Let $R$ be a commutative ring with identity and $M$ be an $R$-module. I have trouble understanding the restriction map in the definition of the sheaf of $\mathcal{O}_X$ modules. Explicitly, let $f,g\in R$ s.t. $D(g)\subseteq D(f)$, then what is the map $M_f\to M_g$. My main difficulty is understanding what kind of module homomorphism to expect, since the left hand side module is over $R_f$ while the one on the right is over $R_g$. Most books I flipped through, said "it is defined similarly" referring to the definition of the restriction maps in the structure sheaf of Spec$R$
restriction map in a Sheaf of $\mathcal{O}_X$ modules
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algebraic-geometry
commutative-algebra
category-theory
1 Answers
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If $D(g) \subset D(f)$, then some power of $g$ is a multiple of $f$. That is due to the fact that the radical of an ideal, is precisely the intersection of all of the prime ideals containing it. Then the restriction map is just the localization map $R_f \to R_{fg} = R_g$. Take a look at the first chapter of "Geometry of Schemes", by Eisenbud and Harris.
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0@Matt E. and @beroal: Thanks. This helps clear things up. – 2011-03-06