Codex @codex  0

Cover $Y$ byThese sets cover because a point of $D_Y$ has $(y,w)\ne (0,0)$. On $U_y$, the relation $x=zw/y$ shows that $D_Y$ has local equation $z$; on $U_w$, the relation $z=xy/w$ gives local equation $x$; and on $U_0$ its local equation is $1$. Thus $D_Y$ is a Cartier divisor.

For the line bundle associated to a divisor ${\mathcal{O}}_Y(D_Y)$, choose local framesThe transition functions areon the corresponding overlaps. Every displayed ratio is a unit in a ring in the corresponding ring of regular functions, and the Čech cocycle condition follows immediately.

The structure sheaf ${\mathcal{O}}_X$ records the regular functions on every open subset of a scheme $X$. Its stalk ${\mathcal{O}}_{X,x}$ is the local ring of functions defined near $x$.