LOCALLY RINGED SPACE Meaning and
Definition
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A locally ringed space is a fundamental concept in algebraic geometry and topology. It is a pair consisting of a topological space together with a sheaf of rings called the structure sheaf. This sheaf associates to each open set in the space a commutative ring, with the compatibility condition that the restriction maps are ring homomorphisms.
Formally, let X be a topological space. A locally ringed space is a pair (X, O_X), where X is the topological space and O_X is a sheaf of rings on X. The sheaf O_X assigns to each open set U in X a commutative ring O_X(U), such that for every inclusion of open sets V ⊆ U, there is a ring homomorphism O_X(U) → O_X(V) called the restriction map. These restriction maps satisfy the usual compatibility conditions, such as the composition of restrictions being the same as the restriction of the composition.
The ring O_X(U) is often referred to as the ring of "locally defined functions" on U, as it consists of functions that are defined locally, meaning they have values at every point in U. These functions can be added, multiplied, and subjected to other ring operations. The structure sheaf O_X captures the algebraic structure and local properties of the space X, providing a key tool for studying the geometry and topology of X.
