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

Save this PDF as:

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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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.

### 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.

### 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

### Néron Models of Elliptic Curves.

Néron Models of Elliptic Curves. Marco Streng 5th April 2007 These notes are meant as an introduction and a collection of references to Néron models of elliptic curves. We use Liu [Liu02] and Silverman

### 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

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### Math 711: Lecture of September 7, Symbolic powers

Math 711: Lecture of September 7, 2007 Symbolic powers We want to make a number of comments about the behavior of symbolic powers of prime ideals in Noetherian rings, and to give at least one example of

### MIT Algebraic techniques and semidefinite optimization February 16, Lecture 4

MIT 6.972 Algebraic techniques and semidefinite optimization February 16, 2006 Lecture 4 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture we will review some basic elements of abstract

### 5 Dedekind extensions

18.785 Number theory I Fall 2016 Lecture #5 09/22/2016 5 Dedekind extensions In this lecture we prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also

### Néron models of abelian varieties

Néron models of abelian varieties Matthieu Romagny Summer School on SGA3, September 3, 2011 Abstract : We present a survey of the construction of Néron models of abelian varieties, as an application of

### THE KEEL MORI THEOREM VIA STACKS

THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for

### 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!

### ALGEBRAIC GEOMETRY I, FALL 2016.

ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of

### ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

### SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the

### 1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism

1 RINGS 1 1 Rings Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism (a) Given an element α R there is a unique homomorphism Φ : R[x] R which agrees with the map ϕ on constant polynomials

### Extension theorems for homomorphisms

Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following

### 14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as

### THE SMOOTH BASE CHANGE THEOREM

THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change

### JOHAN S PROBLEM SEMINAR

JOHAN S PROBLEM SEMINAR SHIZHANG LI I m writing this as a practice of my latex and my english (maybe also my math I guess?). So please tell me if you see any mistake of these kinds... 1. Remy s question

### Recent Applications of Schmidt s Subspace Theorem

Recent Applications of Schmidt s Subspace Theorem Paul Vojta University of California, Berkeley Abstract. In 2002, Corvaja and Zannier obtained a new proof of Siegel s theorem (on integral points on curves)

### 1 Existence of the Néron model

Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S

### DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

### FINITELY GENERATED SIMPLE ALGEBRAS: A QUESTION OF B. I. PLOTKIN

FINITELY GENERATED SIMPLE ALGEBRAS: A QUESTION OF B. I. PLOTKIN A. I. LICHTMAN AND D. S. PASSMAN Abstract. In his recent series of lectures, Prof. B. I. Plotkin discussed geometrical properties of the

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### SEPARATED AND PROPER MORPHISMS

SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.

### 3 Lecture 3: Spectral spaces and constructible sets

3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible

### The p-adic numbers. Given a prime p, we define a valuation on the rationals by

The p-adic numbers There are quite a few reasons to be interested in the p-adic numbers Q p. They are useful for solving diophantine equations, using tools like Hensel s lemma and the Hasse principle,

### 1 Fields and vector spaces

1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

### Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

### TC10 / 3. Finite fields S. Xambó

TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### NORMALIZATION OF THE KRICHEVER DATA. Motohico Mulase

NORMALIZATION OF THE KRICHEVER DATA Motohico Mulase Institute of Theoretical Dynamics University of California Davis, CA 95616, U. S. A. and Max-Planck-Institut für Mathematik Gottfried-Claren-Strasse

### Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

### Commutative Algebra. Andreas Gathmann. Class Notes TU Kaiserslautern 2013/14

Commutative Algebra Andreas Gathmann Class Notes TU Kaiserslautern 2013/14 Contents 0. Introduction......................... 3 1. Ideals........................... 9 2. Prime and Maximal Ideals.....................

### Section VI.33. Finite Fields

VI.33 Finite Fields 1 Section VI.33. Finite Fields Note. In this section, finite fields are completely classified. For every prime p and n N, there is exactly one (up to isomorphism) field of order p n,

### Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

### COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting d-tuples of commuting n n matrices that

### Homework 4 Solutions

Homework 4 Solutions November 11, 2016 You were asked to do problems 3,4,7,9,10 in Chapter 7 of Lang. Problem 3. Let A be an integral domain, integrally closed in its field of fractions K. Let L be a finite

### Some generalizations of Abhyankar lemma. K.N.Ponomaryov. Abstract. ramied extensions of discretely valued elds for an arbitrary (not

Some generalizations of Abhyankar lemma. K.N.Ponomaryov Abstract We prove some generalizations of Abhyankar lemma about tamely ramied extensions of discretely valued elds for an arbitrary (not nite, not

### Elliptic curves, Néron models, and duality

Elliptic curves, Néron models, and duality Jean Gillibert Durham, Pure Maths Colloquium 26th February 2007 1 Elliptic curves and Weierstrass equations Let K be a field Definition: An elliptic curve over

### Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

### Finite Fields: An introduction through exercises Jonathan Buss Spring 2014

Finite Fields: An introduction through exercises Jonathan Buss Spring 2014 A typical course in abstract algebra starts with groups, and then moves on to rings, vector spaces, fields, etc. This sequence

### 6 Lecture 6: More constructions with Huber rings

6 Lecture 6: More constructions with Huber rings 6.1 Introduction Recall from Definition 5.2.4 that a Huber ring is a commutative topological ring A equipped with an open subring A 0, such that the subspace

### A short proof of Klyachko s theorem about rational algebraic tori

A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture

### Fermat s Last Theorem for Regular Primes

Fermat s Last Theorem for Regular Primes S. M.-C. 22 September 2015 Abstract Fermat famously claimed in the margin of a book that a certain family of Diophantine equations have no solutions in integers.

### BASE POINT FREE THEOREMS SATURATION, B-DIVISORS, AND CANONICAL BUNDLE FORMULA

BASE POINT FREE THEOREMS SATURATION, B-DIVISORS, AND CANONICAL BUNDLE FORMULA OSAMU FUJINO Abstract. We reformulate base point free theorems. Our formulation is flexible and has some important applications.

### This is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:

Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties

### THE REGULAR ELEMENT PROPERTY

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2123 2129 S 0002-9939(98)04257-9 THE REGULAR ELEMENT PROPERTY FRED RICHMAN (Communicated by Wolmer V. Vasconcelos)

### SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

### LEGENDRE S THEOREM, LEGRANGE S DESCENT

LEGENDRE S THEOREM, LEGRANGE S DESCENT SUPPLEMENT FOR MATH 370: NUMBER THEORY Abstract. Legendre gave simple necessary and sufficient conditions for the solvablility of the diophantine equation ax 2 +

### A reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

1 1 A reduction of the Batyrev-Manin Conjecture for Kummer Surfaces David McKinnon Department of Pure Mathematics, University of Waterloo Waterloo, ON, N2T 2M2 CANADA December 12, 2002 Abstract Let V be

### CHAPTER 3: THE INTEGERS Z

CHAPTER 3: THE INTEGERS Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets?

### 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

### Normalization of a Poisson algebra is Poisson

Normalization of a Poisson algebra is Poisson D. Kaledin Introduction It is well-known that functions on a smooth symplectic manifold M are equipped with a canonical skew-linear operation called the Poisson

### Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going

### Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

### Standard forms for writing numbers

Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

### Adic Spaces. Torsten Wedhorn. June 19, 2012

Adic Spaces Torsten Wedhorn June 19, 2012 This script is highly preliminary and unfinished. It is online only to give the audience of our lecture easy access to it. Therefore usage is at your own risk.

### Model Theory of Differential Fields

Model Theory, Algebra, and Geometry MSRI Publications Volume 39, 2000 Model Theory of Differential Fields DAVID MARKER Abstract. This article surveys the model theory of differentially closed fields, an

### SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

### 12 Ramification, Haar measure, the product formula

18.785 Number theory I Lecture #12 Fall 2015 10/22/2015 12 Ramification, Haar measure, the product formula 12.1 Ramification in terms of the different and discriminant We conclude our discussion of the

### Assigned homework problems S. L. Kleiman, fall 2008

18.705 Assigned homework problems S. L. Kleiman, fall 2008 Problem Set 1. Due 9/11 Problem R 1.5 Let ϕ: A B be a ring homomorphism. Prove that ϕ 1 takes prime ideals P of B to prime ideals of A. Prove

### NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques

### MINIMAL MODELS FOR ELLIPTIC CURVES

MINIMAL MODELS FOR ELLIPTIC CURVES BRIAN CONRAD 1. Introduction In the 1960 s, the efforts of many mathematicians (Kodaira, Néron, Raynaud, Tate, Lichtenbaum, Shafarevich, Lipman, and Deligne-Mumford)

### Rings and Fields Theorems

Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

### CYCLOTOMIC POLYNOMIALS

CYCLOTOMIC POLYNOMIALS 1. The Derivative and Repeated Factors The usual definition of derivative in calculus involves the nonalgebraic notion of limit that requires a field such as R or C (or others) where

### The Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013

The Kummer Pairing Alexander J. Barrios Purdue University 12 September 2013 Preliminaries Theorem 1 (Artin. Let ψ 1, ψ 2,..., ψ n be distinct group homomorphisms from a group G into K, where K is a field.

### Formally real local rings, and infinitesimal stability.

Formally real local rings, and infinitesimal stability. Anders Kock We propose here a topos-theoretic substitute for the theory of formally-real field, and real-closed field. By substitute we mean that

### On the vanishing of Tor of the absolute integral closure

On the vanishing of Tor of the absolute integral closure Hans Schoutens Department of Mathematics NYC College of Technology City University of New York NY, NY 11201 (USA) Abstract Let R be an excellent

### Tangent spaces, normals and extrema

Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

### NAGATA COMPACTIFICATION FOR ALGEBRAIC SPACES. To the memory of Masayoshi Nagata

NAGATA COMPACTIFICATION FOR ALGEBRAIC SPACES BRIAN CONRAD, MAX LIEBLICH, AND MARTIN OLSSON Abstract. We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact

### Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures

2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon

### 6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not

6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not only motivic cohomology, but also to Morel-Voevodsky

### On a theorem of Ziv Ran

INSTITUTUL DE MATEMATICA SIMION STOILOW AL ACADEMIEI ROMANE PREPRINT SERIES OF THE INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY ISSN 0250 3638 On a theorem of Ziv Ran by Cristian Anghel and Nicolae

### EXISTENCE OF COMPATIBLE SYSTEMS OF LISSE SHEAVES ON ARITHMETIC SCHEMES

EXISTENCE OF COMPATIBLE SYSTEMS OF LISSE SHEAVES ON ARITHMETIC SCHEMES KOJI SHIMIZU Abstract. Deligne conjectured that a single l-adic lisse sheaf on a normal variety over a finite field can be embedded

### Algebraic group actions and quotients

23rd Autumn School in Algebraic Geometry Algebraic group actions and quotients Wykno (Poland), September 3-10, 2000 LUNA S SLICE THEOREM AND APPLICATIONS JEAN MARC DRÉZET Contents 1. Introduction 1 2.

### Height Functions. Michael Tepper February 14, 2006

Height Functions Michael Tepper February 14, 2006 Definition 1. Let Y P n be a quasi-projective variety. A function f : Y k is regular at a point P Y if there is an open neighborhood U with P U Y, and

### Math 210C. The representation ring

Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

### Plane quartics and. Dedicated to Professor S. Koizumi for his 70th birthday. by Tetsuji Shioda

Plane quartics and Mordell-Weil lattices of type E 7 Dedicated to Professor S. Koizumi for his 70th birthday by Tetsuji Shioda Department of Mathematics, Rikkyo University Nishi-Ikebukuro,Tokyo 171, Japan

### Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

### DIHEDRAL GROUPS II KEITH CONRAD

DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite

### MATH 615 LECTURE NOTES, WINTER, 2010 by Mel Hochster. Lecture of January 6, 2010

MATH 615 LECTURE NOTES, WINTER, 2010 by Mel Hochster ZARISKI S MAIN THEOREM, STRUCTURE OF SMOOTH, UNRAMIFIED, AND ÉTALE HOMOMORPHISMS, HENSELIAN RINGS AND HENSELIZATION, ARTIN APPROXIMATION, AND REDUCTION

### Error-Correcting Codes

Error-Correcting Codes HMC Algebraic Geometry Final Project Dmitri Skjorshammer December 14, 2010 1 Introduction Transmission of information takes place over noisy signals. This is the case in satellite

### Polynomial Rings and Group Rings for Abelian Groups. E. L. Lady. (June 16, 2009)

Polynomial Rings and Group Rings for Abelian Groups E. L. Lady (June 16, 2009) A lifetime or so ago, when I was teaching at the University of Kansas, Jim Brewer and I wrote a paper on group rings for finite

### Solutions of exercise sheet 8

D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra

### Algebra SEP Solutions

Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

### EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS. 1. Introduction. Although this paper focuses on a commutative algebra result, we shall begin by

Furman University Electronic Journal of Undergraduate Mathematics Volume 5, 1 9, 1999 EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS DAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKU Abstract. In this paper

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### DONG QUAN NGOC NGUYEN

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS DONG QUAN NGOC NGUYEN Contents 1 Introduction 1 2 Some basic notions 3 21 The Galois group Gal(K /k) 3 22 Representation of integers in O, and the

### Math 201C Homework. Edward Burkard. g 1 (u) v + f 2(u) g 2 (u) v2 + + f n(u) a 2,k u k v a 1,k u k v + k=0. k=0 d

Math 201C Homework Edward Burkard 5.1. Field Extensions. 5. Fields and Galois Theory Exercise 5.1.7. If v is algebraic over K(u) for some u F and v is transcendental over K, then u is algebraic over K(v).

### Boolean Algebras. Chapter 2

Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union