Definition:Composition of Morphisms of Ringed Spaces

Definition

Let $\struct {X, \OO_X}$, $\struct {Y, \OO_Y}$ and $\struct {Z, \OO_Z}$ be ringed spaces.

Let

$\struct {f, f^\sharp: \OO_Y \to f_* \OO_X} : \struct {X, \OO_X} \to \struct {Y, \OO_Y}$

$\struct {g, g^\sharp: \OO_Z \to g_* \OO_Y} : \struct {Y, \OO_Y} \to \struct {Z, \OO_Z}$

be morphisms of ringed spaces.

Let $g_* f^\sharp: g_* \OO_Y \to g_* f_* \OO_X$ denote the pushforward of $f^\sharp$ along $g$.

The composition (of morphisms of ringed spaces) of $\struct {g, g^\sharp}$ with $\struct {f, f^\sharp}$ is defined as:

$\struct {g, g^\sharp} \circ \struct {f, f^\sharp} := \struct {g \circ f, \paren {g_* f^\sharp} \circ g^\sharp}$