# CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

Size: px
Start display at page:

Download "CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998"

Transcription

1 CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic preliminary notation, both mathematical and logistical. Section 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Section 3 gives a few more details on the field of definition of a variety. Section 4 does the same as Section 2 for number theory. The remaining sections of this chapter give slightly longer descriptions of some topics in algebraic geometry that will be needed: Kodaira s lemma in Section 5, and descent in Section General notation The symbols Z, Q, R, and C stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol N signifies the natural numbers, which in this book start at zero: N = {0, 1, 2, 3,... }. When it is necessary to refer to the positive integers, we use subscripts: Z >0 = {1, 2, 3,... }. Similarly, R 0 stands for the set of nonnegative real numbers, etc. The set of extended real numbers is the set R := { } R { }. It carries the obvious ordering. If k is a field, then k denotes an algebraic closure of k. If α k, then Irr α,k (X) is the (unique) monic irreducible polynomial f k[x] for which f(α) = 0. Unless otherwise specified, the wording almost all will mean all but finitely many. Numbers (such as Section 2, Theorem 2.3, or (2.3.5)) refer to the chapter in which they occur, unless they are preceded by a number or letter and a colon in bold-face type (e.g., Section 3:2, Theorem A:2.5, or (7:2.3.5)), in which case they refer to the chapter or appendix indicated by the bold-face number or letter, respectively. 1

2 2 PAUL VOJTA 2. Conventions and required knowledge in algebraic geometry It is assumed that the reader is familiar with the basics of algebraic geometry as given, e.g., in the first three chapters of [H], especially the first two. Note, however, that some conventions are different here. This book will primarily use the language of schemes, rather than of varieties. The reader who prefers the more elementary approach of varieties, however, will often be able to mentally substitute the word variety for scheme without much loss, especially in the first few chapters. With the exception of Appendix B, all schemes are assumed to be separated. We often omit Spec when it is clear from the context; e.g., X(A) means X(Spec A) when A is a ring, P n A means Pn Spec A, and X A B means X Spec A Spec B when A and B are rings. The following definition gives slightly different names for some standard objects. Definition 2.1. Let X be a scheme. Then a vector sheaf of rank r on X is a sheaf that is locally isomorphic to OX r. A line sheaf is a vector sheaf of rank 1. Note that a vector sheaf is what is often called a locally free sheaf, with the additional restriction that its rank be the same everywhere. A line sheaf is also called an invertible sheaf by many authors. Varieties Not all authors use the same definition of variety. Here we use: Definition 2.2. Let k be a field. A variety over k, also called a k-variety, is an integral scheme, of finite type (and separated) over Spec k. If it is also proper over k, then we say it is complete. A curve is a variety of dimension 1. Note that, since k is not assumed to be algebraically closed, the set of closed points of a variety X is the set X( k), modulo the action of Aut k ( k). Also, the residue field K(P ) for a closed point P will in general be a finite extension of k. Note also that we have not assumed a variety to be geometrically integral. The advantage of this approach is that every irreducible closed subset of a variety will again be a variety, so that there is a natural one-to-one correspondence between the set of points of a variety and its set of subvarieties. This also agrees with the general philosophy that definitions should be weak. Finally, note that X being a variety over k is not the same as X being defined over k (Definition 3.2). For example, let k be a number field. Then any variety over k can be transformed into a variety over Q merely by composing with the canonical morphism Spec k Spec Q. On the other hand, not all varieties are defined over Q ; for example take the point in A 1 Q corresponding to ± 2. More details on this situation appear in the next section. 3. The field of definition of a variety This section defines what it means for a variety to be defined over a field. We show

3 PRELIMINARY MATERIAL 3 that there is a well defined minimal field over which this is the case, and that it is true over all larger fields. Moreover, some information on the structure of this field is given. We begin by describing what happens to a variety under base change to a larger field. Proposition 3.1. Let k be a field, let X be a variety over k, and let k be a normal extension of k. Then every irreducible component of X k k dominates X (under the projection X k k X ), and all irreducible components are conjugate under the action of Aut k (k ). Proof. We may restrict to an open affine of finite type over k, so we may assume that X is a closed subvariety of A n k for some n. The first assertion is then obvious from the going-down theorem. The second assertion follows from ([L 1], Ch. I, Prop. 11), which states that if A is an integrally closed entire ring with field of fractions K, if L is a Galois extension of K, if B is the integral closure of A in L, and if p is a maximal ideal in A, then all prime ideals of B lying over p are conjugate under Aut K (L). But by localizing, we may allow p to be any prime ideal, and the restriction that L be Galois over K can be relaxed to L normal over K ; the details of this are left to the reader. This motivates the following definition. Definition 3.2. Let X be a variety over a field k, and let k be an extension field of k. Then we say that X is defined over k if all irreducible components of (X k k ) red are geometrically integral. If this is the case, then we also say that k is a field of definition for X. Remark 3.3. In particular, if k = k, then this reduces to saying that X is defined over k if and only if it is geometrically integral. Lemma 3.4. Let k k be fields, let X be a variety over k, and let {U 1,..., U n } be a cover of X by open subvarieties. Then X is defined over k if and only if all U i are defined over k. Proof. Fix some i, and let U be an irreducible component of (U i k k ) red. Then the closure of U in (X k k ) red is an irreducible component of (X k k ) red ; since that irreducible component is geometrically integral, so is U. Thus all U i are defined over k. Now suppose that all U i are defined over k, and let X be an irreducible component of (X k k ) red. Then X is covered by irreducible components of (U i k k ) red ; hence X is geometrically integral. Thus the converse holds as well. Regular extensions Definition 3.2 can be phrased in algebraic terms using the notion of regular field extensions.

4 4 PAUL VOJTA Definition 3.5. A field extension K/k is regular if k is algebraically closed in K (i.e., any element of K algebraic over k is already contained in k ), and K is separable over k. Lemma 3.6. Let K/k be a field extension, and regard the algebraic closure k of k as a subfield of K. Then the following conditions are equivalent. (i). K is a regular extension of k ; (ii). K is linearly disjoint from k over k ; and (iii). the natural map K k k K k is injective. Proof. The equivalence (i) (ii) follows from ([L 2], Ch. VIII, Lemma 4.10). The equivalence of (ii) and (iii) is immediate from the definitions. Proposition 3.7. A variety X over k is defined over k if and only if K(X) is a regular extension of k. Proof. By Lemma 3.4, we may assume that X is affine. Let A be the affine ring of X, and let K = K(X), so that K is the field of fractions of A. Then X k k is integral if and only if A k k is entire. First suppose that K is regular over k. Consider the composition of maps A k k K k k K k. The first arrow is injective because k is flat over k. Lemma 3.6 implies that the second arrow is also injective, so A k k is entire. Thus X is geometrically integral. Conversely, assume that A k k is entire. Then, for all finite subextensions L of k/k, A L := A k L is entire since it is a subring (by flatness). Let K be its field of fractions. Let θ 1,..., θ d be a basis for L over k ; then (by flatness) θ 1,..., θ d, as elements of A L, are linearly independent over A. Therefore they form a basis for K over K, so [K : K] = d = [L : k]. But it is easy to check that since A L is integral over A, any map A L K extending the injection A K is injective. Thus [KL : K] = [K : K] = [L : k], so K and L are linearly disjoint over k. This holds for all L finite over k, so K and k are linearly disjoint over k. The minimal field of definition We now show that a variety X over k has a unique minimal field of definition. This leads to a slightly different concept of the field of definition, in a different context; this is useful in its own right. Definition 3.8. Let k k k 2 be fields, let X be a scheme of finite type over k, and let Z be a closed subscheme of X k k 2. Then we say that Z is defined over k if there exists a subscheme Z of X k k such that Z k k 2 = Z. If so, then we also say that k is a field of definition for Z. Lemma 3.9. Let k k be fields, and let X be a closed subvariety of A n k. Then X is defined over k (as a variety) if and only if every irreducible component of (X k k ) red is defined over k (as a subscheme of A n ). k Proof. First, we immediately reduce to the case where k = k.

5 PRELIMINARY MATERIAL 5 If X is defined over k, then it is geometrically integral, and the only irreducible component of (X k k)red is X k k itself, which is defined over k since it comes from X. Conversely, suppose some irreducible component of (X k k)red is defined over k. Then there exists a scheme X of A n k such that X k k is this irreducible component. But, by Proposition 3.1, X k k dominates X under the map X k k X, so since X is integral, we must have X = X. Thus X k k is integral, so X is geometrically integral. Lemma Let k k 2 be fields, let n N, and let a be an ideal in k 2 [X 1,..., X n ]. Then there exists a field k 1 such that k k 1 k 2 and, for all fields k with k k k 2, a is generated by elements of k [X 1,..., X n ] if and only if k k 1. Proof. By ([C-L-O S], Ch. 2, 7, Prop. 6), every ideal in a polynomial ring over a field has a unique reduced Gröbner basis. Let k 1 be the field generated over k by the coefficients of the elements of such a basis of a. Then the if part of the lemma is obvious. Conversely, suppose a is generated by elements of k [X 1,..., X n ] for some field k. Let a be the ideal in k [X 1,..., X n ] generated by those elements. Then a has a reduced Gröbner basis with coefficients in k. But the definition of reduced Gröbner basis involves only linear algebra in the coefficients of the polynomials, so the unique reduced Gröbner basis is preserved by enlarging the field of coefficients. In particular the reduced Gröbner bases of a and a coincide; hence k k 1. Proposition Let k k 2 be fields, let X be a scheme of finite type over k, and let Z be a closed subscheme of X k k 2. Then there exists a unique minimal field of definition k 1 of Z : for all fields k with k k k 2, Z is defined over k if and only if k k 1. Proof. In the special case where X = A n k, this follows immediately by translating Lemma 3.10 into geometrical language. The general case follows by covering X by open affines U i, which can then be regarded as closed subvarieties of A n i k. Proposition Let X be a variety over a field k, and let k 2 be a field extension of k containing an algebraic closure of k. Then there exists a unique minimal field of definition k 1 of X : for all fields k with k k k 2, X is defined over k if and only if k k 1. Proof. By Lemma 3.4, we may reduce immediately to the case where X is affine. Thus we may consider X as a closed subvariety of A n k. If we replace k 2 with its algebraic closure, then a minimal field of definition for X exists by Lemma 3.9 and Proposition But it is clear from Definition 3.2 that X is always defined over k, so k 1 is contained in k, which is contained in the original field k 2. The following proposition gives a good idea of the structure of the field k 1.

6 6 PAUL VOJTA Proposition Let k be a field, let p be a prime ideal in k[x 1,..., X n ], and let k be the smallest field such that p is generated by elements of k [X 1,..., X n ]. Let L be the field of fractions of k[x 1,..., X n ]/p, and let K be the subfield of L generated by k and the images of X 1,..., X n. Then k is a purely inseparable extension of the algebraic closure of k in K ; moreover, K is separable over k if and only if k is separable over k. Proof. See ([W], Ch. 1, Prop. 23). Corollary Let X be a variety over a field k, and let k be the minimal field of definition of X. Then k is a purely inseparable extension of the algebraic closure of k in K(X) ; moreover, K(X) is separable over k if and only if k is separable over k. Proof. This follows by translating Proposition 3.13 into geometrical language. 4. Conventions and required knowledge in number theory It is assumed that the reader has mastered the basics of algebraic number theory as presented, for example, in Part I of [L 1], especially the first five chapters. In addition, the following definitions are used. Number fields A number field k has a canonical set of places, denoted M k. This set is in one-to-one correspondence with the disjoint union of: (i). the set of real embeddings σ : k R ; (ii). the set of complex conjugate pairs {σ, σ} of embeddings σ : k C ; and (iii). the set of nonzero prime ideals p in the ring of integers R of k. These places are referred to as real places, complex places, and non-archimedean places, respectively. In addition, an archimedean place is a real or complex place. These places have almost-absolute values v defined by σ(x), if v is real, corresponding to σ : k R; x v = σ(x) 2, if v is complex, corresponding to σ, σ : k C; (R : p) ordp(x), if v is non-archimedean, corresponding to p R (In the last of the above three cases, ord p (x) is the order of x at p ; i.e., the exponent of p in the factorization of the fractional ideal (x). This requires x 0 ; we also define 0 v = 0.) These are not necessarily genuine absolute values, since the almost absolute value associated to a complex place does not satisfy the triangle inequality. Instead of the triangle inequality, however, we have that if a 1,..., a n k, then (4.1) n a i n v N v max a i v, 1 i n i=1

7 PRELIMINARY MATERIAL 7 where 1 if v is real; (4.2) N v = 2 if v is complex; and 0 if v is non-archimedean. In particular, if v is non-archimedean, then v the triangle inequality: obeys something stronger than (4.3) x + y v max( x v, y v ). (4.4) In addition, note that we have v M k N v = [k : Q] for any number field k. In addition, if L is a finite extension of k, v M k, and x k, then (4.5) w M L w v x w = x [L:k] v. Furthermore, these places satisfy a product formula: (4.6) v M k x v = 1 for all x k. The set of archimedean places of a number field k is denoted S (k), or just S if it is clear what k is. Function fields Recall that a function field is the field k := K(X) of rational functions on some nonsingular projective curve X over a field k 0. Fix a constant c > 1. Then define a set M k of places of k by defining, for each closed point P X, a place v with absolute value defined by (4.7) x v = c [K(P ):k 0] ord P x, where ord P is the order of vainishing of the rational function x at P. (Again, (4.7) requires that x be nonzero; we of course define 0 v = 0 for all v M k.) Then these absolute values again satisfy the product formula (4.6), as well as the formula (4.5) for extensions of fields. It is customary to take c = e, so that log x v will take on integral values. Or, with finite fields, one could take c = #k 0, to provide some parallels with the number field case.

8 8 PAUL VOJTA In the number field case, there is really only one choice for M k, and the normalizations of v do not depend on additional choices. This is not the case in the function field case, however. Not only do the normalizations of v depend on c and on k 0, but also the set M k may vary for a given function field k. For example, with the field k := C(X, Y ), we may take C(X) as the field of constants and treat Y as an indeterminate, or vice versa. Therefore, we will assume that, whenever a function field k is given, its set M k of places, and the normalizations of its absolute values, are given with it. Furthermore, an extension L/k of function fields is assumed to be one for which the set M L is the set of all places extending places in M k, and c and k 0 coincide. Then, in the notation of the preceding paragraph, L = K(X ) for some curve X provided with a finite morphism to X. For a function field k, all places v M k are non-archimedean. Therefore, by (4.3), the set {x k x v 1 for all v M k } is a field, called the field of constants of k. If X is defined over k 0, then k 0 coincides with this field of constants, by Proposition 3.7 and Lemma 3.6. Moreover, this remains true if k 0 is enlarged. Therefore it will often be assumed that X is defined over k 0. For a function field k, let N v = 0 for all v M k ; then (4.1) and (4.2) hold also in the function field case. In addition: Convention 4.8. If k is a function field, then we adopt the convention that [k : Q] = 0. With this convention, (4.4) holds for function fields as well. If k is a function field, then we let S (k) =. Global fields Definition 4.9. Recall that a global field is either a number field or a function field. If k is a global field, we let M k denote the disjoint union of M L for all finite extensions L of k. Local fields Definition A local field is the completion of a global field at one of its places. The set of local fields (as defined here) coincides with the disjoint union of: (i). The set of finite extensions of Q p for some rational prime p, (ii). The set of finite extensions of k 0 ((T )) for some field k 0, and (iii). R and C. If k is a global field and v M k, then the almost-absolute value v extends uniquely to an almost-absolute value on the completion k v. In that case, we may drop the subscript v, since the place is implicit from the fact that we are dealing with elements of k v : x for x k v. For a local field k v as above, we let M kv = {v}. Of course, there is no product formula in this case.

9 PRELIMINARY MATERIAL 9 5. Associated points, rational maps, and rational sections When dealing with schemes that are not necessarily reduced, we use a definition of rational map and rational section that is a bit different from the usual definition; see Definition 5.3. This definition is based on schematic denseness. Definition 5.1. Let U be an open subset of a scheme X. We say that U is schematically dense in X if its schematic closure (the schematic image of the open immersion U X, or the smallest closed subscheme of X containing U ) is all of X. Often it is convenient to think of schematic denseness in terms of the associated points of X. Proposition 5.2. An an open subset U of a noetherian scheme X is schematically dense if and only if it contains all associated points of X. Proof. It suffices to show this when X is an affine scheme. Let A be the affine ring of X, let (0) = q 1 q n be a minimal primary decomposition of the ideal (0) in A, and let p i = q i for each i. Then p 1,..., p n are the primes corresponding to the associated points of X. First suppose that U contains all associated points of X. Let a be the ideal corresponding to the schematic closure of U, and let f a. For all i, Spec A/a contains an open neighborhood of p i ; therefore (A/a) pi /a = A p. In particular, f = 0 in A p, so Ann(f) meets A \ p. Thus, for all i there exists x i Ann(f) such that x i / p i. It follows by an easy exercise that there exists x Ann(f) such that x / p 1 p n. But then x is not a zero divisor, by ([L 2], X Prop. 2.9). This can happen only if f = 0, so a = 0 and thus U is schematically dense in X. Conversely, suppose that U does not contain the point corresponding to p i. After renumbering the indices, we may assume that i = n and that {i p i p n } = {1,..., r} for some r < n. Let a = q 1 q r. Since the chosen primary decomposition was minimal, we have a (0). We claim that U Spec A/a. Indeed, let p be a prime ideal corresponding to a point in U. Since U is open and since p n / U, we have p p n, and therefore p p i for all i > r. Let S = A \ p. By ([A-M], Prop. 4.8(i)) and ([A-M], Prop. 4.9), we then have (0) = n S 1 q i = i=1 r S 1 q i = S 1 a. Thus p Spec A/a (since S 1 a = (0) implies a p ), and (A/a) p = A p. This holds for all p U, so U is an open subscheme of Spec A/a. This shows that U is not schematically dense in X. Definition 5.3. Let X be a scheme. i=1 (a). If X is noetherian, then a rational map f : X Y from X to another scheme Y is an equivalence class of pairs (U, f), where U is a schematically

10 10 PAUL VOJTA dense open subset of X, f : U Y is a morphism, and (U, f) is said to be equivalent to (U, f ) if f and f agree on U U. (b). If X is noetherian, then a rational function on X is a class of pairs (U, f), where U is a schematically dense open subset of X, f is a regular function on U, and the equivalence relation is similar to the one in part (a). (c). A rational section of a sheaf F on X is a section of F over a schematically dense open set. (d). A rational section of a line sheaf L on X is said to be invertible if it has an inverse over a schematically dense open subset. Remark. The relations in parts (a) and (b) above are equivalence relations because Y is separated, and because the open sets in question are schematically dense. 6. Cartier divisors and associated line sheaves The above definition of invertible rational section of a line sheaf meshes well with the notion of a Cartier divisor. We begin by recalling the definition of a Cartier divisor. Definition 6.1. Let X be a scheme. (a). The sheaf K, called the sheaf of total quotient rings of O X, is the sheaf associated to the presheaf U S 1 O X (U), where S is the multiplicative system of elements which are not zero divisors. (b). Let O X denote the sheaf of invertible elements of O X. Then a Cartier divisor on X is a global section of the sheaf K /O X. The set of Cartier divisors forms a group, which is written additively instead of multiplicatively by analogy with Weil divisors. By standard properties of sheaves, a Cartier divisor can be described by giving an open cover {U i } of X and sections f i K (U i ) such that f i /f j lies in O X (U i U j ) for all i and j. Such a description is called a system of representatives for the Cartier divisor. If {(U i, f i )} i I is a system of representatives for a Cartier divisor D, then D Ui = (f i ) for all i. If the scheme X is integral, which is true in most applications, then K is just the constant sheaf K(X), and therefore in a system of representatives all f i may be taken to lie in K(X). A Cartier divisor is principal if it lies in the image of the map Γ(X, K ) Γ(X, K /O X). If X is integral, then this is equivalent to the assertion that all f i can be taken equal to the same f K(X). Two Cartier divisors are linearly equivalent if their difference is principal. A Cartier divisor is effective if it has a system of representatives {(U i, f i )} i I such that f i lies in O X (U i ) for all i. If D is a Cartier divisor, its support, denoted Supp D, is the set Supp D = {P X D P / OX,P }

11 PRELIMINARY MATERIAL 11 (where D P K P denotes the germ of D at P ). The support of D is a Zariski-closed subset of X. More concretely, if {(U i, f i )} i I is a system of representatives for D, then (Supp D) U i equals the set of all P U i such that f i is not an invertible element of the local ring O X,P at P. If D is a Cartier divisor with Supp D =, then D = (1). For more information on Cartier divisors and how they compare to Weil divisors, see ([H], II 6). Lemma 6.2. Let A be a commutative noetherian ring. Let X = Spec A, and let U be an open subset of X containing all associated points. Let S be the multiplicative system of (nonzero) elements of A which are not zero divisors. Then O X (U) S 1 A. Conversely, for any s S 1 A, there exists an open subset U, containing all associated points of X, such that s O X (U). Proof. Let s be a section in O X (U). Since the complement of U is a closed subset not containing any associated point of X, there exists f A which vanishes on the complement of U, yet is not contained in any associated prime of A. The former condition implies that s A f, since D(f) U ; the latter implies that f is not a zero divisor, by ([L 2], X Prop. 2.9). Hence f S and s A f S 1 A. Conversely, suppose s S 1 A. Then s A f for some f S. In particular, by ([L 2], X Prop. 2.9), f is not contained in any associated prime of A. Thus s O X (U) for some open U containing all the associated points of X. Definition 6.3. Let X be a noetherian scheme, let L be a line sheaf on X, and let s be an invertible rational section of L. Then, for any open affine subset U = Spec A of X over which L is trivial, the restriction of s to U defines an element of S 1 A, where S is as in Lemma 6.2. Since s is invertible, this element actually lies in (S 1 A). This element depends on the choice of trivialization: changing the trivialization multiplies the element by an element of A. Therefore, this gives a well defined section of K /O over U. These sections glue to give a global section of K /O over X. This section, regarded as a Cartier divisor, is called the associated Cartier divisor, and is denoted (s). Proposition 6.4. Let X be a noetherian scheme. (a). Let f be a rational function on X. Then f may be regarded as an invertible rational section of the trivial line sheaf O X, and the definition of (f) as such coincides with the principal divisor (f). (b). Let L be a line sheaf on X and let s be an invertible rational section of L. Then s is a global (regular) section of L if and only if (s) is effective. (c). Let φ: X X be a morphism of noetherian schemes, let L be a line sheaf on X, and let s be an invertible rational section of L which is defined and nonzero at the images (under φ ) of all associated points of X. Then φ s is an invertible rational section of φ L, and (φ s) = φ (s). (d). Let s 1 and s 2 be invertible rational sections of line sheaves L 1 and L 2, respectively. Then s 1 s 2 is an invertible rational section of L 1 L 2, and (s 1 s 2 ) = (s 1 ) + (s 2 ).

12 12 PAUL VOJTA Proof. These all follow immediately from the definition. Recall from ([H], II 6) the definition of the sheaf O(D) associated to a Cartier divisor D on a scheme X (Hartshorne denotes it L (D) ). It is a subsheaf of K. Definition 6.5. Let D be a Cartier divisor on a noetherian scheme X. The global section 1 K defines a rational section, called the canonical section of O(D). In addition to ([H], II Prop. 6.13), we have the following properties. Proposition 6.6. Let D be a Cartier divisor on a noetherian scheme X. (a). If s is the canonical section of O(D), then (s) = D. (b). If φ: X X is a morphism of noetherian schemes such that none of the associated points of X are taken into the support of D, then O(φ D) = φ O(D). Proof. This is left as an exercise for the reader. 7. Big divisors and Kodaira s lemma Not written yet. 8. Descent Not written yet. Should it go earlier? See Serre, Groupes Alg. et Corps de Classes, p. 108 (Ch. V No. 20). References [C-L-O S] D. Cox, J. Little, and D. O Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, Springer-Verlag, [H] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, [L 1] S. Lang, Algebraic number theory, Addison-Wesley, 1970; reprinted, Springer-Verlag, [L 2], Algebra, 3rd edition, Addison-Wesley, Reading, Mass., [W] A. Weil, Foundations of algebraic geometry, revised and enlarged edition, American Math. Society, Department of Mathematics, University of California, Berkeley, CA 94720

### A MORE GENERAL ABC CONJECTURE. Paul Vojta. University of California, Berkeley. 2 December 1998

A MORE GENERAL ABC CONJECTURE Paul Vojta University of California, Berkeley 2 December 1998 In this note we formulate a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES

AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### Exploring the Exotic Setting for Algebraic Geometry

Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology

### NONSINGULAR CURVES BRIAN OSSERMAN

NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF

ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical

### 9. Integral Ring Extensions

80 Andreas Gathmann 9. Integral ing Extensions In this chapter we want to discuss a concept in commutative algebra that has its original motivation in algebra, but turns out to have surprisingly many applications

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### INTERSECTION THEORY CLASS 6

INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### Algebraic Number Theory Notes: Local Fields

Algebraic Number Theory Notes: Local Fields Sam Mundy These notes are meant to serve as quick introduction to local fields, in a way which does not pass through general global fields. Here all topological

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

### Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:

### (1) is an invertible sheaf on X, which is generated by the global sections

7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

### Math 418 Algebraic Geometry Notes

Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R

### 9. Birational Maps and Blowing Up

72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18 CONTENTS 1. Invertible sheaves and divisors 1 2. Morphisms of schemes 6 3. Ringed spaces and their morphisms 6 4. Definition of morphisms of schemes 7 Last day:

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### DIVISORS ON NONSINGULAR CURVES

DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce

### Math 145. Codimension

Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

### Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 In Lecture 6 we proved (most of) Ostrowski s theorem for number fields, and we saw the product formula for absolute values on

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### SCHEMES. David Harari. Tsinghua, February-March 2005

SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............

### Section Projective Morphisms

Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism

### Arithmetic Algebraic Geometry

Arithmetic Algebraic Geometry 2 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016 2 Contents 1 Preliminaries 1 1.1 Affine Varieties.......................... 1 1.2 Projective Varieties........................

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### Section Divisors

Section 2.6 - Divisors Daniel Murfet October 5, 2006 Contents 1 Weil Divisors 1 2 Divisors on Curves 9 3 Cartier Divisors 13 4 Invertible Sheaves 17 5 Examples 23 1 Weil Divisors Definition 1. We say a

### 7 Orders in Dedekind domains, primes in Galois extensions

18.785 Number theory I Lecture #7 Fall 2015 10/01/2015 7 Orders in Dedekind domains, primes in Galois extensions 7.1 Orders in Dedekind domains Let S/R be an extension of rings. The conductor c of R (in

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### Polynomials, Ideals, and Gröbner Bases

Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields

### LINE BUNDLES ON PROJECTIVE SPACE

LINE BUNDLES ON PROJECTIVE SPACE DANIEL LITT We wish to show that any line bundle over P n k is isomorphic to O(m) for some m; we give two proofs below, one following Hartshorne, and the other assuming

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14 RAVI VAKIL Contents 1. Dimension 1 1.1. Last time 1 1.2. An algebraic definition of dimension. 3 1.3. Other facts that are not hard to prove 4 2. Non-singularity:

### Algebraic Geometry I Lectures 14 and 15

Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### (1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### 2. Prime and Maximal Ideals

18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally

Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free

### A connection between number theory and linear algebra

A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Base-point-free, ample, very ample 2 3. Every ample on a proper has a tensor power that

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Theta divisors and the Frobenius morphism

Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following

### INTERSECTION THEORY CLASS 7

INTERSECTION THEORY CLASS 7 RAVI VAKIL CONTENTS 1. Intersecting with a pseudodivisor 1 2. The first Chern class of a line bundle 3 3. Gysin pullback 4 4. Towards the proof of the big theorem 4 4.1. Crash

### 12. Hilbert Polynomials and Bézout s Theorem

12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of

### n P say, then (X A Y ) P

COMMUTATIVE ALGEBRA 35 7.2. The Picard group of a ring. Definition. A line bundle over a ring A is a finitely generated projective A-module such that the rank function Spec A N is constant with value 1.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

### Another proof of the global F -regularity of Schubert varieties

Another proof of the global F -regularity of Schubert varieties Mitsuyasu Hashimoto Abstract Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally F -regular. We give

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### 1.6.1 What are Néron Models?

18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in

### On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question

### 32 Divisibility Theory in Integral Domains

3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### 1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

### Lecture 7: Etale Fundamental Group - Examples

Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.

ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology

### COHOMOLOGY AND DIFFERENTIAL SCHEMES. 1. Schemes

COHOMOLOG AND DIFFERENTIAL SCHEMES RAMOND HOOBLER Dedicated to the memory of Jerrold Kovacic Abstract. Replace this text with your own abstract. 1. Schemes This section assembles basic results on schemes

### Remarks on the existence of Cartier divisors

arxiv:math/0001104v1 [math.ag] 19 Jan 2000 Remarks on the existence of Cartier divisors Stefan Schröer October 22, 2018 Abstract We characterize those invertible sheaves on a noetherian scheme which are

### Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

### Some remarks on Frobenius and Lefschetz in étale cohomology

Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)

### MANIN-MUMFORD AND LATTÉS MAPS

MANIN-MUMFORD AND LATTÉS MAPS JORGE PINEIRO Abstract. The present paper is an introduction to the dynamical Manin-Mumford conjecture and an application of a theorem of Ghioca and Tucker to obtain counterexamples

### A NOTE ON RETRACTS AND LATTICES (AFTER D. J. SALTMAN)

A NOTE ON RETRACTS AND LATTICES (AFTER D. J. SALTMAN) Z. REICHSTEIN Abstract. This is an expository note based on the work of D. J. Saltman. We discuss the notions of retract rationality and retract equivalence

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!

### Summer Algebraic Geometry Seminar

Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### INTERSECTION THEORY CLASS 2

INTERSECTION THEORY CLASS 2 RAVI VAKIL CONTENTS 1. Last day 1 2. Zeros and poles 2 3. The Chow group 4 4. Proper pushforwards 4 The webpage http://math.stanford.edu/ vakil/245/ is up, and has last day

### CRing Project, Chapter 7

Contents 7 Integrality and valuation rings 3 1 Integrality......................................... 3 1.1 Fundamentals................................... 3 1.2 Le sorite for integral extensions.........................

### 4. Noether normalisation

4. Noether normalisation We shall say that a ring R is an affine ring (or affine k-algebra) if R is isomorphic to a polynomial ring over a field k with finitely many indeterminates modulo an ideal, i.e.,

### Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

### Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

### Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

### The Proj Construction

The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of

### Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014

Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................

### 11. Dimension. 96 Andreas Gathmann

96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for