# ne varieties (continued)

Size: px
Start display at page:

Transcription

1 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 can still define the coordinate ring O(X) as the ring of regular functions. We can again identify it with O(X) =k[x 1,...x n ]/I(X). Since I(X) is radical, we get: Lemma O(X) is reduced (which means that it does not have nonzero nilpotents). Proof. Suppose r 2O(X) satisfied r N = 0. We can lift r to a polynomial r 2 k[x 1,...x n ] which would have to satisfy r N 2I(X). Therefore r 2I(X), which implies r = 0. One can define a category of algebraic sets as before, and the duality theorem can be extended without much trouble. Theorem (Duality version 2). There is an anti-equivalence between the category of algebraic sets and reduced a ne algebras. Given two a ne algebraic sets X A n k and Y Am k. We can form the cartesian product X Y A n+m k. We saw in an exercise that this is algebraic. In fact, we will do that part of it. Let k[y 1,...,y m ] denote polynomials on the second A m k. We have inclusions k[x 1,...x n ] 1! k[x 1,...,x n,y 1...,y m ] 2 k[y 1,...y m ] Expressing X Y = X A m \ A n Y, shows that X Y = V ( 1 (I(X)) + 2 (I(Y )). Therefore Lemma X Y is algebraic with ideal I(X) +I(Y ) ( we suppress the s from now on). 13

2 o o c # O We can jazz this up by observing that k[x 1,...,y m ] = k[x 1,...x n ] k k[y 1,...,y m ] where the isomorphism sends f g on the right to the product fg on the left. Lemma Proof. The left side is isomorphic to O(X Y ) = O(X) k O(Y ) k[x 1,...,y m ]/(I(X)) + I(Y )) from above. This is also isomorphic to the tensor product using the fact that is right exact. k[x 1,...x n ]/I(X) k k[y 1,...,y m ]/I(Y ) We will give a second proof based on universal properties. First, let us consider the product X Y. The two projections p : X Y! X and q : X Y! Y are both morphisms. We have the following universal property. Lemma If Z is any algebraic set equipped with morphisms p 0 : Z! X and q 0 : Z! Y, there is unique morphism r making the diagram Z Y q 0 p 0 / r q p X X Y commute. Proof. The map r(z) =(p 0 (z),q 0 (z)) is clearly regular, and is the unique solution. At first, this may seem to be strange way to think about the product. But one can check that familiar examples of products of sets, groups, topological spaces... are characterized this way. If we take a familiar category and turn the arrows around, we get the dual notion called a coproduct, which less familiar. Let us spell it out. Given objects X, Y in a category, their coproduct (if it exists) is an object X Y with morphisms i : X! X Y and j : Y! X Y such that any diagram with solid lines Z j 0 O Y i 0 r j / X i X Y and be uniquely completed with the dotted line r. The coproduct of two sets is their disjoint union X ` Y. We come to the most important example for us. 14

3 Theorem If A and B are commutative k-algebras, their coproduct is the algebra A k B with i(a) =a 1 and j(b) =1 b. Proof. Consider a diagram, as above, with A, B, C in place of X, Y, Z. Then r( P a i b i )= P i 0 (a i )j 0 (b i ) gives the required homomorphism. We can now give a short second proof of lemma 2.1.4: Second Proof. By duality, products of algebraic sets must correspond to coproducts of their coordinate rings. 2.2 Algebraic Groups An algebraic group is to algebraic geometry what a Lie group is to di erential geometry. More precisely, an a ne 1 algebraic group over k is an a ne algebraic set G which is also a group such that the group operations are regular maps. This means that the product and inverse m : G G! G, m(xy) =xy i : G! G, i(x) =x 1 are morphisms. One reason for not restricting to varieties is because of the following example. Example Any finite group is an algebraic group. Example As we saw earlier, G = GL n (k) can be embedded as a closed set in A n2 +1 by sending A 7! (A, 1/ det A). This means that 1/ det A is a regular function on G Matrix multiplication is clearly regular. So is A 7! A 1 = 1 det A adja. Therefore G is an algebraic group To generate more examples, we observe Lemma If G is an algebraic group and H G is a closed subgroup, then H is an algebraic group on its own. Example SL n (k) GL n (k) is a closed subgroup. Example The orthogonal group O n (k) ={A 2 GL n (k) A 1 = A T } is closed. The next example is tricky. The projective linear group PGL n (k) =GL n (k)/k 1 We will suppress the word a ne now, until we encounter nona ne examples later such as elliptic curves. 15

4 where k is identified with the group of nonzero scalar matrices. There is no question it s a group, but it s not obvious that it is an a ne algebraic set, let alone an algebraic group. So let us first construct try to the coordinate ring S = O(PGL n (k)).this can be identified with ring of regular functions on GL n (k) which is are constant on the orbits of k. This means that they are invariant under the action of k. We can make this explicit by observing that R = O(GL n (k)) = k[x ij, 1/D], where i, j range from 1,...,n and D =det(x ij ). An element t 2 k acts by x ij 7! tx ij. This means that S can be identified with the homogeneous elements of degree 0. A bit of thought shows that S is generated as a k-algebra by ratios m/d where m is a monomial of degree n. Therefore S is finitely generated. Since it is a subring of R, it is also a domain. Therefore S must be the coordinate of an algebraic variety, which, as we ll see in the exercises, is PGL n (k).this takes care of the first problem. However, we still need to make it into an algebraic group. Given an algebraic group G, O(G) is an a ne algebra, with a co-multiplication m : O(G)!O(G) O(G) and a co-iniverse (usually called an antipode) and co-identity (or augmentation) i : O(G)!O(G) e : O(G)! k Let explain what these mean, using g, h for elements of G and Greek letters for functions. If e 2 G is the identity, then e ( )= (e). The second is required to satisfy (i )(g) = (g 1 ) Finally suppose that Then we must have m ( )= X i i (gh) = X i (g) i (h) These maps are required to satisfy various identities dual the axioms of a group. For example the associative law can be expressed as saying that the diagram G G G m 1 / G G 1 m G G commutes. So by duality, we have that the diagram O(G) m m / / G m O(G) O(G) m O(G) O(G) m 1 1 m / O(G) O(G) O(G) 16

5 should commute. O(G) together with these maps, satisfying all of these laws, is called a Hopf algebra. As a corollary of the duality theorem, we see that Proposition Isomorphism classes of algebraic groups correspond to isomorphism classes of Hopf algebras. When G = GL n (k), we can see that on R = O(G) e (x ij )= ij i (x ij )= 1 (ijth cofactor of the generic matrix) D m (x ij )= X` x i` x`j These are ring homomorphisms, so the above formulas determine these maps. The map i sends a homogeneous element of degree d to a homogeneous element of degree d, and m sends a degree element to a degree 2d element. It follows that S = O(PGL n (k)) is preserved under these operations. Therefore it becomes a Hopf algebra. This completes the description of PGL n (k) as an algebraic group. 2.3 Chevalley s theorem, and algebraic groups Before saying more about algebraic groups, we need to talk about regular maps in more detail. If f : X! Y is a regular map of algebraic sets, then we say that preimages of closed sets are closed. By contrast images are more complicated. Example Let : A 2! A 1 be projection onto x (using x, y as coordinates). Then p(v (xy 1)) = A 1 {0} is open, but p(v (x)) = {0} is closed. In the above examples, images can be either open or closed. But for more general maps, we need a new concept. A set is locally closed if it is an open subset of a closed set. More generally a set is constructible if it is a finite union locally closed sets. Theorem (Chevalley). If f : X! Y is a regular map between a ne algebraic sets, then images of constructible sets are constructible. In particular, f(x) is constructible. We will just prove the last statement that f(x) is constructible. Our proof will be long and meandering; it is meant to be instructive, rather than e cient. Let us make a preliminary reduction. Lemma If the last part of theorem is true for varieties, then it is true for algebraic sets. 17

6 Proof. If X = S X i is a decomposition into irreducible components, then it su ces to prove that f(x i ) is constructible. So we may as well assume that X is irreducible. Now suppose that Y = S Y i is a decomposition into irreducible components. Let f i = f f 1 Y i.thenf(x) = S im f i. So we may as well assume that Y is irreducible. So from now on until we say otherwise, assume that X and Y are irreducible, so that A = O(X) and B = O(Y ) are domains. Then f corresponds to a homomorphism = f : B! A. Let us catalogue the various kinds of maps we will encounter. A closed immersion is an inclusion X Y where X is closed. In terms of rings, we have the canonical map B! A = B/I, wherei = I(X). The converse is also clear. Therefore Lemma The map f is a closed immersion if and only if B! A is surjective. A basic open immersion is an inclusion X = D(g) Y. We saw earlier that D(g) is a ne. In fact D(g) can be viewed as the closed subset V (gt 1) Y A 1 where t is the coordinate of the second factor. Therefore A = B/(gt 1) = B[1/g]. Lemma The map f is a basic open immersion if and only if inclusion of the form B B[1/g]. is an A map f : X! Y is dominant if f(x) isdense. Lemma f is dominant if and only if B! A is injective. Proof. If f is dominant, the nonzero regular function g on Y has nonzero restriction to f(x), and therefore f g 6= 0. In the other direction, if is not injective, it factors through A/ ker. This implies that f factors through a closed immersion, so can t be dominant. In the next couple of items, we will work backwards from the algebra. Lemma The map is the inclusion B A = B[x 1,...,x n ] if and only if f : X = A n k Y! Y is the projection. Proof. The key point is that if X = A n Y then A = O(A n ) k B = k[x 1,...x n ] k B = B[x 1,...,x n ]. Recall that an inclusion B A is an integral extension if A is finite as a B-module. Definition f : X! Y is called finite, if B A is an integral extension. A terminology is explained by the following Lemma A finite map is surjective and set theoretically finite to one. 18

7 Proof. Remember that points of X (or Y ) correspond to maximal ideals of A (or B). If m 2 Max B, then the preimage f 1 (m) can be identified with the set of maximal ideals of A containing m. This is nonempty by the so called going up theorem. We can also idenity f 1 (m) =MaxC where C =(B/m B A). This a finite dimensional k algebra, so it has only finitely many maximal ideals. Here is the key proposition. Proposition f(x) contains an open subset of f(x). Proof. By factoring f as X! f(x) Y, we can reduce to the case where f is dominant. So we have an inclusion B A. Let K be the fraction field of B. Since A is finitely generated over k, K B A is necessarily finitely generated as a K-algebra. We can express the generators as a i /b for some b 2 B. By the Noether normalization, we can write K A B as an integral extension of a polynomial ring K[x 1,...,x n ]. It follows that A[1/b] =B[1/b] B A is an integral extension of B[1/b][x 1,...,x n ]. This implies that over D(b), the restriction of f is the composite of finite map and a projection, so it is surjective. Therefore D(b) f(x). To finish the proof, we use dimensions. Recall that dim X is the transcendence degree of the field of fractions of A. It is also equal to the Krull dimension of A. This implies that any (closed) subvariety of X has strictly small dimension. Proof of last part lof theorem We can assume, by induction, that the result holds when X is replaced by aset whose irreducible components have dimension less than dim X. By the last proposition, f(x) =U [ Z, whereu f(x) is open and Z =(f(x) U). The set U is locally closed in Y. By induction Z = f f 1 Z is constructible. Therefore f(x) is constructible. Let us return to the study of algebraic groups. A homomorphism of algebraic groups is a homomorphism which is also regular. Theorem Let G be an algebraic group (a) If U, V G are dense open sets, U V = G. (b) The closure of a subgroup is a subgroup (c) A constructible subgroup of G is a closed subgroup. (d) If f : G 1! G 2 is a homomorphism, then both ker f and im f are closed subgroups of G 1 and G 2 respectively. Proof. Suppose g 2 G. Then gv 1 is dense open because x 7! gx 1 is a homeomorphism. Therefore this has a nonempty intersection with U. Let h 2 U \ gv 1. This means that h = g` 1 for some ` 2 V. Therefore g = h` 2 UV. This proves (a). (b) Suppose that H G is a subgroup. Then since inversion is a homeomorphism which takes H itself, it takes H to itself. If h 2 H, then left or right 19

8 multiplication by h also a homeomorphism preserving H, hh,hh H. This implies that if h 2 H, thenhh H and Hh H. The same reasoning shows that hh H and Hh H. SoH is a subgroup. (c) Let H be a constructible subgroup. Then H is a subgroup by (b). We also know that H contains a subset U open and dense in H. By (a), U U = H. This forces H = H. (d) Finally if f is a homomorphism, it s continuous. Therefore ker f is closed. By Chevalley, im f is a constructible subgroup of G 2. So it s closed by (c). 2.4 Exercises 1. Prove that the product of two irreducible a ne algebraic sets is irreducible. 2. Let X ` Y denote the disjoint union of two a ne algebraic sets. Give two proofs that O(X ` Y )=O(X) O(Y ), an direct elementary proof, and a proof using universal properties. 0 I 3. Let J = where the blocks n n. Prove that Sp I 0 2n (k) ={A 2 GL 2n (k) A T JA = J} is a closed subgroup. 4. Given an algebraic group G, prove that the center Z(G) ={g 2 G 8h 2 G, hg = gh} is a closed subgroup. 5. Give an example of a regular map of varieties, where the image is neither open nor closed. 6. Given an algebraic group G and an a ne (for now) algebraic set X, we say that G acts X is there is a regular map : G X! X satisfying the usual conditions for an action (e, x) =x and (g 1, (g 2,x)) = (g 1 g 2,x). Usually, we just write gx = (g, x). Given x 2 X, show that its orbit {gx g 2 G} is constructible 7. Show that the converse to lemma fails. 20

### ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

### 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

### 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

### Exercises on chapter 0

Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that

### 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

### 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.

### 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

### MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to

### 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

### ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

### ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

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

### 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.

### 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

### 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

### Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

### 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 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

### 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

### 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

### 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.

### Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

### 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

### 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,

### 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

### A Crash Course in Topological Groups

A Crash Course in Topological Groups Iian B. Smythe Department of Mathematics Cornell University Olivetti Club November 8, 2011 Iian B. Smythe (Cornell) Topological Groups Nov. 8, 2011 1 / 28 Outline 1

### MATH 326: RINGS AND MODULES STEFAN GILLE

MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called

### Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

### 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

### Representations and Linear Actions

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

### 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

### Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

### 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

### TROPICAL SCHEME THEORY

TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),

### Exercises on chapter 1

Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

### 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

### 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

### Math 40510, Algebraic Geometry

Math 40510, Algebraic Geometry Problem Set 1, due February 10, 2016 1. Let k = Z p, the field with p elements, where p is a prime. Find a polynomial f k[x, y] that vanishes at every point of k 2. [Hint:

### 3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at

### MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

### 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

### 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 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

### 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

### Homework 3: Relative homology and excision

Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior

### 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

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

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

### Math 210C. The representation ring

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

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

### Background on Chevalley Groups Constructed from a Root System

Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described

### (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

### Boolean Algebras, Boolean Rings and Stone s Representation Theorem

Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to

### Representations. 1 Basic definitions

Representations 1 Basic definitions If V is a k-vector space, we denote by Aut V the group of k-linear isomorphisms F : V V and by End V the k-vector space of k-linear maps F : V V. Thus, if V = k n, then

### THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

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

### ADVANCED TOPICS IN ALGEBRAIC GEOMETRY

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of

### 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,

### 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

### 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)

### V (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =

20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F -vector subspaces of F n+.

### 12. Hilbert Polynomials and Bézout s Theorem

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

### 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

### 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)

### Polynomials, Ideals, and Gröbner Bases

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

### Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

### Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

### LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

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

### 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

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

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

### ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add

### 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

### Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

### Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

### 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

### 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

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

### 11 Annihilators. Suppose that R, S, and T are rings, that R P S, S Q T, and R U T are bimodules, and finally, that

11 Annihilators. In this Section we take a brief look at the important notion of annihilators. Although we shall use these in only very limited contexts, we will give a fairly general initial treatment,

### 8. Prime Factorization and Primary Decompositions

70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings

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

### COHOMOLOGY AND DIFFERENTIAL SCHEMES. 1. Schemes

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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

### CW-complexes. Stephen A. Mitchell. November 1997

CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,

### 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

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

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

### 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

### GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

### Subgroups of Linear Algebraic Groups

Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.

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

### 15. Polynomial rings Definition-Lemma Let R be a ring and let x be an indeterminate.

15. Polynomial rings Definition-Lemma 15.1. Let R be a ring and let x be an indeterminate. The polynomial ring R[x] is defined to be the set of all formal sums a n x n + a n 1 x n +... a 1 x + a 0 = a

### Review of Linear Algebra

Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

### Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

### POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS

POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS Z. REICHSTEIN AND N. VONESSEN Abstract. We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining

### NOTES ON ALGEBRAIC GEOMETRY MATH 202A. Contents Introduction Affine varieties 22

NOTES ON ALGEBRAIC GEOMETRY MATH 202A KIYOSHI IGUSA BRANDEIS UNIVERSITY Contents Introduction 1 1. Affine varieties 2 1.1. Weak Nullstellensatz 2 1.2. Noether s normalization theorem 2 1.3. Nullstellensatz

### REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

### The Proj Construction

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

### A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In

### SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT

SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.

### 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