Antoine Chambert-Loir's Algebraic Geometry of Schemes [Lecture notes] PDF

By Antoine Chambert-Loir

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Extra resources for Algebraic Geometry of Schemes [Lecture notes]

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Let Q = (V, E, s, t) be a quiver and let C be a category. A Q-diagram A in C consists in a family (Av )v∈V of objects of C and in a family ( f e )e∈E of morphisms of C such that for every arrow e ∈ E, f e ∈ C (As(e) , At(e) ). 3. Limits. — A cone on a diagram A is the datum of an object A of C and of morphisms fv ∶ A → Av , for every v ∈ V, such that f e ○ fs(e) = f t(e) for every e ∈ E. 3. LIMITS AND COLIMITS 55 diagram A , there exists a unique morphism g∶ B → A in C such that gv = fv ○ g for every v ∈ V.

Consequently, the three following partially ordered sets are isomorphic: – The set C of closed irreducible subsets of X, ordered by inclusion; – The set of all prime ideals of A, ordered by containment; – The set Spec(A), ordered by the relation x ≺ y if and only if x ∈ {y}. It follows that the dimension of X is equal to the supremum of the lengths of chains of prime ideals of A, the Krull dimension dim(A) of the ring A. For every prime ideal p of A, the codimension of V(p) in Spec(A) is equal to the height ht(p) of p, defined as the supremum of the lengths of chains of prime ideals of A ending at p.

Xn ] is equal to n. Observe that (0) ⊂ (X1 ) ⊂ ⋅ ⋅ ⋅ ⊂ (X1 , . . , Xn ) is a chain of prime ideals of K[X1 , . . , Xn ]; since its length is equal to n, this shows that dim(K[X1 , . . , Xn ]) ⩾ n. Conversely, let (0) ⊊ p1 ⊊ ⋅ ⋅ ⋅ ⊊ pm be a chain of prime ideals of K[X1 , . . , Xn ] and let us set A′ = K[X1 , . . , Xn ]/p1 . Then A′ is a finitely generated K-algebra and dim(A′ ) ⩾ m − 1. Since p1 is a prime ideal, A′ is an integral domain; let F′ be its field of fractions. Any non-zero polynomial P ∈ p1 furnishes gives a non-trivial algebraic dependence relation between the classes x1 , .

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Algebraic Geometry of Schemes [Lecture notes] by Antoine Chambert-Loir


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