# CHEVALLEY S THEOREM AND COMPLETE VARIETIES

Size: px
Start display at page:

Transcription

1 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 in terms of existence of extensions of morphisms from nonsingular curves, and conclude that projective varieties are complete. As a prelude to this, we also prove Chevalley s theorem on images of morphisms. 1. Chevalley s theorem We have already seen that the image of a morphism of a variety need not be a subvariety (that is, it need not be a closed subset of an open subset). We recall the example: Example 1.1. Consider the morphism A 2 A 2 determined by (x, y) (x, xy). A 2 Z(x) {(0, 0)}. Its image is Chevalley s theorem asserts that this example is typical: the image of a morphism is always a finite union of subvarieties. Theorem 1.2 (Chevalley). Let ϕ : X Y be a morphism of varieties. Then ϕ(x) can be written as a finite union of (not necessarily closed) subvarieties of Y. Note that ϕ(x) must be irreducible, so there is a unique subvariety of Y contained in ϕ(x) which is dense in ϕ(x); the other subvarieties of the theorem are all in its closure. The key statement to prove is: Proposition 1.3. If ϕ : X Y is a dominant morphism of prevarieties, then ϕ(x) contains an open subset of Y. In the proof we will need the following statement, which isn t hard to prove, and we already used implicitly in our results on normalizations. Proposition 1.4. Let ϕ : X Y be a dominant morphism of affine varieties which makes A(X) into an integral extension of A(Y ). Then ϕ is surjective. Although the proof is not difficult, we (continue to) omit it. The next step is to show that any dominant morphism factors as a composition of two particular types of dominant morphisms, which will be easier to analyze. Lemma 1.5. If ϕ : X Y is a dominant morphism of affine varieties, and r is the transcendence degree of the induced field extension K(X)/K(Y ), then ϕ factors as a composition of dominant morphisms X Y A r Y, where the second morphism is the projection morphism. Proof. Let f 1,..., f m be generators of A(X) over A(Y ). Then the f i also generate K(X) over K(Y ), so we can reorder indices such that f 1,..., f r are algebraically independent over K(Y ). Let R = A(Y )[f 1,..., f r ] A(X). Since the f i are algebraically independent over K(Y ) they are algebraically independent over A(Y ), so R is isomorphic to an r-variable polynomial ring, which is to say that R = A(Y A r ). Then the inclusions A(Y ) R A(X) induce the desired factorization. We can now prove that the image of a dominant morphism contains an open subset. 1

2 Proof of Proposition 1.3. Let V be an affine open subset of Y, and U an affine open subset of X such that ϕ(u) V. Then it clearly suffices to prove ϕ(u) contains an open subset of V, so we have reduced to the affine case. Applying Lemma 1.5, it suffices to prove that the image of X in Y A r contains an open subset, and that the projection morphism Y A r Y is open. For the first assertion, we have that the transcendence degrees of K(X) and K(Y A r ) are equal, so the morphism makes K(X) into an algebraic extension of K(Y A r ). Suppose f 1,..., f m generate A(X) over A(Y A r ). Then each f i is a root of some polynomial g i = j c i,jt j over K(Y A r ), which can assume to be monic. Let h be the product over all i, j of the denominators of c i,j (considering K(Y A r ) as the fraction field of A(Y A r )). We have the induced morphism of open subsets X h (Y A r ) h, and we see that A(X h ) = A(X) h is still generated by the f i over A((Y A r ) h ) = A(Y A r ) h. But each f i is integral over A(Y A r ) h by construction, so by Proposition 1.4, we have that X h surjects onto (Y A r ) h, and thus the image of X contains the open subset (Y A r ) h Y A r. It remains to prove that for any U Y A r open, the image of U under the projection morphism Y A r Y is open (we only need to prove it contains an open subset, but the stronger statement is no harder). Any such U is a union of open subsets of the form (Y A r ) f, for some nonzero f A(Y A r ) = A(Y )[t 1,..., t r ], so we may thus assume that U is of this form. Let I A(Y ) be the ideal generated by the coefficients of f. We claim that the image of U is precisely A(Y ) Z(I). Indeed, given Q Y, if we let Z Q be the preimage of Q in Y A r, we have Z Q = A r, and the inclusion Z Q Y A r is induced by the ring homomorphism A(Y )[t 1,..., t r ] k[t 1,..., t r ] obtained by sending g A(Y ) to g(q). We thus see that f ZQ is obtained by evaluating the coefficients of f at Q, and so Z Q U = Z Q Z(f) is non-empty if and only if f remains nonzero when its coefficients are evaluated at Q, which is precisely equivalent to the condition that Q I. But Q is in the image of U if and only if Z Q U, so we conclude that the image of U is the complement of Z(I), as desired. Chevalley s theorem then follows easily. Proof of Theorem 1.2. Given ϕ : X Y, let Z Y be the closure of ϕ(x); our proof is by induction on dim(z). If dim(z) = 0, then ϕ is constant and there is nothing to show. Otherwise, assume dim(z) = d > 0, and we have the theorem already for dimensions smaller than d. Now, we have a dominant morphism X Z, so let U Z be the maximal open subset of Z contained in ϕ(x); this exists by Proposition 1.3. Then let Z = Z U; this has dimension less than d, and is not necessary a variety, but can be write it as a finite union of varieties Z 1,..., Z m. Similarly, ϕ 1 (Z i ) is not necessarily a variety, but can be written as a finite union of varieties X i,1,..., X i,mi X. We have ϕ(x) = U ϕ(ϕ 1 (Z 1 )) ϕ(ϕ 1 (Z m )) = U i,j ϕ(x i,j ), and by the induction hypothesis each ϕ(x i,j ) is a finite union of subvarieties of Z i Y, so we conclude the theorem. 2. Completeness and limits We now apply our discussion of curves to give a definition for varieties which is analogous to the notion of compactness for topological spaces. We have the opposite problem that we had with the Hausdorff condition: every variety is compact in the Zariski topology, because its underlying topological space is Noetherian. However, the fix is the same as before: we give a rephrasing of the 2

3 compactness condition in topology which will turn out to agree better with our intuition when we apply it to varieties. Exercise 2.1. A topological space X is compact if and only if for every topological space Y, the projection map X Y Y is a closed map. Motivated by this, we define: Definition 2.2. A variety X is complete if for all varieties Y, the projection morphism X Y Y is a closed map. Remark 2.3. One can apply the definition of completeness to prevarieties as well, but it is traditional to reserve the term complete for varieties. This is related to the French tradition that compactness should incorporate the Hausdorff condition as well. One reason for this definition is its relation to closed morphisms: Proposition 2.4. If X is complete, any closed subvariety of X is complete. If we also have Y an arbitrary variety, and ϕ : X Y a morphism, then ϕ is closed. Proof. The first assertion is immediate from the definition, since if Z is closed in X, we have Z Y closed in X Y for any Y. For the second assertion, let Γ = {(x, ϕ(x) : x X} X Y be the graph of ϕ. We can express Γ as the preimage of the diagonal (Y ) Y Y under the morphism ϕ id : X Y Y Y. Since Y is a variety, (Y ) is closed, so Γ is closed. But ϕ(x) is precisely the image of Γ under the projection X Y Y, so we conclude from the completeness hypothesis on X that ϕ(x) is closed. Remark 2.5. This is a natural property for complete varieties to have, since a continuous map from a compact topological space to a Hausdorff space is closed. In fact, this property characterizes complete varieties Nagata proved that every variety can be realized as an open subset of a complete variety, so in particular if X is not complete, the inclusion as an open subset of a complete variety is not a closed mapping. However, the proof of Nagata s theorem is beyond the scope of this course. We wish to give a more intuitive necessary and sufficient criterion for completeness. Because the ideas are closely related, we will also give a more intuitive criterion for a prevariety to be a variety. In the informal language of limits we introduced, we will show that a prevariety is a variety if and only if limits are unique when they exist, and that a variety is complete if and only if limits always exist. An important lemma is the following: Lemma 2.6. Suppose ϕ : X Y is a morphism of prevarieties, and Q Y is in the closure of ϕ(x). Then there exists a nonsingular curve C and P C, and morphisms ψ : C {P } X and ψ : C Y such that ϕ ψ = ψ C {P }, and and ψ(p ) = Q. An intermediate lemmas is the following. Lemma 2.7. Suppose X is a prevariety, U a nonempty open subset, and P X U. Then there exists a subprevariety Z of X which is a curve, such that P Z, and U Z. Proof. It suffices to produce such a Z inside any open neighborhood of P in X, so let V X be an affine open neighborhood of Q. We prove the statement by induction on dim X; if dim X = 1, we simply let Z = X. For dim X > 1, let m P be the maximal ideal of A(V ) corresponding to P, and let I A(V ) be the radical ideal with V U = Z(I). Choose Q 1,..., Q r P in Z(I), with one Q i in each irreducible component of Z(I) having codimension 1 (we may have r = 0). We claim there exists f such that f(p ) = 0, but f(q i ) 0 for all i. 3

4 We construct f as follows: for each i j, first find f i,j such that f i,j (Q i ) = 0, but f i,j (Q j ) 0. Such an f i,j exists because m Qj m Qi. Also choose g i such that g i (P ) = 0, but g i (Q i ) 0, which exists for the same reason. Let f i = g i j i f i,j. Then f i (Q i ) 0, but f i (Q j ) = f i (P ) = 0 for all j i. We can then let f i f i. In the case r = 0, we can choose any nonzero f m P. Now, Z(f) a closed subset of X containing P ; and not containing any component of Z(I) having codimension 1. Let X be an irreducible component of Z(f) which contains P. Then dim X = dim X 1, and since Z(f) doesn t contain any component of Z(I) of codimension 1, we find that X cannot be contained in Z(I), so X U. Replacing X by X and U by X U, we apply the induction hypothesis to find the desired curve. We can now prove the first lemma. Proof of Lemma 2.6. Our first claim is that there is a subprevariety D Y which is a curve containing Q, and such that D ϕ(x) is dense in D. But by Proposition 1.3, if Y is the closure of ϕ(x) in Y, we know that ϕ(x) contains an open subset of Y, so this follows immediately from Lemma 2.7. Now, let Z X be the preimage of D under ϕ. Since D ϕ(x) is dense in D, some irreducible component X of Z maps dominantly to D. Choose Q D ϕ(x ); then ϕ 1 (Q ) X is closed in X, and cannot be all of X. Again using Lemma 2.7, there is a curve C in X which meets ϕ 1 (Q ) X but is not contained in it. We then see that C maps dominantly to D, since its image must be irreducible and strictly contains Q. Let C be the normalization of D inside K(C ), and let P C be a point mapping to Q. By construction, K( C) = K(C ), so we get a birational map C C commutes with the given morphisms to D. Let U C be the open subset on which the rational map is defined. We can set C = U {P }, which is still an open subset of C, and we obtain morphisms C D Y and C {P } C X X satisfying the desired conditions. Before stating the theorem, we give one more lemma. Lemma 2.8. Let ϕ : C D be a birational morphism of curves, with D nonsingular. Then ϕ is an isomorphism of C onto the open subset ϕ(c) of D. Proof. Let ν : C C be the normalization, and let C and D be the nonsingular projective curves having C and D as open subsets, respectively. Then ϕ ν induces a birational map C D, which is necessarily an isomorphism. But then both C and D are identified as open subsets of a given curve, compatibly with the morphism ϕ ν, so we conclude that ϕ ν must map C isomorphically onto an open subset of D. By the surjectivity of ν, this open subset is ϕ(c), and then the morphism ϕ(c) C C show that ϕ is an isomorphism onto ϕ(c), as desired. We now give the promised more intuitive geometric description in terms of algebraic limits of completeness, as well as of the separation property distinguishing varieties from prevarieties. In a related but more general form, this theorem gives what are typically called valuative criteria. Theorem 2.9. A prevariety X is a variety if and only if for all nonsingular curves C, and points P C, and morphisms ϕ : C {P } X, there is at most one extension of ϕ to a morphism C X. A variety X is complete if and only if for all nonsingular curves C, and points P C, and morphisms ϕ : C {P } X, there exists a (necessarily unique) extension of ϕ to a morphism C X. Proof. For the first statement, we already know that if X is a variety, then the stated condition holds. Conversely, suppose the condition holds, and consider the diagonal morphism : X X X. Let Q X X be in the closure of (X). Then by Lemma 2.6, there is a nonsingular curve C and a point P C, with morphisms ψ : C {P } X and ψ : C X X such that ψ(p ) = Q, and ψ = ψ on C {P }. We then get two extensions of ψ to all of C by composing ψ 4

5 with the projection morphisms p 1, p 2. By hypothesis, these extensions are unique, so we conclude that p 1 (ψ(p )) = p 2 (ψ(p )), so Q = ψ(p ) (X), and (X) is closed. Now, suppose X is a complete variety. Given C, the point P C, and ϕ : C {P } X, consider the product X C. Let Z be the closure of {(ϕ(p ), P ) : P C} X C. Then we have dominant morphisms C {P } Z C, and the image of Z is closed in C by hypothesis, so we must have Z C surjective. By Lemma 2.8, we conclude that Z C is an isomorphism. Inverting the isomorphism, it follows that we have a morphism C Z extending C {P } Z, and taking the first projection we get the desired extension of ϕ to all of C. Conversely, suppose X satisfying the stated condition. Given any variety Y, let Z be a closed subset of X Y. Let Q Y be in the closure of the image of Z under the projection morphism p 2. By Lemma 2.6, there is a nonsingular curve C and a point P C, with morphisms ψ : C {P } Z and ψ : C Y such that ψ(p ) = Q, and ψ = p 2 ψ on C {P }. By hypothesis, we can extend p 1 ψ to a morphism ψ : C X, and we see that if we take the product morphism ψ ψ : C X Y, it extends ψ, and must therefore have image contained in Z, since Z is closed. Moreover, by definition we have that Q = p 2 ((ψ ψ)(p )), so Q p 2 (Z), and we conclude p 2 (Z) is closed. We immediately conclude: Corollary Any projective variety is complete. From Proposition 2.4 we then conclude: Corollary Any morphism from a projective variety to an arbitrary variety is closed. Remark There are direct, algebraic proofs of Corollary See for instance Theorem 2 of Chapter I, 5.2 of [2]. However, the point of our approach is to show that if one builds up enough foundational tools, one can start to prove interesting results more geometrically, without resorting to going back to definitions and using algebra to prove each result. Exercise In this exercise, we give a proof of Chow s Lemma. It is clear that every complete variety is birational to some projective variety. However, a much stronger statement is true: given a complete variety X, there exists a projective variety X together with a birational morphism X X (which is necessarily surjective, by Corollary 2.11). Let {U i } be an affine open cover of X, and let Y i be the closures of the U i in projective space. Let U be the intersection of the U i, and ϕ : U X Y 1 Y n the morphism induced by the inclusions of U into X and the Y i. Let X be the closure of ϕ(u). Let p 1 : X X be the morphism induced by the first projection, and p 2 : X Y 1 Y n be the morphism induced by projection to the remaining factors. (a) Show that p 1 gives an isomorphism p 1 1 (U) U. Hint: first prove that ϕ(u) is an open subset of X. (b) Show that p 2 induces an isomorphism of X onto a closed subvariety of Y 1 Y n. Hint: first prove that X X Y 1 U i Y n = X U i Y 1 Y n, by considering the projections to X and to Y i for each. (c) Conclude Chow s lemma. We see that even if X is not quasiprojective, it remains true that there are no nonconstant functions which are globally regular on X. Exercise Prove that if X is a complete variety, then O(X) = k. 5

6 References 1. Robin Hartshorne, Algebraic geometry, Springer-Verlag, Igor Shafarevich, Basic algebraic geometry 1. varieties in projective space, second ed., Springer-Verlag,

### 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

### 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

### 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

### 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.

### 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

### VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS

VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where

### 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

### VALUATIVE CRITERIA BRIAN OSSERMAN

VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not

### Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local

### 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

### 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

### (dim Z j dim Z j 1 ) 1 j i

Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the

### 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,

### 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:

### 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:

### Math 203A - Solution Set 3

Math 03A - Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)

### Algebraic Varieties. Brian Osserman

Algebraic Varieties Brian Osserman Preface This book is largely intended as a substitute for Chapter I (and an invitation to Chapter IV) of Hartshorne [Har77], to be taught as an introduction to varieties

### PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset

### 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

### 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

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### INTRODUCTION TO ALGEBRAIC GEOMETRY. Throughout these notes all rings will be commutative with identity. k will be an algebraically

INTRODUCTION TO ALGEBRAIC GEOMETRY STEVEN DALE CUTKOSKY Throughout these notes all rings will be commutative with identity. k will be an algebraically closed field. 1. Preliminaries on Ring Homomorphisms

### 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

### 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

### Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

### ALGEBRAIC GEOMETRY CAUCHER BIRKAR

ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasi-projective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction

### 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

### 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

### 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.

### 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

### ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.

### INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

### 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

### Math 203A - Solution Set 1

Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in

### DEFORMATIONS VIA DIMENSION THEORY

DEFORMATIONS VIA DIMENSION THEORY BRIAN OSSERMAN Abstract. We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring

### ALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30

ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects

### Math 203A, Solution Set 6.

Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P

### 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

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### NOTES ON FIBER DIMENSION

NOTES ON FIBER DIMENSION SAM EVENS Let φ : X Y be a morphism of affine algebraic sets, defined over an algebraically closed field k. For y Y, the set φ 1 (y) is called the fiber over y. In these notes,

### Math 203A - Solution Set 1

Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in

### 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

### 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

### 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,

### Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 As usual, k is a perfect field and k is a fixed algebraic closure of k. Recall that an affine (resp. projective) variety is an

### 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

### Algebraic varieties. Chapter A ne varieties

Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point

### MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1

MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1 CİHAN BAHRAN I discussed several of the problems here with Cheuk Yu Mak and Chen Wan. 4.1.12. Let X be a normal and proper algebraic variety over a field k. Show

### 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

### 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

### Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.

Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y

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

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

### 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.............

### 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

### ne varieties (continued)

Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we

### MATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties

MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,

### A course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011

A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September

### 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!

### Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points

Chapter 7 Schemes III 7.1 Functor of points Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of R-valued points X(R) =Hom Schemes (Spec R, X) This is

### A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties

A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties Lena Ji February 3, 2016 Contents 1. Algebraic Sets 1 2. The Zariski Topology 3 3. Morphisms of A ne Algebraic Sets 5 4. Dimension 6 References

### 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

### 10. Noether Normalization and Hilbert s Nullstellensatz

10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.

### Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

### D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties

D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski Solutions Sheet 1 Classical Varieties Let K be an algebraically closed field. All algebraic sets below are defined over K, unless specified otherwise.

### Lecture 3: Flat Morphisms

Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

### Algebraic Geometry. Instructor: Stephen Diaz & Typist: Caleb McWhorter. Spring 2015

Algebraic Geometry Instructor: Stephen Diaz & Typist: Caleb McWhorter Spring 2015 Contents 1 Varieties 2 1.1 Affine Varieties....................................... 2 1.50 Projective Varieties.....................................

### 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

### Dedekind Domains. Mathematics 601

Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K/F is a finite

### 4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset

4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z

### 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

### LECTURE Affine Space & the Zariski Topology. It is easy to check that Z(S)=Z((S)) with (S) denoting the ideal generated by elements of S.

LECTURE 10 1. Affine Space & the Zariski Topology Definition 1.1. Let k a field. Take S a set of polynomials in k[t 1,..., T n ]. Then Z(S) ={x k n f(x) =0, f S}. It is easy to check that Z(S)=Z((S)) with

### 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)

### 1. Valuative Criteria Specialization vs being closed

1. Valuative Criteria 1.1. Specialization vs being closed Proposition 1.1 (Specialization vs Closed). Let f : X Y be a quasi-compact S-morphisms, and let Z X be closed non-empty. 1) For every z Z there

### Yuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99

Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by

### 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

### 4.4 Noetherian Rings

4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)

### Institutionen för matematik, KTH.

Institutionen för matematik, KTH. Contents 7 Affine Varieties 1 7.1 The polynomial ring....................... 1 7.2 Hypersurfaces........................... 1 7.3 Ideals...............................

### 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

### 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

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then

### 2. Intersection Multiplicities

2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.

### π X : X Y X and π Y : X Y Y

Math 6130 Notes. Fall 2002. 6. Hausdorffness and Compactness. We would like to be able to say that all quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties.

### Margulis Superrigidity I & II

Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd

### k k would be reducible. But the zero locus of f in A n+1

Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along

### 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

### 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

### 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......................

### 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

### Algebraic Varieties. Chapter Algebraic Varieties

Chapter 12 Algebraic Varieties 12.1 Algebraic Varieties Let K be a field, n 1 a natural number, and let f 1,..., f m K[X 1,..., X n ] be polynomials with coefficients in K. Then V = {(a 1,..., a n ) :

### Projective Varieties. Chapter Projective Space and Algebraic Sets

Chapter 1 Projective Varieties 1.1 Projective Space and Algebraic Sets 1.1.1 Definition. Consider A n+1 = A n+1 (k). The set of all lines in A n+1 passing through the origin 0 = (0,..., 0) is called the

### 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

### 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

### Algebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003

Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties

### 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)

### Infinite-Dimensional Triangularization

Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector

### MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS

MATH 63: ALGEBRAIC GEOMETRY: HOMEWORK SOLUTIONS Problem. (a.) The (t + ) (t + ) minors m (A),..., m k (A) of an n m matrix A are polynomials in the entries of A, and m i (A) = 0 for all i =,..., k if and

### ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS

ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS UZI VISHNE The 11 problem sets below were composed by Michael Schein, according to his course. Take into account that we are covering slightly different material.

### CHOW S LEMMA. Matthew Emerton

CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there