By Denis S. Arnon, Bruno Buchberger
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Extra info for Algorithms in Real Algebraic Geometry
When U is equal to X, we call the corresponding sections global sections. b. General sheaves We will need more general sheaves than function sheaves. d. 3. Let X be a topological space. A presheaf on X is given by the following data: • • For every open set U in X, a set F(U ); For every pair of open sets U and V with V ⊂ U , a map rV,U : F(U ) → F(V ) called the restriction map, such that the two following conditions are satisﬁed: i) If W ⊂ V ⊂ U , then rW,U = rW,V rV,U , ii) We have rU,U = IdF (U ) ; We set rV,U (f ) = f |V .
Such an ideal is said to be homogeneous. Proof. It is clear that 2) implies 1). Conversely, assume that I is generated by homogeneous elements Gi of degrees αi . Consider F = F0 + · · · + Fr ∈ I, where Fi is homogeneous of degree i. By induction, it will be enough to show Ui Gi , and on identifying terms of that Fr ∈ I. But we can write F = highest degree, we get Fr = Ui,r−αi Gi , so Fr is contained in I. 3. Let R be a graded k-algebra and let I be a homogeneous ideal of R. Let S be the quotient k-algebra S = R/I and p the canonical projection.
Ym ) ∈ I(W ) and x ∈ V . We calculate F (ϕ(x)) = F (θ(η1 ), . . , θ(ηm ))(x). Since θ is a morphism of algebras, F (θ(η1 ), . . , θ(ηm )) = θ(F (η1 , . . , ηm )), and since F (η1 , . . , ηm ) is the image in Γ (W ) of F (Y1 , . . , Ym ) ∈ I(W ), it vanishes and we are done. 8. Let ϕ : V → W be a morphism. Then ϕ is an isomorphism if and only if ϕ∗ is an isomorphism. It follows that V and W are isomorphic if and only if their algebras Γ (V ) and Γ (W ) are isomorphic. 9. The morphism ϕ : k → V = V (Y 2 − X 3 ) given by ϕ(t) = (t2 , t3 ) is not an isomorphism.
Algorithms in Real Algebraic Geometry by Denis S. Arnon, Bruno Buchberger