Arithmetic Algebraic Geometry


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1 Arithmetic Algebraic Geometry
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3 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016
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5 Contents 1 Preliminaries Affine Varieties Projective Varieties General Properties of Schemes Spectrum of a Ring Ringed Topological Spaces Schemes Reduced Schemes and Integral Schemes Dimension Morphisms and Base Change Base Change Applications to Algebraic Varieties Global Properties of Morphisms Categorical Notions Local Properties Normal Schemes Regular Schemes Flat and Smooth Morphisms Zariski s Main Theorem Coherent Sheaves and Čech Cohomology Coherent Sheaves on a Scheme Čech Cohomology Cohomology of Projective Schemes Sheaves of Differentials Kähler Differentials Differential Study of Smooth Morphisms Local Complete Intersection Duality Theory i
6 CONTENTS 7 Divisors and Applications to Curves Cartier Divisors Weil Divisors RiemannRoch Theorem Algebraic Curves Singular Curves, Structure of Pic 0 (X) ii
7 Chapter 1 Preliminaries 1.1 AFFINE VARIETIES Definition Let X be an arbitrary set. A topology T is a collection of subsets of X such that i) X, T, ii) if G α T for each α in a nonempty index set I, then α I G α T, iii) if G 1,..., G n T, then n i=1g i T. Definition A subcollection B of T is a base if every element of T is a union of sets in B. Definition Suppose (X, T ) and (Y, U) are two topological spaces. A function f : X Y is continuous if f 1 (G) T whenever G U. The function f is open if f(h) is in U whenever H T. A homeomorphism between X and Y is a function that is onetoone, onto, continuous, and open, meaning f 1 is continuous. Definition Let X be a topological space. We say that X is reducible if it can be written as X = X 1 X 2 for closed subsets X 1, X 2 X. Otherwise X is called irreducible. Definition The space X is called disconnected if it can be written as X = X 1 X 2 for closed subsets X 1, X 2 X with X 1 X 2 =. Otherwise X is called connected. Definition Let k be a fixed algebraically closed field. We define affine n space over k, denoted A n k or simply An, to be the set of all ntuples of elements of k. An element P A n will be called a point, and if P = (a 1,..., a n ) with a i k, then the a i will be called the coordinates of P. 1
8 CHAPTER 1. PRELIMINARIES Definition For a subset S K[x 1,..., x n ] of polynomoials we call the zero locus of S. Z(S) = {x A n : f(x) = 0 for all f S} A n Definition A subset Y of A n is an algebraic set if there exists a subset T A such that Y = Z(T ). An algebraic subset V (S) of k n is the set of common zeros of some collection S of polynomials in k[x 1,..., X n ]. The zero set of S is the same as the zero set of the ideal generated by S. Therefore the algebraic subsets of k n can also be described as the zero sets of ideals in k[x 1,..., X n ]. Definition We define the Zariski topology on A n by taking the open subsets to be the complements of the algebraic sets. Definition An affine variety is an irreducible closed subset of A n. An open subset of an affine variety is a quasiaffine variety. Definition For any subset Y A n, we define the ideal of Y in A = k[x 1,..., x n ] by I(Y ) = {f A : f(p ) = 0 for all P Y }. Theorem (Hilbert s Nullstellensatz). Let k be an algebraically closed field, let a be an ideal in A = k[x 1,..., x n ], and let f A be a polynomial which vanishes at all points of Z(a). Then f r a for some integer r > 0. Corollary There is a onetoone inclusionreversing correspondence between algebraic sets in A n and radical ideals (i.e., ideals which are equal to their own radical) in A, given by Y I(Y ) and a Z(a). Furthermore, an algebraic set is irreducible if and only if its ideal is a prime ideal. Notice that A n is irreducible since it corresponds to the zero ideal in A which is prime. Definition Let f be an irreducible polynomial in A = k[x, y]. Then f generates a prime ideal in A, since A is a UFD, so the zero set Y = Z(f) is irreducible. We call it the affine curve defined by the equation f(x, y) = 0. If f has degree d, we say that Y is a curve of degree d. Definition More generally, if f is an irreducible polynomial in A = k[x 1,..., x n ], we obtain an affine variety Y = Z(f), which is called a surface if n = 3, or a hypersurface if n > 3. 2
9 CHAPTER 1. PRELIMINARIES A maximal ideal m of A = k[x 1,..., x n ] corresponds to a minimal irreducible closed subset of A n, which must be a point P = (a 1,..., a n ). This shows that every maximal ideal of A is of the form m = (x 1 a 1,..., x n a n ), for some a 1,..., a n k. Definition If Y A n is an affine algebraic set, we define the affine coordinate ring A(Y ) of Y, to be A/I(Y ). If X is an affine variety then two polynomials f, g k[x 1,..., x n ] define the same polynomial function on X, i.e., f(x) = g(x) for all x X, if and only if f g I(X). So the quotient ring k[x 1,..., x n ]/I(X) can be thought of as the ring of polynomial functions on X Theorem (Hilbert Basis Theorem). The ring k[x 1,..., X n ] is noetherian. This shows that every algebraic set is the zero set of a finite set of polynomials. For the Zariski topology on k, the closed subsets are just the finite sets and the whole space, and so the topology is not Hausdorff. The proper closed subsets of k 2 are finite unions of points and curves. Classifying the irreducible algebraic sets V of A 2, if dim V = 2 then it cant be proper, so it is all of A 2. If dim V = 1, then V = V (f) where f ia any irreducible polynomial in I(V ). If dim V = 0, then it is a point. Correspondingly, the following is a complete list of the prime ideals in k[x, Y ]: (0), (f) with f irreducible, (X a, Y b) with a, b k. Theorem Every proper ideal a in k[x 1,..., X n ] has a zero in k n. Lemma (Zariski s Lemma). Let k K be fields, not necessarily algebraically closed. If K is finitely generated as an algebra over k, then K is algebraic over k. (Hence K = k if k is algebraically closed.) Definition An ideal is said to be radical if it equals its radical. Thus a is radical if and only if the ring A/a is reduced, i.e., without nonzero nilpotent elements. Since integral domains are reduced, prime ideals (and hence maximal ideals) are radical. Theorem (Strong Nullstellensatz). For every ideal a in k[x 1,..., X n ], IV (a) = rad(a); in particular, IV (a) = a if a is a radical ideal. 3
10 CHAPTER 1. PRELIMINARIES To summarize, there are onetoone correspondences (ideals in k[x 1,..., X n ]; algebraic subsets of A n ): radical ideals algebraic subsets prime ideals irreducible algebraic subsets maximal ideals onepoint sets. Let f k[x 1,..., X n ]. We know that k[x 1,..., X n ] is a UFD, and so (f) is prime iff f is irreducible. Thus f irreducible V (f) is irreducible. On the other hand, suppose f factors as Then f = f m i i, f i distinct irreducible polynomials. (f) = (f m i i ) (f m i i ) distinct primary ideals rad(f) = (f i ) (f i ) distinct prime ideals V (f) = V (f i ) V (f i ) distinct irreducible algebraic sets. Definition Let Y be a quasiaffine variety in A n. A function f : Y k is regular at a point P Y if there is an open neighborhood U with P U Y, and polynomials g, h A = k[x 1,..., x n ], such that h is nowhere zero on U, and f = g/h on U. We say that f is regular on Y if it is regular at every point of Y. Definition Let k be a fixed algebraically closed field. A variety over k (or simply variety) is any affine, quasiaffine, projective, or quasiprojective variety. If X, Y are two varieties, a morphism φ : X Y is a continuous map such that for every open set V Y and for every regualr function f : V k, the function f φ : φ 1 (V ) k is regular. Definition Let Y be a variety. We denote by O(Y ) the ring of all regular functions on Y. If P is a point of Y, we define the local ring of P on Y, O P to be the ring of germs of regular functions on Y near P. In other words, an element of O P is a pair U, f where U is an open subset of Y containing P, and f is a regular function on U, and where we identify two such pairs U, f and V, g if f = g on U Y. The maximal ideal m of O P is the set of germs of regular functions which vanish at P. The residue field O P /m is isomorphic to k. Definition If Y is a variety, we define the function field K(Y ) of Y as follows: an element of K(Y ) is an equivalence class of pairs U, f where U is a nonempty, open subset of Y, f is a regular function on U, and where we identify two pairs U, f and V, g if f = g on U V. The elements of K(Y ) are called rational functions on Y. 4
11 CHAPTER 1. PRELIMINARIES By restricting functions we obtain natural maps O(Y ) O P K(Y ) which in fact are injective. Hence we will treat O(Y ) and O P as subrings of K(Y ). Lemma Let g, h A(V ) with h 0. The function D(h) k P g(p )/h(p ) m is zero if and only if gh = 0 in A(V ). This shows that the map A(V ) h O V (D(h)) sending g/h m to the regular function P g(p )/h(p ) m is well defined and injective. Proposition The above map A(V ) h O V (D(h)) is an isomorphism of kalgebras. Theorem Let Y A n be an affine variety with affine coordinate ring A(Y ). Then: i) O(Y ) = A(Y ); ii) for each point P Y, let m P A(Y ) be the ideal of functions vanishing at P. Then P m P gives a 11 correspondence between the points of Y and the maximal ideals of A(Y ); iii) for each P, O P = A(Y )mp, and dim O P = dim Y ; iv) K(Y ) is isomorphic to the quotient field of A(Y ), and hence K(Y ) is a finitely generated extension field of k, of transcendence degree = dim Y. Let V be an algebraic set and let P be a point on V. We know that there is a 11 correspondence between the prime ideals of A(V ) contained in m P and the prime ideals of O P. In geometric terms, this says that there is a 11 correspondence between the irreducible closed subsets of V passing through P and the prime ideals in O P. The ideal p corresponding to an irreducible closed subset Z consists of the elements of O P represented by a pair (U, f) with f Z U = 0. Let V be an algebraic subset of k n, and let A = A(V ). We can describe (V, O V ) purely in terms of A: i) V is the set of maximal ideals in A. ii) For each f A, let D(f) = {m : f / m}; the topology on V is that for which the sets D(f) form a base. iii) For f A h and m D(h), let f(m) denote the image of f in A h /ma h = k; in this way A h becomes identified with a kalgebra of functions D(h) k, and O V is the unique sheaf of kvalued functions on V such that Γ(D(h), O V ) = 5
12 CHAPTER 1. PRELIMINARIES A h for all h A. When V is irreducible, all the rings attached to it can be identified with subrings of the field k(v ). For example, Γ(D(h), O V ) = {g/h n k(v ) : g k(v ), n N} O P = {g/h k(v ) : h(p ) 0} Γ(U, O V ) = P U O P = Γ(D(h i ), O V ) if U = D(h i ). Definition An affine kalgebra is a reduced finitely generated kalgebra. For such an algebra A, there exist x i A such that A = k[x 1,..., x n ], and the kernel of the homomorphism X i x i : k[x 1,..., X n ] A is a radical ideal. Also the intersection of the maximal ideals in A is 0. Also, Zariski s lemma implies that for every maximal ideal m A, the map k A A/m is an isomorphism. Thus we can identify A/m with k. For f A, we write f(m) for the image of f in A/m = k, i.e., f(m) mod m. We attach a ringed space (V, O V ) to A by letting V be the set of maximal ideals in A. For f A, let D(f) = {m : f(m) 0} = {m : f / m}. Definition Let X, Y be varieties. A rational map φ : X Y is an equivalence class of pairs U, φ U where U is a nonempty open subset of X, φ U is a morphism of U to Y, and where U, φ U and V, φ V are equivalent if φ U and φ V agree on U V. The rational map φ is dominant if for some (and hence every) pair U, φ U, the image of φ U is dense in Y. The next result shows that if X and Y are affine varieties, then X is isomorphic to Y if and only if A(X) is isomorphic to A(Y ) as a kalgebra. Proposition Let X be any variety and let Y be an affine variety. Then there is a natural bijective mapping of sets α : Hom(X, Y ) Hom(A(Y ), O(X)) where the left is morphisms of varieties, and the right is homomorphisms of k algebras. 6
13 CHAPTER 1. PRELIMINARIES Corollary The functor X A(X) induces an arrowreversing equivalence of categories between the category of affine varieties over over k and the category of finitely generated integral domains over k. 1.2 PROJECTIVE VARIETIES Definition Let k be our fixed algebraically closed field. We define projective nspace over k, denoted P n k or simply Pn, to be the set of equivalence classes of (n+1)tuples (a 0,..., a n ) of elements of k, not all zero, under the equivalence relation given by (a 0,..., a n ) (λa 0,..., λa n ) for all λ k, λ 0. Another way of saying this is that P n as a set is the quotient of the set A n+1 {(0,..., 0)} under the equivalence relation which identifies points lying on the same line through the origin. Definition If P is an element/point of P n, then any (n+1)tuple (a 0,..., a n ) in the equivalence class P is called a set of homogeneous coordinates for P. Obviously, a 1dimensional linear subspace of K n+1 is uniquely determined by a nonzero vector in K n+1, with two such vectors spanning the same linear subspace if and only if they are scalar multiples of each other. In other words, P n = (K n+1 \{0})/ or also as (K n+1 \{0})/K. For a geometric interpretation, consider the map f : A n P n such that (x 1,..., x n ) (1 : x 1 : : x n ). We can embed the affine space A n in K n+1 at the height x 0 = 1 and think of f as mapping a point to the 1dimensional linear subspace spanned by it. Usually, it is more helpful to think of P n as the affine space A n compactified by adding some points at infinity. Definition Let B be a commutative ring (with unit). A grading on B consists of a decomposition of B in subgroups B = d 0 B d, such that B d B e B d+e. The elements of B d are called the homogeneous elements of degree d. If B is an algebra over a ring A, we also ask that the image of A in B be contained in B 0. We then call B a graded algebra over A. 7
14 CHAPTER 1. PRELIMINARIES The polynomial ring R = K[x 0,..., x n ] is a graded ring with { } R d = a i0,,i n x i 0 0 x in n : a i0,...,i n K for all i 0,..., i n. i 0 + +i n=d Definition A subset Y of P n is an algebraic set if there exists a set T of homogeneous elements of a graded ring S such that Y = Z(T ). Definition We define the Zariski topology on P n by taking the open sets to be the complements of algebraic sets. Definition A projective variety is an irreducible algebraic set in P n, with the induced topology. An open subset of a projective variety is a quasiprojective variety. If Y P n, we define the homogeneous ideal of Y in the graded ring S, denoted I(Y ), to be the ideal generated by {f S : f is homogeneous and f(p ) = 0 for all P Y }. If Y is an algebraic set, we define the homogeneous coordinate ring of Y to be S(Y ) = S/I(Y ). Projective nspace has an open covering by affine nspaces, and hence every projective (resp. quasiprojective) variety has an open covering by affine (resp. quasiaffine) varieties. If f S is a linear homogeneous polynomial, then the zero set of f is called a hyperplane. In particular, we denote the zero set of x i by H i, for i = 0,..., n. Let U i be the open set P n H i. Then P n is covered by the open sets U i, because if P = {a 0,..., a n } is a point, then at least one a i 0, hence P U i. We define a mapping φ i : U i A n as follows: if P U i, then φ i (P ) = Q, where Q is the point with affine coordinates ( a0,, a ) n a i a i with a i /a i ommited. Proposition The map φ i is a homeomorphism of U i with its induced topology to A n with its Zariski topology. Corollary If Y is a projective (resp. quasiprojective) variety, then Y is covered by the open sets Y U i, i = 0,..., n, which are homeomorphic to affine (resp. quasiaffine) varieties via the mapping φ i defined above. Proposition Let U i P n be the open set defined by the equation x i 0. Then the mapping φ i : U i A n, is an isomorphism of varieties. 8
15 CHAPTER 1. PRELIMINARIES If S is a graded ring, and p is a homogeneous prime ideal in S, then we denote by S (p) the subring of elements of degree 0 in the localization of S with respect to the multiplicative subset T consisting of the homogeneous elements of S\p. Note that T 1 S has a natural grading given by deg(f/g) = deg f deg g for f homogeneous in S and g T. S (p) is a local ring, with maximal ideal (pt 1 S) S (p). In particular, if S is a domain, then for p = 0 we obtain a field. Similarly, if f S is a homogeneous element, we denote by S (f) the subring of elements of degree 0 in the localized ring S f. Theorem Let Y P n be a projective variety with homogeneous coordinate ring S(Y ). Then i) O(Y ) = k; ii) for any point P Y, let m P S(Y ) be the ideal generated by the set of homogeneous f S(Y ) such that f(p ) = 0. Then O P = S(Y ) (mp ); iii) K(Y ) = S(Y ) ((0)). 9
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17 Chapter 2 General Properties of Schemes 2.1 SPECTRUM OF A RING To a geometric object we can associate algebra via functions, and the reverse construction, via the notion of a spectrum. Definition Let A be a (commutative) ring (with unit). We let Spec A denote the set of prime ideals of A. We call it the spectrum of A. By convention, the unit ideal is not a prime ideal. Thus Spec{0} =. We will now endow Spec A with a topological structure. For any ideal I of A, let V (I) = {p Spec A : I p}. If f A, let D(f) = Spec A\V (fa). Proposition Let A be a ring. We have the following properties: i) For any pair of ideals I, J of A, we have V (I) V (J) = V (I J). ii) Let (I λ ) λ be a family of ideals of A. Then λ V (I λ ) = V ( λ I λ). iii) V (A) = and V (0) = Spec A. Definition Let A be a ring. We call the topology defined above, the Zariski topology on Spec A. An open set of the form D(f) is called a principal open subset, while its complement V (f) = V (fa) is called a principal closed subset. Definition Let p Spec A. Then the singleton {p} is closed for the Zariski topology if and only if p is a maximal ideal of A. We will then say that p is a closed point of Spec A. More generally, a point x of a topological space is said to be closed if the set {x} is closed. Definition The Zariski closure of any Z Spec(R), is just its closure 11
18 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES in the Zariski topology: Z = V (f). f R;f Z =0 Definition Let I be an ideal of A. The radical of I, I is the set of elements a A such that a n I for some n 1. We have that V (I) = V ( I). Lemma Let A be a ring. Let I, J be two ideals of A. The following properties are true. i) The radical I equals the intersection of the ideals p V (I). ii) We have V (I) V (J) if and only if J I. Let φ : A B be a ring homomorphism. Then we have a map of sets Spec φ: Spec B Spec A defined by p φ 1 (p) for every p Spec B. Lemma Let φ : A B be a ring homomorphism. Let f = Spec φ be the map above. The following properties are true. i) The map f is continuous. ii) If φ is surjective, then f induces a homeomorphism from Spec B onto the closed subset V (Kerφ) of Spec A. iii) If φ is a localization morphism A S 1 A, then f is a homeomorphism from Spec(S 1 A) unto the subspace {p Spec A : p S = } of Spec A. Definition Let k be a field. A finitely generated kalgebra A, is an algebra which is the quotient of a polynomial ring over k. Definition We say that a ring homomorphism φ : A 0 A is integral or that A is integral over A 0, if every a A is integral over A 0, that is to say that there exists a monic polynomial i α it i A 0 [T ] such that i φ(α i)a i = 0. Such an equality is called an integral equation for a over A 0. The set of elements of A that are integral over A 0 form a subring of A. We say that A is finite over A 0 if it is a finitely generated A 0 module. It is easy to see that A is finite over A 0 if and only if it is integral and finitely generated over A 0. Proposition (Noether Normalization Lemma) Let A be a nonzero finitely generated algebra over a field k. Then there exist an integer d 0 and a finite injective homomorphism k[t 1,..., T d ] A. Proposition Let k be a field, and I a proper ideal of k[x 1,..., X n ]. Then there exist a polynomial subkalgebra k[s 1,..., S n ] of k[x 1,..., X n ] and an integer 0 r n such that: i) k[x 1,..., X n ] is finite over k[s 1,..., S n ]; ii) k[s 1,..., S n ] I = (S 1,..., S r ) (this is the zero ideal if r = 0); iii) k[s r+1,..., S n ] k[x 1,..., X n ]/I is finite injective. 12
19 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Corollary Let A be a finitely generated algebra over a field k. Let m be a maximal ideal of A. Then A/m is a finite algebraic extension of k. Corollary (Weak Nullstellensatz) Let k be an algebraically closed field. Then for any maximal ideal m of k[t 1,..., T n ], there exists a unique point (α 1,..., α n ) k n such that m is the ideal generated by T 1 α 1,..., T n α n. Definition Let k be an algebraically closed field. Let P 1 (T ),..., P m (T ) be polynomials in k[t 1,..., T n ]. Let Z(P 1,..., P m ) = {(α 1,..., α n ) k n : P j (α 1,..., α n ) = 0, 1 j m}. We call such a set an algebraic set. It is the set of solutions of a system of polynomial equations. It is also the set of common zeros of the polynomials belonging to the ideal generated by the P j (T ). Corollary Let k be an algebraically closed field. Let A = k[t 1,..., T n ]/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) = {(α 1,..., α n ) k n : P (α 1,..., α n ) = 0, for every P (T ) I}. The object of algebraic geometry is the study of solutions of systems of polynomial equations over a field k. The above corollary explains why such a study corresponds to that of the spectra of finitely generated algebras over k. Lemma Let A be a nonzero finitely generated algebra over a field k. Then the intersection of the maximal ideals of A is equal to the nilradical 0 of A. Proposition Let k be an algebraically closed field. Let I be an ideal of k[t 1,..., T n ]. If a polynomial F k[t 1,..., T n ] is such that F (α) = 0 for every α Z(I), then F I. If k is a field and R is a kalgebra, then R/I is also a kalgebra, so in particular if I is maximal, then k R/I is a map of fields, and therefore a field extension. Thus, if k is algebraically closed and R is a finitely generated k algebra, then maximal ideals of R are in bijection with homomorphisms R k. Thus, given a ring R, we ll associate a set MSpec(R), the set of maximal ideals of R, such that R should be its ring of functions. To do this, we ll say that an r R is a function on MSpec(R) by acting on an m x R as r mod m x. This is a number, since it s in a field, but the notion may be different at every point in MSpec(R). For example, if R = Z, then MSpec(Z) is the set of primes, and n Z is a function which at 2 is n mod 2, at 3 is n mod 3, and so on. Another example is when R = R[x], which has maximal ideals (x t) for all t R. Here, 13
20 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES evaluation sends f(x) f(x) mod (x t) = f(t). That is, this is really evaluation, and here the quotient field is R. So these look like normal realvalued functions, but these aren t all the maximal ideals: (x 2 + 1) is also maximal, and R[x]/(x 2 + 1) = C. Then, we do get a kind of evaluation again, but we have to identify points and their complex conjugates. If m R is maximal, that s the same as a surjection R k. But if p is a prime ideal, then the surjection R R/p is onto an integral domain. So prime ideals are surjections onto integral domains. If p is a prime ideal, then R R/p F rac(r/p), and the composite map may not be surjective, but its image generates the field F rac(r/p). In other words, a prime ideal is the same as a homomorphism R k which generates k as a field. The field associated to a prime ideal is called its residue field. Now, suppose r R and p R is prime (we ll think of it as a point x Spec(R)). Then there s an evaluation map r(x) = r mod p R/p, or even inside F rac(r/p). So we can think of R as the set of regular functions on Spec(R). As another example, for Spec(Z), Z initial in the category of rings Spec(Z) is final in the category of affine schemes. The picture is a point for every prime p Z, with residue field F p, but also the zero ideal, corresponding to the generic point, whose residue field is Q. The intuition is that every rational number is a function at all but finitely many points: 19/15 Q, so we can evaluate (19/15)(7) = 5 mod 7, and do this everywhere except 3 and 5, where are its poles (its value at the generic point is 19/15 again). Since we have a map Z Q, then we d better have a nice map Spec(Q) Spec(Z), corresponding to morphisms of residue fields. Since Spec(Q) is a point, we can just send it to the generic point, whose residue field is Q. This is why we need prime ideals (and generic points as a consequence). We also have affine nspace over a ring R, A n R = Spec(R[x 1,..., x n ]). If R = k is a field, then the affine line over k is A 1 k = Spec(k[x]). This ring is a PID, and in particular primes correspond to irreducible polynomials, which correspond to Galois orbits of points in k, along with one generic point. There are two types of localization we usually deal with. Given f R, there s a multiplicative subset S = {1, f, f 2,...}, and the localization S 1 R is denoted R f. Then Spec(R f ) = {p R : f / p}. For example, Spec(Z 15 ) is the same as Spec(Z) but with the points (3) and (5) removed. The generic point is still in this set since 15 mod (0) = 15. Similarly, Spec ((x 2 + y 2 1) 1 C[x, y]) is the affine plane minus the circle. We also define localization at a prime, letting S = R\p and denote S 1 R as R p. This removes everything except the things that are inside of p. That is, Spec(R p ) is the prime ideals contained in p. For example, Spec(Z (5) ) contains (5) and (0). 14
21 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES 2.2 RINGED TOPOLOGICAL SPACES Definition Let X be a topological space. groups) on X consists of the following data:  an Abelian group F(U) for every open subset U of X, and A presheaf F (of Abelian  a group homomorphism (restriction map) ρ UV : F(U) F(V ) for every pair of open subsets V U which verify the following conditions: i) F( ) = 0; ii) ρ UU = Id; iii) if we have three open subsets W V U, then ρ UW = ρ V W ρ UV. An element s F(U) is called a section of F over U. We let s V denote the element ρ UV (s) F(V ), and we call it the restriction of s to V. Definition We say that a presheaf F is a sheaf if we have the following properties: iv) (Uniqueness) Let U be an open subset of X, s F(U), {U i } i a covering of U by open subsets U i. If s Ui = 0 for every i, then s = 0. v) (Glueing local sections) Keeping the previous notation, let s i F(U i ), i I, be sections such that s i Ui U j = s j Ui U j. Then there exists a section s F(U) such that s Ui = s i (this section s is unique by condition iv). We can define in the same way, sheaves of rings etc. Definition We have that F is a subsheaf of F when F (U) is a subgroup of F(U), and the restriction ρ UV is induced by ρ UV. Definition If U is an open subset of X, every presheaf F on X induces a presheaf F U on U by setting F U (V ) = F(V ) for every open subset V of U. This is the restriction of F to U. If F is a sheaf, then so is F U. Definition Let B be a base of open subsets on X (a set of open subsets of X such that any open subset of X is a union of open subsets in B, and B is stable by finite intersection). We define Bpresheaves/sheaves by replacing open subset U of X by open set U belonging to B in the definition above. Every Bsheaf F 0 extends in a unique way to a sheaf F on X. A sheaf is completely determined by its sections over a base of open sets. 15
22 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition Let F be a presheaf on X, and let x X. The stalk of F at x is the group F x = lim U x F(U), the direct limit being taken over the open neighborhoods U of x. If F is a presheaf of rings, then F x is a ring. Let s F(U) be a section; for any x U, we denote the image of s in F x by s x. We call s x the germ of s at x. The map F(U) F x defined by s s x is clearly a group homomorphism. We can write the stalk F x as {f U F(U) where x U}/ ( f U f V if res W U f U = res W V f V for some W U V ). We identify functions that agree on a neighborhood of x. Lemma Let F be a sheaf on X. Let s, t F(X) be sections such that s x = t x for every x X. Then s = t. Definition Let F, G be two presheaves on X. A morphism of presheaves α : F G consists of, for every open subset U of X, a group homomorphism α(u) : F(U) G(U) which is compatible with the restrictions ρ UV. A morphism of presheaves α is called injective if for every open subset U of X, α(u) is injective. An isomorphism is an invertible morphism, this amounts to saying that α(u) is an isomorphism for every open U of X. Let α : F G be a morphism of presheaves on X. For any x X, α canonically induces a group homomorphism α x : F x G x such that (α(u)(s)) x = α x (s x ) for any open subset U of X, s F(U), and x U. We say that α is surjective if α x is surjective for every x X. By definition of the direct limit, this means that for every t x G x, there exist an open neighborhood U of x and a section s F(U) such that (α(u)(s)) x = t x. Proposition Let α : F G be a morphism of sheaves on X. Then α is an isomorphism if and only if α x is an isomorphism for every x X. Corollary Let α : F G be a morphism of sheaves. Then it is an isomorphism if and only if it is injective and surjective. Definition Let F be a presheaf on X. We define the sheaf associated to F to be the sheaf F together with a morphism of presheaves θ : F F verifying the following universal property: For every morphism α : F G, where G is a sheaf, there exists a unique morphism α : F G such that α = α θ. Proposition Let F be a presheaf on X. Then the sheaf F associated to F exists and is unique up to isomorpism. Moreover, θ x : F x F x is an isomorphism for every x X. 16
23 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition Let F be a sheaf on X and F a subsheaf. Then U F(U)/F (U) is a presheaf on X. The associated sheaf F/F is called the quotient sheaf. Let α : F G be a morphism of sheaves. Then U Ker(α(U)) is a subsheaf of F, denoted by Kerα. This sheaf is called the kernel of α. On the other hand, U Im(α(U)) is in general, only a presheaf, and the associated sheaf Imα is called the image of α. Lemma Let F, G be sheaves on X, F a subsheaf of F, α : F G a morphism. We have (F/F ) x = F x /F x, (Imα) x = Im(α x ), and Imα is a subsheaf of G. Definition We say that a sequence of sheaves F α G β H is exact if Imα = Kerβ. Definition We can transfer sheaves from one space to another. Let f : X Y be a continuous map of topological spaces. Let F be a sheaf on X and G a sheaf on Y. Then V F(f 1 (V )) defines a sheaf f F on Y which we call the direct image of F. On the other hand U lim G(V ) V f(u) defines a presheaf on X; the associated sheaf is called the inverse image of G and is denoted by f 1 G. We have the property (f 1 G) x = G f(x), for every x X. Note that if V is an open subset of Y and if we let i : V Y denote the canonical injection, then i 1 G = G V. Definition A ringed topological space (locally ringed in local rings) consists of a topological space X endowed with a sheaf of rings O X on X such that O X,x is a local ring for every x X. We denote it (X, O X ). The sheaf O X is called the structure sheaf of (X, O X ). Let m x be the maximal ideal of O X,x ; we call O X,x /m x the residue field of X at x, and we denote it k(x). Evaluating at x is exactly quotienting by m x. Definition A morphism of ringed topological spaces (f, f # ) : (X, O X ) (Y, O Y ) consists of a continuous map f : X Y and a morphism of sheaves of rings f # : O Y f O X such that for every x X, the stalk f x # : O Y,f(x) O X,x is a local homomorphism (i.e., f x # 1 (m x ) = m f(x) or, equivalently, f x # (m f(x) ) m x ). 17
24 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition An open subscheme of a scheme X is a scheme U, whose topological space is an open subset of X, and whose structure sheaf O U is isomorphic to the restriction O X U of the structure sheaf of X. An open immersion is a morphism f : X Y which induces an isomorphism of X with an open subscheme of Y. Definition A closed immersion is a morphism f : Y X of schemes such that f induces a homeomorphism of sp(y ) onto a closed subset of sp(x), and furthermore the induced map f # : O X f O Y of sheaves on X is surjective. A closed subscheme of a scheme X is an equivalence class of closed immersions, where we say f : Y X and f : Y X are equivalent if there is an isomorphism i : Y Y such that f = f i. Definition We say that a morphism (f, f # ) : (X, O X ) (Y, O Y ) is an open immersion (resp. closed immersion) if f is a topological open immersion (resp. closed immersion) and if f # x is an isomorphism (resp. if f # x is surjective) for every x X. Let J be a sheaf of ideals of O X (i.e., J (U) is an ideal of O X (U) for every open subset U). Let V (J ) = {x X : J x O X,x }. Lemma Let X be a ringed topological space. Let J be a sheaf of ideals of O X. Let j : V (J ) X denote the inclusion. Then V (J ) is a closed subset of X, (V (J ), j 1 (O X /J )) is a ringed toplogical space, and we have a closed immersion (j, j # ) of this space into (X, O X ), where j # is the canonical surjection O X O X /J = j (j 1 (O X /J )). Proposition Let f : Y X be a closed immersion of ringed topological spaces. Let Z be the ringed topological space V (J ) where J = Kerf # O X. Then f factors into an isomorphism Y = Z followed by the canonical closed immersion Z X. 2.3 SCHEMES Let us consider the space X = Spec A endowed with its Zariski topology. We will construct a sheaf of rings O X on X. Let D(f) be a princiapl open subset of 18
25 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES X. Let us set O X (D(f)) = A f. Let D(g) D(f), in which case g fa. Hence there exist m 1 and b A such that g m = fb. Thus f is invertible in A g. This canonically induces a restriction homomorphism A f A g which maps af n A f to (ab n )g nm A g. If D(f) = D(g), we easily verify that A f A g is an isomorphism. Therefore O X (D(f)) does not depend on the choice of f. We have thus constructed a Bpresheaf, where B is the base of open subsets on X made up of the principal open sets. Proposition Let A be a ring and X = Spec A. We have the following properties: i) O X is a Bsheaf of rings. It therefore induces a sheaf of rings O X on Spec A, and we have O X (X) = A. ii) For any p X, the stalk O X,p is canonically isomorphic to A p. In particular, (X, O X ) is a ringed topological space. Definition We define an affine scheme to be a ringed topological space isomorphic to some (Spec A, O Spec A ) constructed as above. Lemma Let A be an integral domain, with field of fractions K. Let ξ be the point of X = Spec A corresponding to the prime ideal 0. Then O X,ξ = K. Moreover, for every nonempty openn subset U of X, we have ξ U, and the canonical homomorphism O X (U) O X,ξ is injective. If V U, then the restriction O X (U) O X (V ) is injective. For an integral domain A, O Spec A is a subsheaf of the constant presheaf K = F rac(a) (which in this case is a sheaf). Lemma Let X = Spec A be an affine scheme, and let g A. Then the open subset D(g), endowed with the structure of a ringed topological space induced by that of X, is an affine scheme isomorphic to Spec A g (as a ringed topological space). Definition A scheme is a ringed topological space (X, O X ) admitting an open covering {U i } i such that (U i, O X Ui ) is an affine scheme for every i. We will denote it by X when there is no confusion possible. Let U be an open subset of a scheme X. The elements of O X (U) are called regular functions on U. An affine scheme is a scheme. If a ringed topological space X admits an open covering {U i } i such that (U i, O X Ui ) is a scheme for every i, then X is a scheme. Proposition Let X be a scheme. Then for any open subset U of X, the ringed topological space (U, O X U ) is also a scheme. 19
26 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition Let X be a scheme. Let U be an open subset of X. We will say that (U, O X U ) (or more simply that U) is an open subscheme of X. We will say that U is an affine open subset if (U, O X U ) is an affine scheme. Definition Let X be a scheme and f O X (X). We denote by X f the set of x X such that f x O X,x. Let us consider the following condition: X admits a covering by a finite number of affine open subsets {U i } i such that U i U j also admits a finite covering by affine open subsets. Affine schemes always verify this condition, as do Noetherian schemes. Proposition Let X be a scheme and f O X (X). Then X f is an open subset of X. Moreover, if X verifies the above condition, then the restriction O X (X) O X (X f ) induces an isomorphism O X (X) f = OX (X f ). Definition A morphism of schemes f : X Y is a morphism of ringed topological spaces. An isomorphism of schemes is an isomorphism of ringed topological spaces. An open/closed immersion of schemes is an open or closed immersion of ringed topological spaces. The proposition that follows makes it possible to construct morphisms of schemes. Let φ : A B be a ring homomorphism. We have defined a continuous map Spec φ : Spec B Spec A. Let us denote it by f φ. Proposition Let φ : A B be a ring homomorphism. Then there exists a morphism of schemes (f φ, f # φ ) : Spec B Spec A such that f # φ (Spec A) = φ. Lemma Let A be a ring. Let I be an ideal of A. Then the morphism of schemes i : Spec A/I Spec A, induced by the canonical surjection A A/I, is a closed immersion of schemes whose image is V (I). Moreover, for any principal open subset D(g) of Spec A, we have (Keri # )(D(g)) = I A A g. Definition A closed subscheme of X is a closed subset Z of X endowed with the structure (Z, O Z ) of a scheme and with a closed immersion (j, j # ) : (Z, O Z ) (X, O X ), where j : Z X is the canonical injection. Proposition Let X = Spec A be an affine scheme. Let j : Z X be a closed immersion of schemes. Then Z is affine and there exists a unique ideal J of A such that j induces an isomorphism from Z onto Spec A/J. 20
27 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition Let S be a scheme. An Sscheme or a scheme over S is a scheme X endowed with a morphism of schemes π : X S. The morphism π is then called the structural morphism and S the base scheme. When S = Spec A, we will also say scheme over A or Ascheme instead of scheme over S, and A will be called base ring. If π : X S, ρ : Y S are Sschemes, a morphism of Sschemes f : X Y is a morphism of schemes that is compatible with the structural morphisms (i.e., ρ f = π). Proposition Let k be an algebraically closed field. There is a natural fully faithful functor t : V(k) S(k) from the category of varieties over k to schemes over k. For any variety V, its topological space is homeomorphic to the set of closed points of sp(t(v )), and its sheaf of regular functions is obtained by restricting the structure sheaf of t(v ) via this homeomorphism. Let Mor(X, Y ) denote the set of morphisms of schemes from X to Y, and A, B be rings. We have a canonical map ρ : Mor(X, Y ) Hom rings (O Y (Y ), O X (X)) which to (f, f # ) associates f # (Y ) : O Y (Y ) f O X (Y ) = O X (X). This map is functorial in X. Lemma Let X, Y be affine schemes. Then the above canonical map, is bijective. This lemma expresses the fact that the category of affine schemes is equivalent to that of (commutative) rings (with unit). Proposition Let Y be an affine scheme. For any scheme X, the canonical map ρ : Mor(X, Y ) Hom rings (O Y (Y ), O X (X)) is bijective. Corollary Let A be a ring. Then giving a scheme X over A is equivalent to giving a scheme X and the structure of a sheaf of Aalgebras on O X. Definition Let π : X S be an Sscheme. A section of X is a morphism of Sschemes σ : S X. This amounts to saying that π σ = Id S. The set of sections of X is denoted X(S) (and also by X(A) if S = Spec A). Definition Let X be a scheme over a field k. With the notation above, we call the points of X(k) (k)rational points of X. The notion of rational points is fundamental in arithmetic geometry. 21
28 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Let Y be an open or closed subscheme of X. For any point y Y, the residue fields of O Y and of O X at y are isomorphic. So if X is a scheme over a field k, then Y (k) = X(k) Y. Also, solving a system of polynomial equations in k amounts to determining the set of rational points of a scheme of the type Spec k[t 1,..., T n ]/I. Lemma (Glueing Schemes) Let S be a scheme. Let us consider a family {X i } i of schemes over S. We suppose given open subschemes X ij of X i and isomorphisms of Sschemes f ij : X ij X ji such that f ii = Id Xi, f ij (X ij X ik ) = X ji X jk, and f ik = f jk f ij on X ij X ik. Then there exists an Sscheme X, unique up to isomorphism, with open immersions (of Sschemes) g i : X i X such that g i = g j f ij on X ij, and that X = i g i (X i ). The scheme X is called the glueing of the X i via the isomorphisms f ij, or along the X ij. Definition Fix a ring A. Let B = d 0 B d be a graded Aalgebra. An ideal I of B is said to be homogeneous if it is generated by homogeneous elements, i.e., I = d 0 (I B d ). The quotient B/I then has a natural grading (B/I) d = B d /(I B d ). We let Proj B denote the set of homogeneous prime ideals of B which do not contain the ideal B + = d>0 B d. For any homogeneous ideal I of B, we let V + (I) denote the set of p Proj B containing I. We have the trivial equalities: i V + (I i ) = V + ( i I i ), V + (I) V + (J) = V + (I J), V + (B) =, V + (0) = Proj B. This makes it possible to endow Proj B with the topology whose closed sets are of the form V + (I). We call it the Zariski topology on Proj B. Let I be an arbitrary ideal of B. We can associate to it a homogeneous ideal I h = d 0 (I B d ). I is homogeneous if and only if I = I h. Lemma Let I, J be ideals of a graded ring B. Then the following are true: i) If I is prime, then the associated homogeneous ideal I h is prime. ii) Let us suppose I and J are homogeneous. Then V + (I) V + (J) if and only if J B + I. iii) We have Proj B = if and only if B + is nilpotent. Let f B be a homogeneous element, and write D + (f) = Proj B\V + (fb). The open sets of this form are called principal open sets. They are a base of open sets of Proj B. In fact we can restrict ourselves to the open sets D + (f) 22
29 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES with f B +. Indeed, = V + (B + ) = i V + (f i ) where the f i are homogeneous elements that generate B +, so Proj B = i D + (f i ). So for every homogeneous g B, we have D + (g) = i D + (gf i ) with gf i B +. If f B is homogeneous, we let B (f) denote the subring of B f made up of the elements of the form af N, N 0, deg a = N deg f. These are the elements of degree 0 of B f. Lemma Let f B + be a homogeneous element of degree r. i) There exists a canonical homeomorphism θ : D + (f) Spec B (f). ii) Let D + (g) D + (f) and α = g r f deg g B (f). Then θ(d + (g)) = D(α). iii) We have a canonical homomorphism B (f) B (g) which induces an isomorphism (B (f) ) α = B(g). iv) Let I be a homogeneous ideal of B. Then the image under θ of V + (I) D + (f) is the closed set V (I (f) ), where I (f) = IB f B (f). v) Let {h 1,..., h n } be homogeneous elements generating I. Then for any f B 1, I (f) is generated by the h i /f deg h i. Proposition Let A be a ring. Let B be a graded algebra over A. Then we can endow Proj B with a unique structure of an Ascheme such that for any homogeneous f B +, the open set D + (f) is affine and isomorphic to Spec B (f). Definition The quotients of A[T 0,..., T n ] by a homogeneous ideal, are called homogeneous algebras over A. Lemma Let B = A[T 0,..., T n ]/I. Then Proj B is isomorphic to a closed subscheme of P n A, with support (i.e., with underlying topological space) V + (f). Definition Let A be a ring. A projective scheme over A is an Ascheme that is isomorphic to a closed subscheme of P n A for some n 0. Let k be a field and V a vector space of finite dim over k. The quotient P(V ) = (V \{0})/k represents the set of lines in V passing through the origin. Fix an isomorphism V = k n+1. If (α 0,..., α n ) k n+1 \{0} is a representative of a point α P(k n+1 ), then the α i are called the homogeneous coordinates of α. Lemma Let k be a field. Let α P(k n+1 ) be a point with homogeneous corrdinates (α 0,..., α n ). Then the ideal ρ(α) k[t 0,..., T n ] generated by the α j T i α i T j, 0 i, j n, is an element of P n k. Moreover, ρ : P(kn+1 ) P n k is a bijection from P(k n+1 ) onto the set P n k (k) of rational points of Pn k. Corollary Let k be a field and P 1 (T ),..., P m (T ) homogeneous polynomials in k[t 0,..., T n ]. Then there exists a bijection between Z + (P 1,..., P m ) (the set of solutions) and the set of krational points of the scheme Proj k[t 0,..., T n ]/I, where I is teh ideal generated by the P j (T ). 23
30 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES Definition A scheme X is said to be Noetherian if it is a finite union of affine open X i such that O X (X i ) is a Noetherian ring for every i. We say that a scheme is locally Noetherian if every point has a Noetherian open neighborhood. Proposition Let X be a Noetherian scheme. i) Any open or closed subscheme of X is Noetherian. For any point x X, the local ring O X,x is Noetherian. ii) For any affine open subset U of X, O X (U) is Noetherian. Definition Let k be a field. An affine variety over k is the affine scheme associated to a finitely generated algebra over k. An algebraic variety over k is a kscheme X such that there exists a covering by a finite number of affine open subschemes X i which are affine varieties over k. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties. A morphism of algebraic varieties over k is a morphism of kschemes. An algebraic variety is a Noetherian scheme. An open/closed subscheme of an alg variety is an alg variety. We will then say subvariety instead of subscheme. On the other hand, Spec O X,s is not an alg variety unless x is an isolated point in X (i.e., {x} is an open subset of X). For X an alg var, the set of closed points of X is dense in X. 2.4 REDUCED SCHEMES AND INTEGRAL SCHEMES Definition A ring A is called reduced if 0 is the only nilpotent element of A. Definition Let X be a scheme, x X. We say that X is reduced at x if the ring O X,x is reduced. We say that X is reduced if it is reduced at every point of X. Proposition Let X be a scheme. The following are true: i) X is reduced if and only if O X (U) is reduced for every open subset U of X. ii) Let {X i } i be a covering of X by affine open subsets X i. If the O X (X i ) are reduced, then X is a reduced scheme. iii) There exists a unique reduced closed subscheme i : X red X having the same underlying topological space as X. Moreover, if X is quasicompact, then the kernel of i # (X) : O X (X) O X (X red ) is the nilradical of O X (X). 24
31 CHAPTER 2. GENERAL PROPERTIES OF SCHEMES iv) Let Y be a reduced scheme. Then any morphism f : Y X factors in a unique way into a morphism Y X red followed by i : X red X. v) Let Z be a closed subset of X. Then there exists a unique structure of reduced closed subschemes on Z. Lemma Let X 0 denote the topological subspace made up of the closed points of X, and X be an algebraic variety over a field k, and let U be an open subset of X. Then U 0 = U X 0. Let X be an algebraic variety over a field k. We compare the regular functions on X (i.e., elements of O X (X)) to functions in the usual sense. Let X 0 denote the topological subspace made up of the closed points of X. Fix an algebraic closure k of k. Let f O X (X). We associate a function f : X 0 k defined by f(x) = the image of f x in k(x) (k(x) is a subextension of k). This induces a group homo from O X (X) to the set F(X 0, k) of maps from X 0 to k. Let F X be the sheaf U F (U 0, k). Then more generally we have a homo of sheaves from O X to F X. Proposition Let X be an algebraic variety over a field k. Then the homomorphism O X F X is injective if and only if X is reduced. Definition A topological space X is called irreducible if the condition X = X 1 X 2 with closed subsets X i implies that X 1 = X or X 2 = X. This amounts to saying that the intersection of two nonempty open subsets of X is nonempty. The set of irreducible subspaces of X admits maximal elements for the inclusion relation. Such a maximal element is a closed subset since the closure of an irr subset is irr. We call these maximal elements the irreducible components of X. Their union ix equal to X. Proposition Let X be a topological space. i) If X is irreducible, then any nonempty open subset of X is dense in X and is irreducible. ii) Let U be an open subset of X. Then the irreducible components of U are the {X i U} i, where the X i are the irreducible components of X which meet U. iii) Suppose that X is a finite union of irreducible closed subsets Z j. Then every irreducible component Z of X is equal to one of the Z j. If, moreover, there is no inclusion relation between the Z j, then the Z j are exactly the irreducible components of X. Definition We say that a scheme X is irreducible if the underlying topological space of X is irreducible. Proposition Let X = Spec A. Let I be an ideal of A. i) The space V (I) is irreducible if and only if I is prime. 25
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