By Ulrich Gortz, Torsten Wedhorn

ISBN-10: 3834806765

ISBN-13: 9783834806765

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**Extra info for Algebraic geometry 1: Schemes**

**Sample text**

40 we have OX (D(f )) = Γ(X)f . As two aﬃne varieties are isomorphic if and only if their coordinate rings are isomorphic, it suﬃces to show that (D(f ), OX |D(f ) ) is an aﬃne variety. Let X ⊆ An (k) and a = I(X) ⊆ k[T1 , . . , Tn ] be the corresponding radical ideal. We consider k[T1 , . . , Tn ] as a subring of k[T1 , . . , Tn+1 ] and denote by a ⊆ k[T1 , . . , Tn+1 ] the ideal generated by a and the polynomial f Tn+1 −1. Then the aﬃne coordinate ring of Y is Γ(Y ) = Γ(X)f ∼ = k[T1 , . .

Then the following assertions are equivalent. (i) X is an aﬃne cone. (ii) I(X) is generated by homogeneous polynomials. (iii) There exists a closed subset Z ⊆ Pn (k) such that X = C(Z). If in this case I(X) is generated by homogeneous polynomials f1 , . . , fm ∈ k[X0 , . . , Xn ], then Z = V+ (f1 , . . , fm ). Proof. We have already seen that (iii) implies (i). Let us show that (i) implies (ii). To show that I(X) is generated by homogeneous elements, we use that an ideal a ⊆ k[T ] is generated by homogeneous elements if and only if for each g ∈ a its homogeneous components are again in a.

As we can decompose uniquely every polynomial into its homogeneous parts, we have R[X0 , . . , Xn ] = R[X0 , . . , Xn ]d . 58. Let i ∈ {0, . . , n} and d ≥ 0. There is a bijective R-linear map (d) ∼ Φi = Φi : R[X0 , . . , Xn ]d → { g ∈ R[T0 , . . , Ti , . . , Tn ] ; deg(g) ≤ d }, f → f (T0 , . . , 1, . . , Tn ). ) Proof. We construct an inverse map. Let g be a polynomial in the right hand side set d and let g = j=0 gj be its decomposition into homogeneous parts (with respect to T for = 0, .

### Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn

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