6.1 Sheaves of modules

Sheaves of modules are a key concept in algebraic geometry, extending vector bundles on manifolds. They allow us to study local properties of geometric objects and encode important algebraic and geometric information.

These sheaves are defined on topological spaces, with each open set assigned a module over a fixed ring. Understanding sheaves of modules is essential for advanced topics like cohomology and duality in algebraic geometry.

Sheaves of modules

• Sheaves of modules are a fundamental concept in algebraic geometry that generalize the notion of vector bundles on manifolds
• They provide a way to study local properties of geometric objects and encode important algebraic and geometric information
• Understanding sheaves of modules is crucial for advanced topics in algebraic geometry such as cohomology and duality

Definition of sheaves of modules

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• A sheaf of modules is a sheaf $\mathcal{F}$ on a topological space $X$ such that for each open set $U \subseteq X$, $\mathcal{F}(U)$ is a module over a fixed ring $R$
• The restriction maps $\mathcal{F}(U) \to \mathcal{F}(V)$ for open sets $V \subseteq U$ are required to be module homomorphisms
• Sheaves of modules satisfy the gluing axiom: if $\{U_i\}$ is an open cover of $U$ and $s_i \in \mathcal{F}(U_i)$ are sections such that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$, then there exists a unique section $s \in \mathcal{F}(U)$ with $s|_{U_i} = s_i$

Sheaves of modules over ringed spaces

• A ringed space is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ on $X$
• A sheaf of $\mathcal{O}_X$-modules is a sheaf $\mathcal{F}$ on $X$ such that for each open set $U \subseteq X$, $\mathcal{F}(U)$ is an $\mathcal{O}_X(U)$-module and the restriction maps are $\mathcal{O}_X(U)$-module homomorphisms
• Examples of ringed spaces include manifolds with the sheaf of smooth functions and schemes with their structure sheaf

Modules over structure sheaf

• On a scheme $(X, \mathcal{O}_X)$, the structure sheaf $\mathcal{O}_X$ is a sheaf of rings
• A sheaf of $\mathcal{O}_X$-modules is called a quasi-coherent sheaf if it locally looks like the sheaf associated to a module over the ring of functions
• Coherent sheaves are quasi-coherent sheaves that are locally finitely generated as $\mathcal{O}_X$-modules
• The category of quasi-coherent sheaves on a scheme is an abelian category with enough injectives

Locally free sheaves

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is called locally free of rank $n$ if there exists an open cover $\{U_i\}$ of $X$ such that $\mathcal{F}|_{U_i} \cong \mathcal{O}_{U_i}^n$ for each $i$
• Locally free sheaves of rank $1$ are called line bundles and play a crucial role in algebraic geometry
• The dual of a locally free sheaf is again locally free
• The tensor product of two locally free sheaves is locally free

Quasi-coherent sheaves

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ and an exact sequence $\mathcal{O}_U^I \to \mathcal{O}_U^J \to \mathcal{F}|_U \to 0$ for some sets $I$ and $J$
• Quasi-coherent sheaves form an abelian category $QCoh(X)$ with enough injectives
• The global section functor $\Gamma(X, -): QCoh(X) \to Mod(\mathcal{O}_X(X))$ is right exact and has a left derived functor $H^i(X, -)$ called sheaf cohomology

Coherent sheaves

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is coherent if it is quasi-coherent and for every open set $U \subseteq X$ and every morphism $\varphi: \mathcal{O}_U^n \to \mathcal{F}|_U$, the kernel of $\varphi$ is finitely generated
• Coherent sheaves form an abelian category $Coh(X)$ which is a full subcategory of $QCoh(X)$
• On a noetherian scheme, a sheaf is coherent if and only if it is locally finitely presented
• Coherent sheaves have finite-dimensional cohomology groups

Sheaf Hom and tensor product

• For sheaves of $\mathcal{O}_X$-modules $\mathcal{F}$ and $\mathcal{G}$, the sheaf Hom $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is the sheaf defined by $U \mapsto Hom_{\mathcal{O}_X|_U}(\mathcal{F}|_U, \mathcal{G}|_U)$
• The tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is the sheaf associated to the presheaf $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$
• Sheaf Hom and tensor product satisfy adjointness properties similar to their module counterparts

Sheaf Hom as left adjoint functor

• The sheaf Hom functor $\mathcal{H}om_{\mathcal{O}_X}(-, \mathcal{G}): (QCoh(X))^{op} \to QCoh(X)$ is left adjoint to the tensor product functor $- \otimes_{\mathcal{O}_X} \mathcal{G}: QCoh(X) \to QCoh(X)$
• This adjointness is expressed by the natural isomorphism $Hom_{QCoh(X)}(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}, \mathcal{H}) \cong Hom_{QCoh(X)}(\mathcal{F}, \mathcal{H}om_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H}))$
• As a consequence, sheaf Hom preserves colimits and tensor product preserves limits

Tensor product of sheaves of modules

• The tensor product of sheaves of modules satisfies associativity and commutativity up to natural isomorphism
• For locally free sheaves $\mathcal{F}$ and $\mathcal{G}$ of ranks $m$ and $n$, respectively, the tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is locally free of rank $mn$
• The tensor product is right exact in each variable and has a left derived functor called the tor functor

Sheaf Ext functors

• The sheaf Ext functors $\mathcal{E}xt^i_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ are the right derived functors of sheaf Hom
• They can be computed using injective resolutions of $\mathcal{G}$ or locally free resolutions of $\mathcal{F}$
• Sheaf Ext functors play a crucial role in duality theory and the study of extensions of sheaves

Derived functors of sheaf Hom

• The right derived functors of sheaf Hom are the sheaf Ext functors $\mathcal{E}xt^i_{\mathcal{O}_X}(\mathcal{F}, -)$
• They can be computed using injective resolutions or by taking an injective resolution of $\mathcal{F}$ and applying sheaf Hom to it
• The long exact sequence of sheaf Ext functors is a powerful tool in homological algebra

Injective and flasque resolutions

• An injective resolution of a sheaf $\mathcal{F}$ is an exact sequence $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$ where each $\mathcal{I}^i$ is an injective sheaf
• A flasque resolution is an exact sequence $0 \to \mathcal{F} \to \mathcal{F}^0 \to \mathcal{F}^1 \to \cdots$ where each $\mathcal{F}^i$ is a flasque sheaf (a sheaf for which restriction maps are surjective)
• Flasque resolutions can be used to compute sheaf cohomology and are easier to construct than injective resolutions

Cohomology of sheaves of modules

• The sheaf cohomology groups $H^i(X, \mathcal{F})$ are the right derived functors of the global section functor $\Gamma(X, -)$
• They measure the obstruction to solving global problems locally and provide important invariants of the space $X$ and the sheaf $\mathcal{F}$
• Sheaf cohomology can be computed using injective, flasque, or Čech resolutions

Čech cohomology of sheaves of modules

• Čech cohomology is a computational tool for sheaf cohomology based on open covers of the space $X$
• For an open cover $\mathfrak{U} = \{U_i\}$ of $X$, the Čech complex $\check{C}^\bullet(\mathfrak{U}, \mathcal{F})$ is defined using alternating products of the sections of $\mathcal{F}$ over intersections of open sets
• The Čech cohomology groups $\check{H}^i(\mathfrak{U}, \mathcal{F})$ are the cohomology groups of the Čech complex and are isomorphic to the sheaf cohomology groups for a sufficiently fine cover

Serre duality for sheaves of modules

• Serre duality is a fundamental duality theorem in algebraic geometry relating the cohomology of a coherent sheaf and its dual
• For a smooth projective variety $X$ of dimension $n$ over a field and a coherent sheaf $\mathcal{F}$ on $X$, there are natural isomorphisms $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$, where $\omega_X$ is the canonical sheaf
• Serre duality has numerous applications, including the Riemann-Roch theorem and the study of moduli spaces

Sheaves of differentials

• The sheaf of differentials $\Omega_{X/k}$ on a scheme $X$ over a field $k$ is a coherent sheaf that generalizes the notion of differential forms on manifolds
• For a morphism of schemes $f: X \to Y$, there is an exact sequence $f^*\Omega_{Y/k} \to \Omega_{X/k} \to \Omega_{X/Y} \to 0$, where $\Omega_{X/Y}$ is the relative sheaf of differentials
• The sheaf of differentials is used to define the canonical sheaf $\omega_X$ and plays a role in duality theory

Sheaves of principal parts

• The sheaf of principal parts $\mathcal{P}^n_{X/k}$ is a quasi-coherent sheaf on a scheme $X$ over a field $k$ that generalizes the notion of jet bundles on manifolds
• It fits into an exact sequence $0 \to \Omega_{X/k} \otimes \mathcal{O}_X(n) \to \mathcal{P}^n_{X/k} \to \mathcal{O}_X(n) \to 0$ and can be used to study infinitesimal properties of $X$
• The sheaf of principal parts is related to the study of differential operators and the jet scheme

Koszul complexes for sheaves of modules

• The Koszul complex is a canonical complex associated to a regular sequence of sections of a sheaf of modules
• For a regular sequence $f_1, \ldots, f_n \in \Gamma(X, \mathcal{F})$, the Koszul complex $K_\bullet(f_1, \ldots, f_n; \mathcal{F})$ is a resolution of the quotient sheaf $\mathcal{F}/(f_1, \ldots, f_n)\mathcal{F}$
• Koszul complexes are used in the study of local cohomology and intersection theory

Sheaves of modules on projective spaces

• Projective spaces $\mathbb{P}^n_k$ over a field $k$ are fundamental examples of schemes with a rich theory of sheaves of modules
• The structure sheaf $\mathcal{O}_{\mathbb{P}^n_k}$ admits a decomposition $\oplus_{d \geq 0} \mathcal{O}_{\mathbb{P}^n_k}(d)$, where $\mathcal{O}_{\mathbb{P}^n_k}(d)$ is the $d$-th twisting sheaf
• Coherent sheaves on $\mathbb{P}^n_k$ can be studied using graded modules over the polynomial ring $k[x_0, \ldots, x_n]$ and the Serre correspondence

Sheaves of modules on affine schemes

• An affine scheme is a scheme isomorphic to the spectrum of a commutative ring
• Quasi-coherent sheaves on an affine scheme $Spec(A)$ correspond bijectively to $A$-modules via the global section functor
• The cohomology of quasi-coherent sheaves on an affine scheme vanishes in positive degrees

Sheaves of modules and quasi-coherent sheaves

• Quasi-coherent sheaves form a full abelian subcategory of the category of sheaves of modules on a scheme
• On a noetherian scheme, the category of quasi-coherent sheaves is a Grothendieck category with a generating family consisting of locally free sheaves
• Many constructions and results for sheaves of modules (e.g., sheaf Hom, tensor product, cohomology) have simpler descriptions when restricted to quasi-coherent sheaves

Comparison of categories of sheaves of modules

• The category of sheaves of modules on a ringed space $(X, \mathcal{O}_X)$ is related to the category of $\mathcal{O}_X(X)$-modules via the global section functor $\Gamma(X, -)$
• For a morphism of ringed spaces $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$, there are adjoint functors $f^*$ (inverse image) and $f_*$ (direct image) between the categories of sheaves of modules
• In the case of schemes, the categories of quasi-coherent sheaves are related by the quasi-coherent inverse image functor $Lf^*$ and the quasi-coherent direct image functor $Rf_*$