3.4 Locally ringed spaces and structure sheaves

Locally ringed spaces and structure sheaves are key concepts in algebraic geometry. They provide a framework for studying geometric objects through their local properties, allowing us to connect abstract algebra with geometric intuition.

These ideas are crucial for understanding schemes, which are locally ringed spaces with special properties. The structure sheaf of a scheme encodes important information about its geometry, helping us analyze properties like dimension, regularity, and normality.

Locally Ringed Spaces and Morphisms

Definition and Properties

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• A locally ringed space is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$ on $X$ such that the stalk $\mathcal{O}_{X,p}$ at each point $p \in X$ is a local ring
• The category of locally ringed spaces is a subcategory of the category of ringed spaces, where the objects are locally ringed spaces and the morphisms are morphisms of locally ringed spaces
• Every scheme $(X, \mathcal{O}_X)$ is a locally ringed space, where the stalk at each point $p \in X$ is the local ring $\mathcal{O}_{X,p}$
  • Examples of schemes include affine schemes $(\operatorname{Spec} A, \mathcal{O}_{\operatorname{Spec} A})$ and projective schemes $(\mathbb{P}^n_A, \mathcal{O}_{\mathbb{P}^n_A})$

Morphisms of Locally Ringed Spaces

• A morphism of locally ringed spaces $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair $(f, f^\#)$ where:
  • $f: X \to Y$ is a continuous map
  • $f^\#: \mathcal{O}_Y \to f_*\mathcal{O}_X$ is a morphism of sheaves of rings such that for each $p \in X$, the induced map on stalks $f^\#_p: \mathcal{O}_{Y,f(p)} \to \mathcal{O}_{X,p}$ is a local homomorphism of local rings
• Morphisms of locally ringed spaces preserve the local ring structure of the stalks
  • For example, if $(f, f^\#): (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a morphism of locally ringed spaces and $p \in X$, then $f^\#_p$ maps the maximal ideal of $\mathcal{O}_{Y,f(p)}$ into the maximal ideal of $\mathcal{O}_{X,p}$

Structure Sheaf of a Scheme

Construction for Affine Schemes

• For an affine scheme $\operatorname{Spec} A$, the structure sheaf $\mathcal{O}_{\operatorname{Spec} A}$ is defined by $\mathcal{O}_{\operatorname{Spec} A}(D(f)) = A_f$ for each basic open set $D(f)$, where $A_f$ is the localization of $A$ at the multiplicative set $\{1, f, f^2, \ldots\}$
  • For example, if $A = k[x, y]/(xy)$, then $\mathcal{O}_{\operatorname{Spec} A}(D(x)) = A_x = k[x, y]/(xy)_x$
• The stalk of $\mathcal{O}_{\operatorname{Spec} A}$ at a point $p \in \operatorname{Spec} A$ is the local ring $A_p$, which is the localization of $A$ at the prime ideal $p$

Construction for General Schemes

• For a general scheme $(X, \mathcal{O}_X)$, the structure sheaf $\mathcal{O}_X$ is constructed by gluing the structure sheaves of an affine open cover
  • If $\{U_i\}$ is an affine open cover of $X$ with $U_i = \operatorname{Spec} A_i$, then $\mathcal{O}_X|_{U_i} \cong \mathcal{O}_{\operatorname{Spec} A_i}$ and these isomorphisms satisfy the cocycle condition on overlaps
• The stalk of the structure sheaf $\mathcal{O}_X$ at a point $p \in X$ is the local ring $\mathcal{O}_{X,p}$, which can be computed as the direct limit of the rings $\mathcal{O}_X(U)$ over all open sets $U$ containing $p$
  • For example, if $X = \mathbb{P}^1_k$ and $p = [a:b] \in \mathbb{P}^1_k$, then $\mathcal{O}_{X,p} \cong k[x, y]_{(ax-by)}$, the localization of $k[x, y]$ at the homogeneous prime ideal $(ax-by)$

Schemes vs Locally Ringed Spaces

Schemes as Locally Ringed Spaces

• Every scheme $(X, \mathcal{O}_X)$ is a locally ringed space, where the stalk at each point $p \in X$ is the local ring $\mathcal{O}_{X,p}$
• Morphisms of schemes are precisely morphisms of locally ringed spaces
  • The category of schemes is a full subcategory of the category of locally ringed spaces

Characterization of Schemes

• Not every locally ringed space is a scheme
• A locally ringed space $(X, \mathcal{O}_X)$ is a scheme if and only if it is locally isomorphic to an affine scheme
  • Every point $p \in X$ has an open neighborhood $U$ such that $(U, \mathcal{O}_X|_U)$ is isomorphic to an affine scheme $\operatorname{Spec} A$ for some ring $A$
• Examples of locally ringed spaces that are not schemes include:
  • The real line $(\mathbb{R}, C^\infty_\mathbb{R})$ with the sheaf of smooth functions
  • The complex plane $(\mathbb{C}, \mathcal{O}_\mathbb{C})$ with the sheaf of holomorphic functions

Structure Sheaf for Local Properties

Closed Points and Dimension

• A point $p \in X$ is a closed point if and only if the stalk $\mathcal{O}_{X,p}$ is a field
  • For example, in the affine scheme $\operatorname{Spec} k[x]$, the closed points correspond to maximal ideals $(x-a)$ for $a \in k$, and the stalks at these points are isomorphic to $k$
• More generally, the dimension of $\mathcal{O}_{X,p}$ as a local ring is equal to the dimension of the closure of $p$ in $X$
  • For instance, in the affine scheme $\operatorname{Spec} k[x, y]$, the origin $(x, y)$ has dimension 2, while the generic point $(0)$ has dimension 0

Reducedness and Normality

• A scheme $X$ is reduced if and only if for each open set $U \subseteq X$, the ring $\mathcal{O}_X(U)$ has no nilpotent elements
  • Equivalently, $X$ is reduced if and only if each stalk $\mathcal{O}_{X,p}$ is a reduced local ring
  • For example, the affine scheme $\operatorname{Spec} k[x, y]/(x^2)$ is not reduced, as the ring $k[x, y]/(x^2)$ contains the nilpotent element $x$
• A scheme $X$ is normal if and only if for each open set $U \subseteq X$, the ring $\mathcal{O}_X(U)$ is integrally closed in its total quotient ring
  • Equivalently, $X$ is normal if and only if each stalk $\mathcal{O}_{X,p}$ is an integrally closed local ring
  • For instance, the affine scheme $\operatorname{Spec} k[x, y]/(y^2-x^3)$ is not normal, as the ring $k[x, y]/(y^2-x^3)$ is not integrally closed (it lacks the element $y/x$)

Regularity

• A scheme $X$ is regular if and only if each stalk $\mathcal{O}_{X,p}$ is a regular local ring, i.e., its maximal ideal is generated by $\dim \mathcal{O}_{X,p}$ elements
  • For example, the affine scheme $\operatorname{Spec} k[x, y]/(y^2-x^2(x+1))$ is regular, as each stalk is a regular local ring
  • In contrast, the affine scheme $\operatorname{Spec} k[x, y]/(y^2-x^3)$ is not regular at the origin, as the maximal ideal of the stalk at the origin is generated by $x$ and $y$, but the dimension is 1