Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)} - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:05:00Z http://mathoverflow.net/feeds/question/23182 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23182/restriction-of-proj-s-to-df-is-isomorphic-to-spec-s-f Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)} ashpool 2010-05-01T14:43:48Z 2010-05-02T15:32:25Z <p>$S$ is a graded ring (over non-negative integers), $f \in S_{+}$ is a homogeneous element of positive degree, $D(f)$ the elements of Proj $S$ not containing $f$. I don't see the bijection between $D(f)$ and Spec $S_{(f)}$. Here $S_{(f)}$ is the zero-degree part of $S_{f}$ obtained from $S$ by inverting f. I see the bijection from $D(f)$ to the homogeneous primes in $S_{f}$, but is there 1-1 correspondence between primes in $S_{(f)}$ and homogeneous primes in $S_{f}$?</p> http://mathoverflow.net/questions/23182/restriction-of-proj-s-to-df-is-isomorphic-to-spec-s-f/23183#23183 Answer by Martin Brandenburg for Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)} Martin Brandenburg 2010-05-01T14:52:01Z 2010-05-01T14:52:01Z <p>The homeomorphism $D_+(f) \to \text{Spec } S_{(f)}$ is given by $\mathfrak{p} \mapsto \mathfrak{p} S_f \cap S_{(f)}$ with inverse map $\mathfrak{q} \mapsto \oplus_n \{x \in S_n : x^{|f|} / f^n \in \mathfrak{q}\}$. This can be checked by simple calcuations.</p>