morphism of schemes. If $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the pushforward $f_*\mathcal{F}$ is a quasi-coherent sheaf on $Y$ under certain conditions
  • If $f$ is an affine morphism (e.g., a closed immersion or a finite morphism), then $f_*\mathcal{F}$ is always quasi-coherent
  • If $f$ is a quasi-compact and quasi-separated morphism (e.g., a proper morphism), then $f_*\mathcal{F}$ is quasi-coherent
• For every open affine subset $V \subset Y$, the section $(f_*\mathcal{F})(V)$ is given by the global sections $\mathcal{F}(f^{-1}(V))$
• Pushforwards are functorial: $(f \circ g)_* = f_* \circ g_*$ and $\mathrm{id}_{X*} = \mathrm{id}$

Quasi-coherent sheaves on affine schemes

• On an affine scheme $X = \mathrm{Spec}(A)$, there is an equivalence of categories between quasi-coherent sheaves on $X$ and $A$-modules
  • The functor $\Gamma(X, -)$ associates to each quasi-coherent sheaf $\mathcal{F}$ its global sections $\mathcal{F}(X)$, which is an $A$-module
  • The functor $\widetilde{(-)}$ associates to each $A$-module $M$ the quasi-coherent sheaf $\widetilde{M}$ on $X$
• Under this equivalence, locally free sheaves correspond to projective $A$-modules, and coherent sheaves correspond to finitely generated $A$-modules

Quasi-coherent sheaves on projective schemes

• On a projective scheme $X = \mathrm{Proj}(S)$ over a ring $A$, where $S$ is a graded $A$-algebra, there is an equivalence of categories between quasi-coherent sheaves on $X$ and graded $S$-modules up to a shift in grading
  • The functor $\Gamma_*(X, -)$ associates to each quasi-coherent sheaf $\mathcal{F}$ the graded $S$-module $\bigoplus_{n \in \mathbb{Z}} \Gamma(X, \mathcal{F}(n))$, where $\mathcal{F}(n)$ is the twist of $\mathcal{F}$ by the $n$-th power of the tautological line bundle $\mathcal{O}_X(1)$
  • The functor $\widetilde{(-)}$ associates to each graded $S$-module $M$ the quasi-coherent sheaf $\widetilde{M}$ on $X$
• Serre's theorem states that a coherent sheaf on a projective scheme over a noetherian ring is generated by its global sections

Quasi-coherent sheaves on noetherian schemes

• On a noetherian scheme $X$, every quasi-coherent sheaf is a direct limit of its coherent subsheaves
• The category of quasi-coherent sheaves on a noetherian scheme is a Grothendieck abelian category
  • It has enough injectives, and every object has an injective resolution
  • Derived functors can be defined using injective resolutions
• The cohomology groups $H^i(X, \mathcal{F})$ of a quasi-coherent sheaf $\mathcal{F}$ on a noetherian scheme $X$ are finitely generated modules over the ring of global sections $\Gamma(X, \mathcal{O}_X)$

Cohomology of quasi-coherent sheaves

• The cohomology groups $H^i(X, \mathcal{F})$ of a quasi-coherent sheaf $\mathcal{F}$ on a scheme $X$ are defined as the right derived functors of the global section functor $\Gamma(X, -)$
• For an affine scheme $X = \mathrm{Spec}(A)$, the cohomology groups $H^i(X, \mathcal{F})$ vanish for $i > 0$, and $H^0(X, \mathcal{F}) = \mathcal{F}(X)$ is the $A$-module of global sections
• The cohomology groups can be computed using Čech cohomology with respect to an affine open covering
• The cohomology groups are functorial: a morphism of quasi-coherent sheaves induces a morphism of their cohomology groups

Serre's theorem for quasi-coherent sheaves

• Serre's theorem states that on a projective scheme $X$ over a noetherian ring $A$, for any quasi-coherent sheaf $\mathcal{F}$, there exists an integer $n_0$ such that for all $n \geq n_0$, the twisted sheaf $\mathcal{F}(n)$ is generated by its global sections
  • This means that the evaluation map $H^0(X, \mathcal{F}(n)) \otimes_A \mathcal{O}_X(-n) \to \mathcal{F}$ is surjective for $n \geq n_0$
• Serre's theorem implies that every quasi-coherent sheaf on a projective scheme over a noetherian ring can be obtained as a quotient of a direct sum of twists of the structure sheaf
• The theorem is a key tool in understanding the structure of quasi-coherent sheaves on projective schemes

Quasi-coherent sheaves in algebraic geometry

• Quasi-coherent sheaves provide a bridge between algebraic geometry and commutative algebra
• Many constructions in algebraic geometry, such as the structure sheaf, sheaves of differential forms, and sheaves of ideals, are quasi-coherent sheaves
• Quasi-coherent sheaves can be used to study the geometry of schemes through their cohomological properties
• The category of quasi-coherent sheaves on a scheme is an important invariant of the scheme and captures its geometric and algebraic features

Applications of quasi-coherent sheaves

• Quasi-coherent sheaves are used in the construction of moduli spaces, such as the moduli space of vector bundles or the Picard scheme
• The study of quasi-coherent sheaves and their cohomology is central to the development of sheaf cohomology and its applications in algebraic geometry
• Quasi-coherent sheaves play a role in the formulation and proof of various duality theorems, such as Serre duality and Grothendieck duality
• The theory of quasi-coherent sheaves is used in the study of derived categories and their applications to problems in algebraic geometry and mathematical physics
• Quasi-coherent sheaves provide a framework for studying the geometry of schemes in a way that is compatible with base change and functorial operations