Definition:Scheme

Definition

Let $\struct {X, \OO_X}$ be a locally ringed space.

Let $\struct {X, \OO_X}$ be such that:

every point $x \in X$ has an open neighborhood $U_x$ such that:

the restriction of $\struct {X, \OO_X}$ to $U_x$ is an affine scheme.

Then $\struct {X, \OO_X}$ is called a scheme.

Also known as

A scheme can also be referred to as a commutative scheme.

Also see

• Definition:Morphism of Schemes
• Definition:Open Subscheme
• Definition:Scheme over Ring
• Definition:Separated Scheme
• Definition:Algebraic Variety
• Definition:Non-Commutative Scheme

Sources

• 1977: Robin Hartshorne: Algebraic Geometry $\S \text{II}.2$ Definition