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 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 is a pair (X,OX) consisting of a X and a sheaf of rings OX on X such that the stalk OX,p at each point p∈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,OX) is a locally ringed space, where the stalk at each point p∈X is the local ring OX,p
Examples of schemes include affine schemes (SpecA,OSpecA) and projective schemes (PAn,OPAn)
Morphisms of Locally Ringed Spaces
A morphism of locally ringed spaces (X,OX)→(Y,OY) is a pair (f,f#) where:
f:X→Y is a continuous map
f#:OY→f∗OX is a morphism of sheaves of rings such that for each p∈X, the induced map on stalks fp#:OY,f(p)→OX,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,OX)→(Y,OY) is a morphism of locally ringed spaces and p∈X, then fp# maps the maximal ideal of OY,f(p) into the maximal ideal of OX,p
Structure Sheaf of a Scheme
Construction for Affine Schemes
For an SpecA, the structure sheaf OSpecA is defined by OSpecA(D(f))=Af for each basic open set D(f), where Af is the localization of A at the multiplicative set {1,f,f2,…}
For example, if A=k[x,y]/(xy), then OSpecA(D(x))=Ax=k[x,y]/(xy)x
The stalk of OSpecA at a point p∈SpecA is the local ring Ap, which is the localization of A at the prime ideal p
Construction for General Schemes
For a general scheme (X,OX), the structure sheaf OX is constructed by gluing the structure sheaves of an affine open cover
If {Ui} is an affine open cover of X with Ui=SpecAi, then OX∣Ui≅OSpecAi and these isomorphisms satisfy the cocycle condition on overlaps
The stalk of the structure sheaf OX at a point p∈X is the local ring OX,p, which can be computed as the direct limit of the rings OX(U) over all open sets U containing p
For example, if X=Pk1 and p=[a:b]∈Pk1, then OX,p≅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,OX) is a locally ringed space, where the stalk at each point p∈X is the local ring OX,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,OX) is a scheme if and only if it is locally isomorphic to an affine scheme
Every point p∈X has an open neighborhood U such that (U,OX∣U) is isomorphic to an affine scheme SpecA for some ring A
Examples of locally ringed spaces that are not schemes include:
The real line (R,CR∞) with the sheaf of smooth functions
The complex plane (C,OC) with the sheaf of holomorphic functions
Structure Sheaf for Local Properties
Closed Points and Dimension
A point p∈X is a closed point if and only if the stalk OX,p is a field
For example, in the affine scheme Speck[x], the closed points correspond to maximal ideals (x−a) for a∈k, and the stalks at these points are isomorphic to k
More generally, the dimension of OX,p as a local ring is equal to the dimension of the closure of p in X
For instance, in the affine scheme Speck[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⊆X, the ring OX(U) has no nilpotent elements
Equivalently, X is reduced if and only if each stalk OX,p is a reduced local ring
For example, the affine scheme Speck[x,y]/(x2) is not reduced, as the ring k[x,y]/(x2) contains the nilpotent element x
A scheme X is normal if and only if for each open set U⊆X, the ring OX(U) is integrally closed in its total quotient ring
Equivalently, X is normal if and only if each stalk OX,p is an integrally closed local ring
For instance, the affine scheme Speck[x,y]/(y2−x3) is not normal, as the ring k[x,y]/(y2−x3) is not integrally closed (it lacks the element y/x)
Regularity
A scheme X is regular if and only if each stalk OX,p is a regular local ring, i.e., its maximal ideal is generated by dimOX,p elements
For example, the affine scheme Speck[x,y]/(y2−x2(x+1)) is regular, as each stalk is a regular local ring
In contrast, the affine scheme Speck[x,y]/(y2−x3) 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