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

Size: px
Start display at page:

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

Transcription

1 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 in fact a functor from the category of commutative rings to sets. To get a sense of what this does, let X =Speck[x 1,...,x n ]/(f 1,...,f m )wherek is a field. Suppose L is also a field. Then we know that X(L) =Hom Rings (k[x 1,...x n ]/(f i ),L) To give such a homomorphism is to give an inclusion k L and an assigment x i 7! a i 2 L such that f i (a 1,a 2,...) = 0. Therefore Lemma X(L) =V (f 1,f 2,...) A n L Thus points are points in more or less the original sense. However, we have extended it allow us to work in bigger fields. Also we don t have to restrict to fields. Consider the the double line X =Speck[x, y]/(x 2 ) A 2 k versus the ordinary line Y =Speck[x, y]/(x). For any field L k, X(L) =Y (L), so we can t detect any di erence this way. However, letting R = k[ ]/( 2 ) gives (, a) 2 X(R) Y (R), 8a 2 k So we can see the di erence using these more general rings. If X is nona ne, we can choose an a ne open cover {U i }.Then X(L) = [ U i (L) 1 Unfortunately, Hartshorne omits this important topic, but there are plenty of other sources: EGA I, Demazure-Gabriel s Intro to AG, Eisenbud-Harris Geometry of Schemes, Mumford s Curves on an algebraic surface... 45

2 Let us see how this works for X = P n k =Projk[x 0,...x n ]. We saw that there is an open cover U i = Spec k[x0 /x i,...]. Thus a point of P n k (L) is a point of one of these a ne spaces U i (L). In the general, a given point would lie in several U i s. We can see that p 2 U i (L) and q 2 U j (L) represent the same point if they are projectively the same in the sense that [p] =[q]. Therefore Lemma P n k (L) =Pn L, where the right hand side has the original meaning. Theorem A scheme is determined by its functor of points. We won t prove it, but we indicate the main steps. Given a scheme X, define the generalized functor of points by S 7! Hom Schemes (S, X) The proof reduces to checking two statements 1. The generalized functor of points is determined by the functor of points. 2. A scheme is determined by its generalized functor of points. 1. comes down to the fact the generalized functor applied to Spec R is the original X(R), a general scheme S is covered by a ne schemes, and X( )is determined what happens on a covering. To make the last statement precise, we observe that Lemma X( ) is a sheaf. Sketch. Given F =(f,f # ):2X(S) and an open cover {U i }. We get restrictions F Ui : U i! X given by f Ui and O X (U)!O S (f 1 U \ U i ). It is easy to see that F is determined uniquely by the F Ui. In the other direction, given a collection of morphisms U i! X which agree on intersections, we have to build an F : S! X such that F i = F Ui. For f this is easy, because continuous functions can be patched. To define f # : O X (U)!O S (f 1 U)sendg to the unique section such that f 1 U\U i = f # i (g) 2. follows from a basic fact from category theory Lemma (Yoneda s lemma). If X and Y are objects of a category C such that there is a natural isomorphism Hom C (,X) = Hom C (,Y), then X = Y. Sketch. One has bijections Hom C (X, X) = Hom C (X, Y ) and Hom C (Y,Y ) = Hom C (Y,X), under which the identities map to f 2 Hom C (X, Y ) and g 2 Hom C (Y,X). Naturality can used to show that f and g are inverses. Therefore X = Y. 46

3 7.2 Infinitessimals Let k be a field. Suppose R = k[x 1,...,x n ]/(f 1,...,f m ) and X =SpecR. Givenp 2 X(k), imitating what we do in calculus, define the tangent space T p X = {c 2 k n p (c i )=0} i Observe that this definition depends on the equations f i, and not just X as scheme. We want to give an alternative description which is more intrinsic. The key is the ring D = k[ ]/( 2 ), which came up earlier. For some reason, it is called the ring of dual numbers. We observe a couple of features of this ring. Firstly, D has exactly one prime ideal m =( ). Given a polynomial f(x 1,x 2,...) 2 k[x 1,...x n ], and a =(a 1,...) 2 k n, we have a Taylor expansion f(x i a (x i a i )+... where the... refers to sum of quadratic and higher order homogeneous polynomials in x i a i.ifwesubstitutea = b + c 2 k n then all the higher terms will cancel. Therefore Lemma f(a) i a (c i ) A point a 2 X(D) is given by assigning a i = b i +c i 2 D such that f(a) = 0. From the previous lemma, we see that this equivalent to b 2 X(k) and c 2 T b X. Therefore Lemma There is a bijection between X(D) and pairs b 2 X(k) and c 2 T b X. Corollary There is a bijection T b X = 1 (b), where : X(D)! X(k) is the natural map. The last result shows that the set T b X doesn t depend on the equations. But we still don t see the vector space structure. This can be obtained as follows. Let b 2 X(k), and let m = m b be the maximal ideal of regular function vanishing at b. Set S = R m. This is a local ring with maximal ideal we also call m, and residue field k. Lemma T b X = Hom k (m/m 2,k) Sketch. We can identify T b X with the set of homomorphisms from R! D which send m to the maximal ideal of D. Such a homomorphism extend uniquely to local homomorphisms f : S! D, and conversely. Since k is also a subfield of S, itsplitsintoasums = k m. f is determined by its restriction f m 2 Hom k (m/m 2,k). This defines a map T b X! Hom k (m/m 2,k), which can be seen to be an isomorphism. 47

4 It is not hard to see that the vector space structure on the right agrees with the vector space structure on T b X from (7.1). Given a local ring S with maximal ideal m and residue field k, we define its (Zariski) tangent space by TS = Hom k (m/m 2,k) Observe that m/m 2 is always k-vector space, so this formula makes sense in general. Suppose that S is noetherian, then this space is finite dimensional. From commutative algebra, we get an inequality Theorem If S is noetherian dim TS dim S, where the dimension on the left is the vector space dimension, and on the right it is the Krull dimension. Rather than taking this on faith, we should understand why this holds, at least in the case when S is the local ring of a subscheme X A n k as a above. From the inclusion, we get dim X apple dim A n = n. If it so happens that T b X = k n, then we are done. But this usually doesn t happen. So instead try to replace A n by A dim TbX in such a way that it still contains X or enough of it, to compute the dimension. To make sense of this, we switch to the algebraic viewpoint. By Nakayama s lemma, d =dimt b X can be identified with the minimum number of generators for the ideal m. Choose generators ȳ 1,...,ȳ d 2 m, and define the k-algebra map k[y 1,...,y d ]! S by sending y i 7! ȳ i. This factor through the localization k[y 1,...,y d ] (y1,...,y d )! S If the map was surjective, we would again be done. Unfortunately, it isn t usually surjective, but what is true and not hard to see is that the map surjects onto S/m N for any N. This implies that it gives surjection of completions 2 k[[y 1,...,y d ]]! Ŝ := lim N S/m N So now we have dim S =dimŝ apple d 7.3 Regularity Recall that noetherian local ring S is called regular if we have equality dim TS =dims 2 See Atiyah-Macdonald for the basics about completions. One can view the completion as awayofmakingthings evenmorelocal thanordinarylocalization. Theintuitionisthat going to power series is like working with analytic neigbourhoods, which are finer than Zariski neighbourhoods. 48

5 A point p 2 X variety (scheme) is called nonsingular (regular) if the local ring O X,p is regular. X is called nonsingular/regular if all its points have this property. Note that for varieties, the words are interchangeable. For schemes, nonsingularity is somewhat ambiguous because it might refer to regularity or a stronger condition called smoothness. Theorem Let k be a field. If X A n k is the closed subscheme defined by polynomials f 1,...,f m, a 2 X(k) is nonsingular if and only rank (a) = n dim j Proof. Let J be the Jacobian matrix above. Then T a X =kerj(a) by(7.1). Therefore, the theorem follows from the rank-nullity theorem of linear algebra. Corollary The set of nonsingular points is open. Proof. This theorem and theorem imply that a is nonsingular if and if rank (J(a)) n dimx The complementary condition is closed because it given by vanishing of minors of J. Note that this set might be empty. For example, this is the case for the double line X =Speck[x, y]/(x 2 ). However, for varieties, it s a di erent story. Theorem Let k be an algebraically closed field. Let X A n k be the closed subvariety, or equivalently Spec k[x 1,...x n ]/I, where I is a prime ideal. Then the set of nonsingular points is nonempty and open. Corollary The same conclusion holds for quasiprojective varieties. We already know that this set is open, so we just have to prove that it is nonempty. We start with a special case. Lemma The theorem holds for the hypersurface X = V (f), where f is irreducible and nonconstant. Proof. We prove this by contradiction. So assume all points are singular. Then f xi (a) i (a) = 0 for all a 2 X and i. Therefore f xi 2 (f) by the Nullstellensatz. Since deg f xi < deg f, wemusthavef xi = 0 as a polynomial, and this holds for each i. There two ways this can happen: Either the characteristic is 0, in which case f must be constant, or the characteristic is p>0, and f is a pth power. In either case, this contradicts what we assumed about f. To finish the proof need the following. 49

6 Lemma If X is an a ne variety, then it has a nonempty open set which is isomorphic to a nonempty open subset of some irreducible hypersurface. Proof. Let K be the function field of X. Recall that this is the field of fractions of O(X). Then K is finitely generated extension of k. By field theory, we can find a separating transcendence basis x 1,...x n 2 K. This means that in the sequence k k(x 1,...,x n ) K the first extension is purely transcendental, and the second is finite separable. The primitive element theorem allows us to express K = k(x 1,...,x n,y) for some element y. This is algebraic, so satisfies a nontrivial equation f(x 1,...,x n,y)= X a i (x 1,...,x n )y i =0 After clearing demoninators, we can assume that f is polynomial. Then Y = V (f) A n+1 defines a hypersurface with the same function field as X, whichis to say that X and Y are birational. The final step is observe that two birational varieties have open sets which are isomorphic. This is well known and standard. See for example p 26 of Hartshorne. 7.4 Exercises Exercise One can show that the category of schemes has products. Prove that X Y (R) =X(R) Y (R). (This pretty formal. You won t need the construction of X Y.) AschemeG is called a group scheme (over Z) ifg(r) is a group for each R, and each ring homomorphism R! S induces a group homomorphism G(R)! G(S). A stronger form of Yoneda s lemma than we stated shows that this equivalent to having morphisms m : G G! G, i : G! G and e :SpecZ! G which makes G into group object in the category of schemes. This is similar to the way we defined algebraic groups. Exercise Let G a =SpecZ[T ] (also called A 1 Z ), G m =SpecZ[T,T 1 ],andµ n = Spec Z[T ]/(T n 1). Show that these are group schemes. 3. Show that A n Z is regular. Conclude the same for G a and G m. 4. Show that µ n is not regular. (Look at primes dividing n.) 50

7 5. Prove that an algebraic group G is regular. (Hint: since G is a variety, it contains a maximal nonempty open set U. Use the group structure to show that G = U.) 51

### 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

### 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:

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### NONSINGULAR CURVES BRIAN OSSERMAN

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

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

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

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

### 1. Algebraic vector bundles. Affine Varieties

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

### 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,

### 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

### Arithmetic Algebraic Geometry

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

### 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)

### 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

### 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

### 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 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 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

### = Spec(Rf ) R p. 2 R m f gives a section a of the stalk bundle over X f as follows. For any [p] 2 X f (f /2 p), let a([p]) = ([p], a) wherea =

LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 41 5. Affine schemes The definition of an a ne scheme is very abstract. We will bring it down to Earth. However, we will concentrate on the definitions. Properties

### 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

### 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,

### 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

### 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

### Homework 2 - Math 603 Fall 05 Solutions

Homework 2 - Math 603 Fall 05 Solutions 1. (a): In the notation of Atiyah-Macdonald, Prop. 5.17, we have B n j=1 Av j. Since A is Noetherian, this implies that B is f.g. as an A-module. (b): By Noether

### 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

### 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!

### 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

### 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

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

### 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

### 1 Flat, Smooth, Unramified, and Étale Morphisms

1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q

### 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

### 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

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

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

### 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

### 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

### 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

### 6.1 Finite generation, finite presentation and coherence

! Chapter 6 Coherent Sheaves 6.1 Finite generation, finite presentation and coherence Before getting to sheaves, let us discuss some finiteness properties for modules over a commutative ring R. Recall

### 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

### 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

### Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

### Exercise Sheet 7 - Solutions

Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 7 - Solutions 1. Prove that the Zariski tangent space at the point [S] Gr(r, V ) is canonically isomorphic to S V/S (or equivalently

### Preliminary Exam Topics Sarah Mayes

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

### 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.

### INTRODUCTION TO ALGEBRAIC GEOMETRY CONTENTS

INTRODUCTION TO ALGEBRAIC GEOMETRY JAMES D. LEWIS Abstract. Algebraic geometry is a mixture of the ideas of two Mediterranean cultures. It is the superposition of the Arab science of the lightening calculation

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### 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

### 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

### 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

### Smooth 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,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last

### MATH2810 ALGEBRAIC GEOMETRY NOTES. Contents

MATH2810 ALGEBRAIC GEOMETRY NOTES KIUMARS KAVEH Contents 1. Affine Algebraic Geometry 1 2. Sheaves 3 3. Projective varieties 4 4. Degree and Hilbert Functions 6 5. Bernstein-Kushnirenko theorem 7 6. Degree

### 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

### 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

### Commutative Algebra. Contents. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 4 1.1 Rings & homomorphisms.............................. 4 1.2 Modules........................................ 6 1.3 Prime & maximal ideals...............................

### 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

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### 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

### 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

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

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

### 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

### ON THE REPRESENTABILITY OF Hilb n k[x] (x) Roy Mikael Skjelnes

ON THE REPRESENTABILITY OF Hilb n k[x] (x) Roy Mikael Skjelnes Abstract. Let k[x] (x) be the polynomial ring k[x] localized in the maximal ideal (x) k[x]. We study the Hilbert functor parameterizing ideals

### 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:

### Commutative Algebra. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 2 1.1 Rings & homomorphisms................... 2 1.2 Modules............................. 4 1.3 Prime & maximal ideals....................

### 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

### I(p)/I(p) 2 m p /m 2 p

Math 6130 Notes. Fall 2002. 10. Non-singular Varieties. In 9 we produced a canonical normalization map Φ : X Y given a variety Y and a finite field extension C(Y ) K. If we forget about Y and only consider

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

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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

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

### (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

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

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

### Algebraic 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 ) :

### 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

### 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

### 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.

### MASTER S THESIS MAT CHOW VARIETIES

MASTER S THESIS MAT-2003-06 CHOW VARIETIES David Rydh DEPARTMENT OF MATHEMATICS ROYAL INSTITUTE OF TECHNOLOGY SE-100 44 STOCKHOLM, SWEDEN Chow Varieties June, 2003 David Rydh Master s Thesis Department

### Spring 2016, lecture notes by Maksym Fedorchuk 51

Spring 2016, lecture notes by Maksym Fedorchuk 51 10.2. Problem Set 2 Solution Problem. Prove the following statements. (1) The nilradical of a ring R is the intersection of all prime ideals of R. (2)

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 51 AND 52

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 51 AND 52 RAVI VAKIL CONTENTS 1. Smooth, étale, unramified 1 2. Harder facts 5 3. Generic smoothness in characteristic 0 7 4. Formal interpretations 11 1. SMOOTH,

### 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

### 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,

### 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.

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### Math 762 Spring h Y (Z 1 ) (1) h X (Z 2 ) h X (Z 1 ) Φ Z 1. h Y (Z 2 )

Math 762 Spring 2016 Homework 3 Drew Armstrong Problem 1. Yoneda s Lemma. We have seen that the bifunctor Hom C (, ) : C C Set is analogous to a bilinear form on a K-vector space, : V V K. Recall that

### Another way to proceed is to prove that the function field is purely transcendental. Now the coordinate ring is

3. Rational Varieties Definition 3.1. A rational function is a rational map to A 1. The set of all rational functions, denoted K(X), is called the function field. Lemma 3.2. Let X be an irreducible variety.

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### 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)

### HILBERT FUNCTIONS. 1. Introduction

HILBERT FUCTIOS JORDA SCHETTLER 1. Introduction A Hilbert function (so far as we will discuss) is a map from the nonnegative integers to themselves which records the lengths of composition series of each

### My way to Algebraic Geometry. Varieties and Schemes from the point of view of a PhD student. Marco Lo Giudice

My way to Algebraic Geometry Varieties and Schemes from the point of view of a PhD student Marco Lo Giudice July 29, 2005 Introduction I began writing these notes during the last weeks of year 2002, collecting

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

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

### HYPERSURFACES 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 quasi-projective morphism over an affine base. We develop in this article a technique

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

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

### 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

### 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