INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

28 Mezzetti4. If Y ⊂ X is closed, X is irreducible, dim X is finite and dimX = dim Y ,then Y = X.Proof.1. Let Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y n be a chain of irreducible closed subsets of Y . Thentheir closures are irreducible and form the following chain: Y 0 ⊆ Y 1 ⊆ . . . ⊆ Y n .Note that for all i Y i ∩ Y = Y i , because Y i is closed into Y , so if Y i = Y i+1 , thenY i = Y i+1 . Therefore the two chains have the same length and we can concludethat dim Y ≤ dimX.2. Let X 0 ⊂ X 1 ⊂ . . . ⊂ X n be a chain of irreducible closed subsets of X. LetP ∈ X 0 be a point: there exists an index i ∈ I such that P ∈ U i . So ∀k = 0, . . ., nX k ∩ U i ≠ ∅: it is an irreducible closed subset of U i , irreducible because open inX k which is irreducible. Consider X 0 ∩U i ⊂ X 1 ∩U i ⊂ . . . ⊂ X n ∩U i ; it is a chainof length n, because X k ∩ U i = X k : in fact X k ∩ U i is open in X k hence dense.Therefore, for all chain of irreducible closed subsets of X, there exists a chain ofthe same length of irreducible closed subsets of some U i . So dim X ≤ sup dim U i .By 1., equality holds.3. Any chain of irreducible closed subsets of X is completely contained in anirreducible component of X. The conclusion follows as in 2.4. If Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y n is a chain of maximal length in Y , then it is amaximal chain in X, because dim X = dim Y . Hence X = Y n ⊂ Y . □7.3. Corollary. dim P n = dim A n .Proof. Because P n = U 0 ∪ . . . ∪ U n , and U i is homeomorphic to A n for all i.□If X is noetherian and all its irreducible components have the same dimensionr, then X is said to have pure dimension r.Note that the topological dimension is invariant by homeomorphism. Bydefinition, a curve is an algebraic set of pure dimension 1; a surface is an algebraicset of pure dimension 2.We want to study the dimensions of affine algebraic sets. The following definitionresults to be very important.7.4. Definition. Let X ⊂ A n be an algebraic set. The coordinate ring of X isK[X] := K[x 1 , . . ., x n ]/I(X).It is a finitely generated K–algebra without non–zero nilpotents, because I(X)is radical. There is the canonical epimorphism K[x 1 , . . ., x n ] → K[X] such thatF → [F]. The elements of K[X] can be interpreted as polynomial functions onX: to a polynomial F, we can associate the function f : X → K such thatP(a 1 , . . ., a n ) → F(a 1 , . . ., a n ).Two polynomials F, G define the same function on X if, and only if, F(P) =G(P) for every point P ∈ X, i.e. if F − G ∈ I(X), which means exactly that Fand G have the same image in K[X].