Algebraic Geometry Lecture notes 14 February 2005 Noetherian topological spaces (end of §1) In these notes, k is an algebraically closed field, and for n ≥ 0 we let An = An(k) be kn with the Zariski topology. Definition. A topological space X is called Noetherian if for every descending chain Y1 ⊃ Y2 ⊃ · · · of closed subsets of X there exists n > 0 such that Yn = Yn+1 = ···. Equivalently, X is Noetherian iff every non-empty collection S of closed subsets of X contains a minimal element Y with respect to inclusion, i.e. ∀ Z ∈ S: Z ⊂ Y ⇒ Z = Y . Example. An is Noetherian because of the bijection n {closed subsets of A } ↔ {radical ideals of k[x1, . , xn]}. Furthermore, all (quasi-)affine algebraic varieties are Noetherian. Proposition 1.5. Let X be a Noetherian topological space, and let Y ⊂ X be a closed subset. Then there exist r ≥ 0 and closed subsets Y1,...,Yr of Y such that Y = Y1 ∪ · · · ∪ Yr. Moreover, if we demand Yi 6⊂ Yj for i 6= j, then the Yi are unique (up to permutation). Proof . Hartshorne, Prop. 1.5. §2. Projective varieties For n ≥ 0, we define projective n-space over k to be Pn = Pn(k) := (kn+1 − {0})/ ∼, where the equivalence relation ∼ is defined by x ∼ y ⇐⇒ ∃λ ∈ k×: y = λx (equivalently, x ∼ y ⇐⇒ kx = ky). More generally, if V is a k-vector space, then k× acts on V − {0} by scalar multiplication, and we can define P(V ) := (V − {0})/k×. This means that our definition of Pn n n+1 n+1 n amounts to P = P(A ). The equivalence class of a point (x0, . , xn) ∈ A − {0} in P is denoted by (x0 : x1 : ··· : xn). The projective spaces can be described as follows in terms of their set structure (` means disjoint union): P0 = {(1)} (a single point) 1 P = {(x0 : x1) | (x0, x1) 6= (0, 0)} ` = {(x0 : 1) | x0 ∈ k} {(1 : 0)} = A1 ` {∞} ··· n P = {(x0 : ··· : xn−1 : 1) | x0, . , xn−1 ∈ k} ` {(x0 : ··· : xn−1 : 0) | (x0, . , xn−1) 6= 0} = An ` Pn−1, so that Pn = An ` An−1 ` ··· ` A0. 1 This set-theoretic description of Pn has no topological meaning. A better way to do geometry is to use charts. For 0 ≤ i ≤ n, we define Ui := {(x0 : ··· : xn) | xi 6= 0} = {(x0 : ··· : xi−1 : 1 : xi+1 : ··· : xn) | x0, . , xi−1, xi+1, . , xn ∈ k} and map it bijectively to An: n φi: Ui −→ A x0 xi−1 xi+1 xn (x0 : ··· : xn) 7−→ ,..., , ,..., . xi xi xi xi n n Note that P = ∪i=0Ui. The Zariski topology of Pn n n+1 In this section we view P in relation to A , and we take A = k[x0, . , xn]. The Zariski topology on Pn is defined as the quotient topology on (An+1 − {0})/k×. Here An+1 − {0} has the induced topology from An+1 with its Zariski topology. Let q: An+1 − {0} −→ Pn be the quotient map. Then Y ⊂ Pn is closed if and only if q−1Y is closed in An+1 − {0}, i.e. if there exists an ideal a ⊂ A such that Z(a) ∩ (An+1 − {0}) = q−1Y. In order to make a similar connection between closed subsets of Pn and ideals of A as we did for affine space, we note that we have the following bijections, sending Y ⊂ Pn to q−1Y ⊂ An+1 − {0}, and to (q−1Y ∪ {0}) ⊂ An+1: {closed subsets of Pn} ↔ {closed k×-invariant subsets of An+1 − {0}} ↔ {closed k×-invariant subsets of An+1 containing 0} × ↔ {k -invariant radical ideals of A contained in (x0, . , xn)}. × × To clarify what a k -invariant ideal is, we let k act on A by (λ, f(x0, . , xn)) 7→ f(λx0, . , λxn). An ideal a ⊂ A is k×-invariant iff × f(λx0, . , λxn) ∈ a for all f ∈ A, λ ∈ k . We will see that these ideals are precisely the homogeneous ideals of A, which shall be defined below. Corollary. Pn is a Noetherian topological space. Proof . This follows directly from the fact that A is a Noetherian ring. Definition. A projective algebraic variety is an irreducible closed subset of Pn with the induced topology. A quasi-projective algebraic variety is a non-empty open subset of a projective algebraic variety. Proposition. The quotient map q: An+1 − {0} −→ Pn is open. (It is not closed unless n = 0.) Proof . This is a general fact about continuous group actions. Let U ⊂ An+1 − {0} be open. Then [ q−1qU = λU λ∈k× is open, hence qU is open. Proposition. Let Y ⊂ Pn be closed. Then Y is irreducible if and only if q−1Y is irreducible. Proof . ⇐: Since q is surjective, we have qq−1Y = Y , and the image of an irreducible space is irreducible (exercise). 2 ⇒: Let Y ⊂ Pn be closed and irreducible, so q−1Y 6= ∅. Let U, V ⊂ q−1Y be non-empty open subsets of q−1Y . Then qU, qV ⊂ Y are also non-empty and open, hence qU ∩ qV 6= ∅. This means that we can choose a ∈ An+1 − {0} such that q(a) ∈ qU ∩ qV . Then k×a = q−1{q(a)} intersects both U and V . Consider the map φ: k× ,→ An+1 λ 7→ λa, where k× has the induced topology from A1. This map is continuous, because every closed set of An+1 is an intersection of sets of the form Z(f) with f ∈ k[x0, x1, . , xn] and the inverse image of such a set under φ is −1 × φ Z(f(x0, x1, . , xn)) = Z(f(λx0, . , λxn) ∈ k[λ]) ∩ k , which is closed. Hence φ−1U and φ−1V are non-empty open subsets of k×, and from the irreducibility of k× it follows that U ∩ V 6= ∅. n Proposition. The map φi: Ui → A is a homeomorphism. Proof . Hartshorne, Prop. 2.2. Let Y ⊂ Pn be a closed subset. Then we write S(Y ) = A/I(q−1Y ∪ {0}) for the homogeneous coordinate ring of Y . The elements of S(Y ) are not functions on Y , as in the affine case, but they are functions on the cone of Y : q−1Y ∪ {0}. Graded rings and homogeneous ideals The ring A = k[x0, . , xn] is a graded ring: it can be written as a direct sum ∼ M A = Ad d≥0 of k-vector spaces, where Ad := {f ∈ A | f is homogeneous of degree d}. P This means that for any f ∈ A, there is a unique way to write f = d≥0 fd with fd ∈ Ad for all d (note that fd = 0 for almost all d). We write prd for the canonical projection A → Ad which maps f to fd. For × n+1 d all f ∈ Ad, λ ∈ k and a ∈ A , we have f(λa) = λ f(a). Proposition. Let a ⊂ A be an ideal. Then the following are equivalent: 1. a is k×-invariant. 2. ∀ f ∈ a ∀ d ≥ 0: fd ∈ a. L 3. a = prd a. Ld≥0 4. a = a ∩ Ad. d≥0 S 5. a is generated by homogeneous elements, i.e. by a subset of d≥0 Ad. An ideal satisfying these equivalent conditions is called a homogeneous ideal. P × Proof . 1 ⇒ 2: Let f ∈ a. Take N > 0 sufficiently large such that f = 0≤d<N fd. Then for all λ ∈ k , we P d × have 0≤d<N λ fd ∈ a. Now take λ1, λ2, . , λN ∈ k distinct (this is possible because k is algebraically P d closed, hence infinite), and set gi := 0≤d<N λi fd for 1 ≤ i ≤ N. Then N−1 g1 1 λ1 ··· λ1 f0 . . .. . . = . . N−1 gN 1 λN ··· λN fN−1 and because the gi are in a and the (Vandermonde) matrix formed by the powers of the λi is invertible, the fd are also in a. 3 L L 2 ⇒ 3 and 2 ⇒ 4: The inclusions d≥0(a ∩ Ad) ⊂ a ⊂ d≥0 prd a are automatic, and the reverse inclusions follow directly from 2. 3 ⇒ 5 and 4 ⇒ 5: Both implications are trivially verified. P 5 ⇒ 1: Suppose T is a set of homogeneous generators for a, i.e. a = t∈T A · t with t ∈ Adt . Then for all × P dt λ ∈ k we have λ · a = t∈T A · λ t = a. Definition. For a homogeneous element f ∈ A we define n Zproj(f) := {(a0 : ··· : an) ∈ P | f(a0, . , an) = 0}. S For T ⊂ d≥0 Ad a subset of A consisting of homogeneous elements, we set \ Zproj(T ) := Z(f). f∈T Corollary. The closed subsets of Pn are the sets of the form n+1 Zproj(a) = q Z(a) ∩ (A − {0}) = {(a0 : ··· : an) | f(a0, . , an) = 0 for all homogeneous f ∈ a}, where a is a homogeneous ideal of A. 4.