Lemma 27.8.1. Let $S$ be a graded ring. Let $f \in S$ homogeneous of positive degree.

If $g\in S$ homogeneous of positive degree and $D_{+}(g) \subset D_{+}(f)$, then

$f$ is invertible in $S_ g$, and $f^{\deg (g)}/g^{\deg (f)}$ is invertible in $S_{(g)}$,

$g^ e = af$ for some $e \geq 1$ and $a \in S$ homogeneous,

there is a canonical $S$-algebra map $S_ f \to S_ g$,

there is a canonical $S_0$-algebra map $S_{(f)} \to S_{(g)}$ compatible with the map $S_ f \to S_ g$,

the map $S_{(f)} \to S_{(g)}$ induces an isomorphism

\[ (S_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong S_{(g)}, \]these maps induce a commutative diagram of topological spaces

\[ \xymatrix{ D_{+}(g) \ar[d] & \{ \mathbf{Z}\text{-graded primes of }S_ g\} \ar[l] \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(S_{(g)}) \ar[d] \\ D_{+}(f) & \{ \mathbf{Z}\text{-graded primes of }S_ f\} \ar[l] \ar[r] & \mathop{\mathrm{Spec}}(S_{(f)}) } \]where the horizontal maps are homeomorphisms and the vertical maps are open immersions,

there are compatible canonical $S_ f$ and $S_{(f)}$-module maps $M_ f \to M_ g$ and $M_{(f)} \to M_{(g)}$ for any graded $S$-module $M$, and

the map $M_{(f)} \to M_{(g)}$ induces an isomorphism

\[ (M_{(f)})_{g^{\deg (f)}/f^{\deg (g)}} \cong M_{(g)}. \]

Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \bigcup _{i = 1}^ n D_{+}(g_ i)$.

Let $g_1, \ldots , g_ n \in S$ be homogeneous of positive degree. Then $D_{+}(f) \subset \bigcup D_{+}(g_ i)$ if and only if $g_1^{\deg (f)}/f^{\deg (g_1)}, \ldots , g_ n^{\deg (f)}/f^{\deg (g_ n)}$ generate the unit ideal in $S_{(f)}$.

## Comments (3)

Comment #128 by Fred Rohrer on

Comment #129 by Fred Rohrer on

Comment #135 by Johan on

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