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Algebraic Geometry Notes 16, 17 November 5, 2008

Summary: Algebraic Geometry Notes 16, 17
Dan Li
November 5, 2008
Recall: If A is a ring, the spectrum of A, (SpecA, O), is the pair consisting
of the topological space SpecA = { prime ideals of A } together with the
sheaf of rings O (i.e. the structure sheaf). What should be the morphisms?
If f is a continuous map f : SpecB SpecA, let V be an open subset of
SpecA, we have an induced map
f : OSpecA(V ) OSpecB(f-1
(V )) = (fOSpecB)(V ),
i.e. f : OSpecA fOSpecB.
1 Schemes
1.1 Schemes
It turns out that we need some restriction on f . Suppose : R S is a
homomorphism of rings, this induces
f : SpecS SpecR; p -1
and f is induced by "pullback of regular functions".
Remark: By convention, a homomorphism of rings takes identity to iden-
tity. In particular -1(p) = R, because if it were, then 1R -1(p). So


Source: Agashe, Amod - Department of Mathematics, Florida State University


Collections: Mathematics