By Qing Liu

ISBN-10: 0198502842

ISBN-13: 9780198502845

ISBN-10: 0199202494

ISBN-13: 9780199202492

ISBN-10: 1435653149

ISBN-13: 9781435653146

Vector PDF. this is often the paperback version from 2006; loads of typos from the 2002 version are corrected right here.

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**Extra info for Liu - Algebraic Geometry**

**Example text**

Tn ]/I be a ﬁnitely generated algebra over k. Then there is a bijection between the closed points of Spec A and the algebraic set Z(I) := {(α1 , . . , αn ) ∈ k n P (α1 , . . , αn ) = 0, for every P (T ) ∈ I}. Proof We can identify the closed points of Spec A with the maximal ideals of k[T1 , . . , Tn ] containing I. Let m = (T1 − α1 , . . , Tn − αn ) be a maximal ideal of k[T1 , . . , Tn ]. Let P (T ) ∈ k[T1 , . . , Tn ]. Using the Taylor expansion of P (T ) at α := (α1 , . . , αn ), we see that P (T ) ∈ m if and only if P (α) = 0.

B) Let V = {Vk }k be another open covering of X. We deﬁne a partial ordering: V U if every Vk is contained in a Ui . Show that we have a canonical homomorphism FU (W ) → FV (W ) for every open subset W if V U. (c) Show that the ordering makes the set of open coverings of X into a direct system. Let lim FV (W ), F (W ) = −→ V the direct limit being taken over the open coverings of W . 15). 4. Let 0 → F → F → F be an exact sequence of sheaves on X. Show that the sequence of Abelian groups 0 → F (X) → F(X) → F (X) is exact.

10. Let A = k[X1 , X2 ]/(X1 X2 ). Then the homomorphism k[T ] → A deﬁned by T → X1 + X2 is ﬁnite and injective. 9 in fact allows us to obtain a somewhat better result. 30 2. 11. Let k be a ﬁeld, and I a proper ideal of k[X1 , . . , Xn ]. Then there exist a polynomial sub-k-algebra k[S1 , . . , Sn ] of k[X1 , . . , Xn ] and an integer 0 ≤ r ≤ n such that: (a) k[X1 , . . , Xn ] is ﬁnite over k[S1 , . . , Sn ]; (b) k[S1 , . . , Sn ] ∩ I = (S1 , . . , Sr ) (this is the zero ideal if r = 0); (c) k[Sr+1 , .

### Liu - Algebraic Geometry by Qing Liu

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