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

By Antoine Chambert-Loir

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B) Let q ⊂ q′ be prime ideals of B such that q ∩ A = q′ ∩ A. Then q = q′ . c) The canonical map from Spec(B) to Spec(A) is surjective: for every prime ideal p of A, there exists a prime ideal q of B such that q ∩ A = p. Proof. — a) Passing to the quotients, one gets an integral extension of integral domains A/p ⊂ B/q. 5, A/p is a field if and only if B/q is a field; in other words, p is maximal in A if and only if q is maximal in B. 34 CHAPTER 1. COMMUTATIVE ALGEBRA b) Let p = q ∩ A and let us consider the integral extension of rings Ap ⊂ Bp induced by localization by the multiplicative subset A p.

G n ) ⊂ p; moreover, any prime ideal q which satisfies this relation and which is contained in p contains p′ , but cannot be equal to p′ , hence is equal to p. This shows that V(p) is an irreducible component of V(g1 , . . , g n ). 4). — The dimension of a local noetherian ring is finite. 46 CHAPTER 1. COMMUTATIVE ALGEBRA Proof. — If A is local, and m is its maximal ideal, one has dim(A) = ht(m). Consequently, dim(A) is finite if A is noetherian. 1. 1. — A category C consists in the following data: – A collection ob(C ) of objects; – For every two objects M, N, a set C (M, N) called morphisms from M to N; – For every three objects M, N, P, a composition map C (M, N) × C (N, P), ( f , g) ↦ g ○ f , so that the following axioms are satisfied: (i) For every object M, there is a distinguished morphism idM ∈ C (M, M), called the identity; (ii) One has idN ○ f = f for every f ∈ C (M, N); (iii) One has g ○ idN = g for every g ∈ C (N, P); (iv) For every four objects M, N, P, Q, and every three morphisms f ∈ C (M, N), g ∈ C (N, P), h ∈ C (P, Q), the two morphisms h ○ (g ○ f ) and (h ○ g) ○ f in C (M, Q) are equal (associativity of composition).

A) There are rings of infinite dimension, for example the ring A = K[T1 , T2 , . . ] of polynomials in infinitely many indeterminates. Worse, while all strictly increasing sequences of ideals in a noetherian ring are finite, their lengths may not be bounded. In fact, Nagata has given the following example of a noetherian ring whose dimension is infinite. Let (m n ) be a stricly increasing sequence of positive integers such that m n+1 − m n is unbounded; for each n, let pn be the prime ideal of A generated by the elements Ti , for m n ⩽ i < m n+1 .

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