6.2 Ringed spaces

Ringed spaces blend topology and algebra, providing a framework to study geometric objects with algebraic structure. They combine a topological space with a sheaf of rings, allowing for a rich interplay between continuity and algebraic operations.

This concept forms the foundation for more specialized objects like schemes and complex manifolds. By examining ringed spaces, we gain insights into how local algebraic properties can be integrated with global topological structures in mathematics.

Definition of ringed spaces

• A ringed space is a fundamental concept in sheaf theory that combines a topological space with a sheaf of rings
• It provides a framework for studying geometric objects with algebraic structure, allowing for a rich interplay between topology and algebra
• Ringed spaces form the basis for more specialized objects like schemes and complex manifolds

Topological space component

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• The topological space $(X, \mathcal{O}_X)$ serves as the underlying space of the ringed space
• The topology on $X$ determines the open sets and the notion of continuity for functions
• Examples of topological spaces include Euclidean spaces, manifolds, and Zariski topologies on algebraic varieties

Sheaf of rings component

• A sheaf of rings $\mathcal{O}_X$ is assigned to each open set $U \subset X$, denoted as $\mathcal{O}_X(U)$
• The sections of the sheaf over an open set form a commutative ring, with restriction maps between them
• The sheaf condition ensures that sections can be glued together consistently on overlapping open sets

Local rings at each point

• For each point $x \in X$, the stalk $\mathcal{O}_{X,x}$ is defined as the direct limit of the rings $\mathcal{O}_X(U)$ over all open sets $U$ containing $x$
• The stalk $\mathcal{O}_{X,x}$ is a local ring, with a unique maximal ideal corresponding to the "vanishing" of sections at $x$
• The local rings capture the algebraic structure of the ringed space near each point

Examples of ringed spaces

• Ringed spaces provide a unifying framework for studying various geometric objects with algebraic structure
• They allow for the application of sheaf-theoretic techniques to a wide range of mathematical contexts

Locally ringed spaces

• A locally ringed space is a ringed space where each stalk $\mathcal{O}_{X,x}$ is a local ring
• Examples include smooth manifolds with the sheaf of smooth functions and complex manifolds with the sheaf of holomorphic functions
• Locally ringed spaces are the most common type of ringed spaces encountered in practice

Complex manifolds as ringed spaces

• A complex manifold $X$ can be viewed as a locally ringed space $(X, \mathcal{O}_X)$, where $\mathcal{O}_X$ is the sheaf of holomorphic functions
• The local rings $\mathcal{O}_{X,x}$ capture the analytic structure near each point
• Sheaf cohomology on complex manifolds plays a crucial role in complex geometry and algebraic geometry

Schemes as ringed spaces

• A scheme is a locally ringed space $(X, \mathcal{O}_X)$ that is locally isomorphic to the spectrum of a commutative ring
• The sheaf $\mathcal{O}_X$ is called the structure sheaf and encodes the algebraic structure of the scheme
• Affine schemes (spectra of rings) and projective schemes are important examples of schemes

Morphisms of ringed spaces

• Morphisms of ringed spaces are the natural maps between ringed spaces that preserve both the topological and algebraic structure
• They allow for the study of relationships between different geometric objects and the functorial properties of constructions

Continuous maps between topological spaces

• A morphism of ringed spaces $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ includes a continuous map $f: X \to Y$ between the underlying topological spaces
• The continuity of $f$ ensures that the preimage of an open set in $Y$ is open in $X$
• Continuous maps allow for the study of topological properties of ringed spaces

Sheaf morphisms

• A morphism of ringed spaces also includes a morphism of sheaves $f^\sharp: \mathcal{O}_Y \to f_*\mathcal{O}_X$, where $f_*\mathcal{O}_X$ is the direct image sheaf
• The sheaf morphism assigns to each open set $V \subset Y$ a ring homomorphism $f^\sharp(V): \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}(V))$
• Sheaf morphisms preserve the algebraic structure of the ringed spaces

Compatibility conditions

• The continuous map $f$ and the sheaf morphism $f^\sharp$ must satisfy certain compatibility conditions
• For each open set $V \subset Y$ and each section $s \in \mathcal{O}_Y(V)$, the pullback $f^\sharp(s)$ should be a section of $\mathcal{O}_X$ over $f^{-1}(V)$
• The compatibility conditions ensure that the topological and algebraic structures are preserved under morphisms

Properties of morphisms

• Various properties of morphisms between ringed spaces capture important geometric and algebraic features
• These properties often reflect the local behavior of the morphisms and their effect on the structure of the ringed spaces

Local morphisms of local rings

• A morphism $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is called a local morphism if for each $x \in X$, the induced map on stalks $f_x^\sharp: \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is a local homomorphism of local rings
• Local morphisms preserve the local structure of the ringed spaces and are essential in the study of locally ringed spaces

Open and closed immersions

• An open immersion is a morphism $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ such that $f: X \to Y$ is an open embedding and $f^\sharp$ is an isomorphism
• A closed immersion is a morphism $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ such that $f: X \to Y$ is a closed embedding and $f^\sharp$ induces an isomorphism between $\mathcal{O}_X$ and the restriction of $\mathcal{O}_Y$ to $f(X)$
• Open and closed immersions allow for the construction of ringed spaces by gluing along subspaces

Étale morphisms

• An étale morphism is a morphism $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ that is locally of finite presentation, flat, and unramified
• Étale morphisms are the algebro-geometric analog of local isomorphisms in differential geometry
• They play a crucial role in the study of schemes and the development of étale cohomology

Constructions with ringed spaces

• Various constructions can be performed on ringed spaces to create new ringed spaces with desired properties
• These constructions often involve the manipulation of the underlying topological spaces and the associated sheaves

Fiber products

• The fiber product of ringed spaces $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ over a base ringed space $(S, \mathcal{O}_S)$ is a ringed space $(X \times_S Y, \mathcal{O}_{X \times_S Y})$ that satisfies a universal property
• The underlying topological space is the fiber product of the topological spaces, and the structure sheaf is constructed using the tensor product of the pullback sheaves
• Fiber products allow for the construction of new ringed spaces by "gluing" along morphisms

Direct and inverse image sheaves

• Given a morphism $f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$, the direct image sheaf $f_*\mathcal{O}_X$ on $Y$ is defined by $(f_*\mathcal{O}_X)(V) = \mathcal{O}_X(f^{-1}(V))$ for open sets $V \subset Y$
• The inverse image sheaf $f^{-1}\mathcal{O}_Y$ on $X$ is defined as the sheafification of the presheaf $U \mapsto \varinjlim_{V \supset f(U)} \mathcal{O}_Y(V)$ for open sets $U \subset X$
• Direct and inverse image sheaves allow for the transfer of sheaves between ringed spaces along morphisms

Sheaf Hom and tensor product

• The sheaf Hom $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ of two sheaves $\mathcal{F}$ and $\mathcal{G}$ on a ringed space $(X, \mathcal{O}_X)$ is defined by $\mathcal{H}om(\mathcal{F}, \mathcal{G})(U) = \text{Hom}_{\mathcal{O}_X|_U}(\mathcal{F}|_U, \mathcal{G}|_U)$ for open sets $U \subset X$
• The tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ of two sheaves of $\mathcal{O}_X$-modules is defined as the sheafification of the presheaf $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$
• Sheaf Hom and tensor product provide tools for constructing new sheaves on ringed spaces with desired properties

Coherent sheaves on ringed spaces

• Coherent sheaves are a class of sheaves on ringed spaces that behave well under certain operations and have finiteness conditions
• They play a central role in algebraic geometry and the study of sheaf cohomology

Definition and examples

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ on a ringed space $(X, \mathcal{O}_X)$ is coherent if it is of finite type and for every open set $U \subset X$ and every morphism $\varphi: \mathcal{O}_X^n|_U \to \mathcal{F}|_U$, the kernel of $\varphi$ is of finite type
• Examples of coherent sheaves include the structure sheaf $\mathcal{O}_X$ itself and the sheaf of sections of a vector bundle on a complex manifold
• Coherent sheaves form an abelian category, which allows for the application of homological algebra techniques

Locally free sheaves and vector bundles

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is locally free of rank $n$ if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $\mathcal{F}|_U \cong \mathcal{O}_X^n|_U$
• Locally free sheaves of rank $n$ correspond to vector bundles of rank $n$ on the ringed space
• Vector bundles and their sheaves of sections are fundamental objects in geometry and topology

Quasi-coherent sheaves

• A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ on a ringed space $(X, \mathcal{O}_X)$ is quasi-coherent if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $\mathcal{F}|_U$ is the cokernel of a morphism between free $\mathcal{O}_X|_U$-modules
• Quasi-coherent sheaves are a generalization of coherent sheaves and include important examples such as the pushforward of a coherent sheaf along a morphism of ringed spaces
• The category of quasi-coherent sheaves on a scheme is an important tool in algebraic geometry

Cohomology of ringed spaces

• Cohomology theories on ringed spaces provide a way to study global properties of sheaves and extract algebraic invariants
• They are essential tools in algebraic geometry, complex geometry, and other areas of mathematics

Derived functors and sheaf cohomology

• The cohomology of a sheaf $\mathcal{F}$ on a ringed space $(X, \mathcal{O}_X)$ is defined using the right derived functors of the global sections functor $\Gamma(X, -)$
• The $i$-th cohomology group of $\mathcal{F}$ is denoted by $H^i(X, \mathcal{F})$ and is computed using an injective resolution of $\mathcal{F}$
• Sheaf cohomology groups measure the obstruction to solving global problems locally and provide important algebraic invariants

Čech cohomology vs derived functor cohomology

• Čech cohomology is another approach to defining cohomology of sheaves on a ringed space, using open covers and Čech cochains
• For certain ringed spaces (e.g., separated schemes), Čech cohomology agrees with derived functor cohomology
• Čech cohomology is often more computationally accessible, while derived functor cohomology provides a more general and conceptual approach

Serre duality for ringed spaces

• Serre duality is a fundamental result in the cohomology of coherent sheaves on proper smooth varieties over a field
• It relates the cohomology of a coherent sheaf $\mathcal{F}$ to the cohomology of its dual sheaf $\mathcal{F}^\vee$ twisted by the canonical sheaf
• Serre duality has important applications in the study of algebraic curves, surfaces, and higher-dimensional varieties