CHEVALLEY S THEOREM AND COMPLETE VARIETIES


 Nickolas Johns
 4 years ago
 Views:
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 rvariable 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 nonempty 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, SpringerVerlag, Igor Shafarevich, Basic algebraic geometry 1. varieties in projective space, second ed., SpringerVerlag,
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
More informationCOMPLEX 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
More informationDIVISORS 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
More informationSEPARATED 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.
More informationSEPARATED 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
More informationVALUATIVE 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
More informationMATH 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
More informationVALUATIVE 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
More informationIntroduction 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
More informationMath 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
More informationHARTSHORNE 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
More information(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 kalgebra. In class we have seen that the dimension theory of A is linked to the
More informationAN 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,
More informationINTRODUCTION 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. Nonsingularity:
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Schemetheoretic closure, and schemetheoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:
More informationMath 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)
More informationAlgebraic 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
More informationPROBLEMS, MATH 214A. Affine and quasiaffine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasiaffine 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
More informationALGEBRAIC 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
More informationABSTRACT 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 nspace the collection A n k of points P = a 1, a,..., a
More informationCHEAT 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
More informationINTRODUCTION 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
More information9. 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
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 610, 2010 1 Introduction In this project, we will describe the basic topology
More informationIntroduction 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
More informationALGEBRAIC GEOMETRY CAUCHER BIRKAR
ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasiprojective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction
More informationSynopsis of material from EGA Chapter II, 5
Synopsis of material from EGA Chapter II, 5 5. Quasiaffine, quasiprojective, proper and projective morphisms 5.1. Quasiaffine morphisms. Definition (5.1.1). A scheme is quasiaffine if it is isomorphic
More informationCHAPTER 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
More informationResolution 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.
More informationSummer 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
More informationALGEBRAIC 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.
More informationINVERSE 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
More informationThis 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
More informationMath 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
More informationDEFORMATIONS 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
More informationALGEBRAIC 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
More informationMath 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
More information3. 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
More informationNOTES 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
More informationNOTES 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,
More informationMath 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
More informationAlgebraic 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
More information10. 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
More informationAN 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,
More informationIntroduction 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
More informationALGEBRAIC 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
More informationAlgebraic varieties. Chapter A ne varieties
Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne nspace 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
More informationMATH 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
More information11. 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
More information2. 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 socalled prime and maximal ideals. Let
More informationReid 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
More informationCHAPTER 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
More informationSCHEMES. David Harari. Tsinghua, FebruaryMarch 2005
SCHEMES David Harari Tsinghua, FebruaryMarch 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............
More informationMath 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
More informationne 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
More informationMATH32062 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,
More informationA 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
More informationFOUNDATIONS 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!
More informationHere is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of Rvalued 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 Rvalued points X(R) =Hom Schemes (Spec R, X) This is
More informationA 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
More informationFOUNDATIONS 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
More information10. 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.
More informationCourse 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...............................
More informationDMATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties
DMATH 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.
More informationLecture 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, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationAlgebraic 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.....................................
More information14. 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
More informationDedekind 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 RiemannRoch theorem. The main theorem shows that if K/F is a finite
More information4. Images of Varieties Given a morphism f : X Y of quasiprojective 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 quasiprojective 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
More information12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism 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 Amorphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s
More informationLECTURE 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
More informationMATH 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)
More information1. 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 quasicompact Smorphisms, and let Z X be closed nonempty. 1) For every z Z there
More informationYuriy 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
More informationAlgebraic 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
More information4.4 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)
More informationInstitutionen 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...............................
More informationLECTURE 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 nonstablerationality based on the decomposition of the diagonal in the Chow group. This criterion
More informationAPPENDIX 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
More informationALGEBRAIC 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
More information2. 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.
More informationπ 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 quasiprojective varieties are Hausdorff and that projective varieties are the only compact varieties.
More informationMargulis Superrigidity I & II
Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and RuhrUniversität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd
More informationk 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
More informationMath 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
More information9. 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
More informationAlgebraic 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......................
More informationExtension 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
More informationAlgebraic 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 ) :
More informationProjective 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
More informationAlgebraic 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
More informationExercises 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 Email address: alex.massarenti@sissa.it These notes collect a series of
More informationAlgebraic 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
More information3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).
3. Lecture 3 3.1. Freely generate qfhsheaves. 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)
More informationInfiniteDimensional Triangularization
InfiniteDimensional 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
More informationMATH 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
More informationADVANCED 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.
More informationCHOW 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
More informationHYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA
HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA OFER GABBER, QING LIU, AND DINO LORENZINI Abstract. Let X/S be a quasiprojective morphism over an affine base. We develop in this article a technique
More informationSmooth morphisms. Peter Bruin 21 February 2007
Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,
More information