Definition:Morphism of Ringed Spaces
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Definition
Let $\struct {X, \OO_X}$ and $\struct {Y, \OO_Y}$ be ringed spaces.
Definition 1
A morphism of ringed spaces from $\struct {X, \OO_X}$ to $\struct {Y, \OO_Y}$ is a pair $\struct {f, f^\sharp}$ where:
- $f: X \to Y$ is continuous
- $f^\sharp: \OO_Y \to f_* \OO_X$ is a morphism of sheaves to the direct image sheaf of $\OO_X$ via $f$
Definition 2
A morphism of ringed spaces from $\struct {X, \OO_X}$ to $\struct {Y, \OO_Y}$ is a pair $\struct {f, f^\sharp}$ where:
- $f : X \to Y$ is continuous
- $f^\sharp: f^{-1} \OO_Y \to \OO_X$ is a morphism of sheaves from the inverse image sheaf of $\OO_Y$ via $f$