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


 Marcus Glenn
 2 years ago
 Views:
Transcription
1 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 equations. Given a set of k polynomials f 1,..., f k K[x], their (algebraic) variety is the set of common zeros: V(f 1,..., f k ) := { p = (p 1,..., p n ) K n : f 1 (p) = = f k (p) = 0 }. Different sets of polynomials can define the same variety. For instance, V(f 1, f 2 ) = V(f 1, f 1 + f 2 ). Thus, instead of thinking about explicit polynomials it is more reasonable to consider the ideal they generate, I = f 1,..., f k, and to define V(I) := V(f 1,..., f k ). Subsets of K n of the form V(I) for some ideal I K[x] are called varieties. Given any ideal I K[x], by Hilbert Basis Theorem, we may always find a finite set of generators. By Exercise 1, the definition of V(I) does not depend on the choice of generators of the ideal I. Remark 1. Two distinct ideals may define the same variety, e.g. V(u) = V(u 2 ) = {0} K 1. Our later lecture on the Nullstellensätze deals with this issue for fields K that are either algebraically closed, like the complex numbers K = C, or real closed, like the reals K = R. Algebraic geometry is the study of the geometry of varieties. As in many branches of mathematics, given a fundamental object  varieties in our case  one considers the basic, irreducible building blocks. A variety V(I) is called irreducible if it cannot be written as a union of proper subvarieties. In symbols, V(I) is irreducible if and only if V(I) = V(J) V(J ) = V(I) = V(J) or V(I) = V(J ) for any ideals J and J. We can turn K n into a topological space, using the Zariski topology, in which varieties are closed sets. In this setting, the definition of an irreducible variety coincides with the definition of an irreducible topological space. Our aim is to study relations between the geometry of V(I) and algebraic properties of I. Consider a maximal ideal m := x 1 p 1,..., x n p n K[x]. Note that (p 1,..., p n ) V(I) if and only if I m. Given any subset V K n, we can consider the set of all polynomials that vanish on V. This set is an ideal, denoted as I(V ) := { f K[x] : f(p) = 0 for all p V }. Note that W is a variety if and only if W = V(I(W )). Furthermore, for varieties V, W, we have V W if and only if I(W ) I(V ). 1
2 Proposition 2. A variety W K n is irreducible if and only if its ideal I(W ) is prime. Proof. Suppose I(W ) is prime and W = V(J) V(J ). If W V(J) then there exists f J and v W such that f(v) 0, i.e. f I(W ). For any g J we know that fg vanishes on V(J) and V(J ), hence on W. Thus fg I(W ). As I(W ) is prime, we have g I(W ), i.e. J I(W ). By Exercise 2 this implies W = V(I(W )) V(J ). Suppose now that W is irreducible and fg I(W ). Hence W = W V(fg) = W (V(f) V(g)) = (W V(f)) (W V(g)). Without loss of generality we may assume W = W V(f), i.e. W V(f), hence f I(W ), which proves that I(W ) is a prime ideal. Many examples of varieties appearing from applications are given as (closures) of images of polynomial maps. Often we think about the domain K n as the space of parameters and the codomain as the space of possible (observable) outcomes cf. Exercise 9. We note that the (Zariski) closure of the image must be irreducible. Example 3. Consider two independent discrete random variables X and Y each one with n states. The probability distribution of X (resp. Y ) may be encoded as a point (p 1,..., p n ) K n (resp. (q 1,..., q n )). The joint distribution of (X, Y ) has n 2 states. The map that associates to a distribution of X and a distribution of Y the joint distribution is given as: K n K n (p 1,..., p n, q 1,..., q n ) (p 1 q 1, p 1 q 2,..., p 1 q n, p 2 q 1,..., p n q n ) K n2. (1) Further, as p i = q i = 1, we may in fact restrict the domain and obtain a map that we write explicitly for n = 3: (p 1, p 2, q 1, q 2 ) (2) (p 1 q 1, p 1 q 2, p 1 (1 q 1 q 2 ), p 2 q 1, p 2 q 2, p 2 (1 q 1 q 2 ), (1 p 1 p 2 )q 1, (1 p 1 p 2 )q 2, (1 p 1 p 2 )(1 q 1 q 2 )) In Exercise 9 we ask for the description of the ideal of the closure of the image of these maps. Prime ideals play a central role in algebraic geometry. This motivates the following definition. We now take R to be any commutative ring with unity. The primary example is the polynomial ring R = K[x], or its quotient R = K[x]/I for some ideal I. Definition 4. The spectrum of the ring R is the set of all (proper) prime ideals: Spec(R) := { p R : p is a prime ideal }. The set Spec(R) comes with an induced Zariski topology, where the closed set V(I) given by an (arbitrary) ideal I is defined as V(I) = {p Spec R : I p}. We note that the spectrum of the ring remembers a lot of information: all prime ideals. In particular, Spec K[x] has points corresponding to all irreducible subvarieties of K n  not 2
3 only to usual points (p 1,..., p n ) K n, which correspond to maximal ideals of the form x 1 p 1,..., x n p n. One could say that K n is a subset of Spec K[x]. In Exercise 4 you will prove that in fact the Zariski topology on K n is the induced one from the Zariski topology on Spec K[x]. Proposition 5. Any variety can be uniquely represented as a finite union of irreducible varieties (pairwise not contained in each other). Proof. We start by proving the existence of such a decomposition. Any variety W is either irreducible or may be represented as a union W 1 V 1. We may continue presenting W 1 as a union W 2 V 2 etc. We obtain an ascending chain I(W 1 ) I(W 2 )... which stabilizes as the ring is noetherian by Hilbert Basis Theorem. Thus the decomposition procedure must finish. Suppose we have two decompositions V 1 V k = W 1 W s. As each W i0 is irreducible and covered by j (V j W i0 ) we have W i0 V j0. But similarly V j0 W i1 for some i 1. As we cannot have W i0 W i1 it follows that W i0 = V j0. Hence, for every component W i0 there exists a (unique) component V j0 equal to it. We recall that the ring K[x] represents the (polynomial) functions on K n. We now would like to represent (polynomial) functions on a variety W K n. They will form a ring K[W ]. Clearly, as we are interested in polynomial functions, we have a surjection K[x] K[W ]. Two functions coincide on W if and only if their difference vanishes on W. Thus the kernel of the above map equals I(W ) and we have an isomorphism K[W ] := K[x]/I(W ), that we may consider as a definition of the ring of functions on W. The advantage of this approach is that we may consider the ring K[W ] as an object representing W, without referring to any embedding. As before, we identify points p = (p 1,..., p n ) in W with maximal ideals x 1 p 1,..., x n p n K[W ]. The Zariski topologies on W and Spec(K[W ]) are compatible. We have defined our basic objects  affine varieties W and associated rings K[W ]. Following a category theory approach, our aim is to define morphisms of varieties. Given two geometric objects X, Y and a map f : X Y between them, one may pullback functions on Y. Explicitly, given g : Y K we define the pullback f (g) = g f. As we are dealing with algebraic varieties, we would like the pullbacks of polynomials to be polynomials. Hence, given an algebraic map f : W 1 W 2 between varieties, we would like the induced map f : K[W 2 ] K[W 1 ] to be a welldefined ring morphism. In Exercise 5 you will show that any ring morphism K[W 2 ] K[W 1 ] induces a map Spec K[W 1 ] Spec K[W 2 ]. Hence, we may think about algebraic maps between varieties as morphisms among their rings of functions in the opposite direction. Using slightly more sophisticated language there is a contravariant functor, inducing an equivalence of categories of affine irreducible varieties (over K) and finitely generated integral Kalgebras. We note that (algebraic) maps between varieties are continuous in Zariski topology. Remark 6. One may define affine algebraic varieties more generally as Spec R for any (commutative, with unity) ring R, not only finitely generated Kalgebra. However, in these lectures all affine varieties will come from zero sets of polynomials defined over K. 3
4 In next examples we note that the dependence on the field is crucial for many properties of ideals. We start with the map f : K[x] K[y], given by f(x) = y 2. This corresponds to the map K 1 λ λ 2 K 1. If K = C the latter map is surjective. If K = R the image is the set of nonnegative real numbers. In both cases the Zariski closure is the whole space. If K = F p and p 2, the image is a proper subset of K 1 and coincides with its Zariski closure. Another important example is the ideal I = (x 2 + 1). The reader is asked to provide the description of V(I) in Exercise 6. Example 7. We consider three ideals I 1 = (x 2 y 2 ), I 2 = (x 2 2y 2 ) and I 3 = (x 2 + y 2 ) in K[x, y]. The first one is not prime for any K. The second one is not prime for K = R or K = C. However, it is a prime ideal when K = Q. The last I 3 is not prime for K = C, but is a prime ideal for K = Q or K = R. Here we only prove the last statement and leave the others as an exercise. Suppose fg I 3 R[x, y]. This means that fg = (x 2 + y 2 ) h, where f, g, h R[x, y]. By the fundamental theorem of algebra every homogeneous polynomial p in two variables has a unique (up to multiplication by constants) representation as a product of linear forms with complex coefficients p = l i. In particular, if p has real coefficients, the decomposition must be stable under conjugation, i.e. for every i, either l i has real coefficients or l i must also appear in the decomposition. We have x 2 + y 2 = (x + iy)(x iy). In the ring C[x, y], without loss of generality, we may assume (x + iy) f. But then, by the above argument also (x iy) f. Thus f = (x + iy)(x iy) i l i for l i C[x, y]. However, i l i is stable under conjugation, i.e. defines a real polynomial. Thus x 2 + y 2 f in R[x, y]. As we have already seen, the image of a variety does not have to be closed, even if K = C or dense in its Zariski closure if K = R. The following theorem shows however that one can always provide an algebraic description of the image. We start with a definition. Definition 8. A subset A K n is (Zariski) constructible if it can be described as a finite union of (settheoretic) differences of two varieties. A subset B R n is semialgebraic if it can be described as a set of solutions of a finite system of polynomial (weak and strong) inequalities or a finite union of such. Theorem (Chevalley) If K is algebraically closed, then the image of a variety is a constructible set. 2. (TarskiSeidenberg) If K = R then the image of a variety is a semialgebraic set. Proof. The first part can be found e.g. in [6.4][4] and of the second part e.g. in [1.4] [1]. We define the dimension of an irreducible variety V as the maximal length r := dim V of the chain of irreducible varieties V 0 V 1 V r = V. If V is reducible then its dimension is equal to the maximum dimension of all irreducible components from Proposition 5. The dimension is a basic invariant of a variety. This invariant has very nice properties: 4
5 the dimension of the (closure of the) image of a variety V is at most dim V, if V 1 V 2 then dim V 1 dim V 2. If V 2 is irreducible, then the inequality is strict. So far all the geometric objects we encountered were contained in K n. We called them varieties, but more precisely we should refer to them as affine varieties. We now change our perspective with the aim of understanding projective algebraic varieties. We start by recalling the construction of a projective space P(V ) over the vector space V of dimension n + 1. Points of P(V ) correspond to lines in V. Hence [a 0 : : a n ] P(V ) represents a line going through the point (a 0,..., a n ) V, where we assume that not all a i are equal to zero. Formally, P(V ) is the set of equivalence classes [v] for v V \ {0} modulo the relation v 1 v 2 if and only if there exists a nonzero scalar λ such that v 1 = λv 2. For the topological construction over R or C, we note that each line in V intersects the unit sphere precisely in two points. Thus P(V ) may be regarded as a quotient of the sphere, identifying two antipodal points. In particular, it is always compact, with respect to the usual topology. If we look at the subset S i of P(V ) where a i 0 we may always rescale and assume a i = 1. This way we may identify S i = K n. As for any p P(V ) some coordinate is nonzero, the affine spaces S i = K n cover P(V ), as i = 0,..., n. In fact, we may start from the affine spaces S i = K n and glue them together to obtain P(V ). As before we are interested in polynomial functions on P(V ). The first problem we encounter is that for a polynomial f it does not make sense to evaluate it on [a 0 : : a n ], as the result depends on the choice of the representative. It may even happen that f vanishes for some representatives, while it does not for others. Thus, from now on we focus on homogeneous polynomials, i.e. linear combinations of monomials of fixed degree. If f is a homogeneous polynomial of degree d in n+1 variables, then f(ta 0,..., ta n ) = t d f(a 0,..., a n ). In particular, f vanishes on some representative of [a 0 : : a n ] if and only if it vanishes on any representative. Given homogeneous polynomials f 1,..., f k, possibly of distinct degrees, we define the associated projective variety: V(f 1,..., f k ) = {[a 0 : : a n ] P(V ) : f 1 (a 0,..., a n ) = = f k (a 0,..., a n ) = 0}. An ideal is called homogeneous if it may be generated by homogeneous polynomials. In analogy to the affine case we define V(I) = V(f 1,..., f k ) for an ideal I generated by homogeneous polynomials f i. Remark 10. We note that homogeneous ideals contain (many) nonhomogeneous polynomials. In particular, x + y 2, y is a homogeneous ideal. For more characterisations and examples see Exercise 11. In theory, instead of considering a projective variety X P(V ) one can consider the affine cone ˆX over it, i.e. the variety defined by the same ideal, but considered in V. However, in almost all cases, if possible it is preferable to work with projective varieties. The reason is that projective varieties are simpler  they behave better with respect to many properties. Below we present just a few of them. First we note that (if X is not a projective subspace) the affine cone ˆX is always singular at the point 0 V. Second, for K = C or K = R Zariski closed sets are closed in the usual 5
6 topology. In particular, projective varieties are compact. Thus, the image of any projective variety X is closed. Theorem 11. Over an algebraically closed field, the image of a projective variety X is Zariski closed. Proof. The first idea, discussed in the next lecture, is to describe the image as a projection of the graph of the map. Then one can apply Nullstellensatz  see Lecture 5  to turn the problem into one from linear algebra. Details can be found in e.g. [4, ]. One of the important invariants of projective varieties, just as for affine varieties, is the dimension. The second one is the degree. There are several ways to define it. For example, when K is algebraically closed, a general projective subspace L P(V ) of dimension equal to the codimension of V = V(I) P(V ) will intersect V only in finitely many, say d, points. This is the degree of V. If I = (f) is principal and radical then the degree of V(I) equals the degree of f. One of the nicest properties of projective varieties over algebraically closed fields is their behavior under intersection. Theorem 12. [3, 6.2 Theorem 6] Let X, Y P(V ) be two projective varieties of dimensions respectively d 1 and d 2. The intersection X Y has dimension at least d 1 + d 2 dim P(V ). Exercises Exercise 1. Prove that the definition of V(I) does not depend on the choice of the generators of I. Exercise Show that J I implies V(I) V(J). 2. Show that for any subsets A, B K n if A B then I(B) I(A). 3. Give counterexamples to both opposite implications. Exercise 3. Prove that varieties (in K n ) satisfy the axioms of closed sets. Exercise 4. By identifying the point (p i ) K n with the prime ideal x 1 p 1,..., x n p n consider K n as a subset of Spec K[x]. Show that the Zariski topology induced from Spec K[x] to K n is the Zariski topology on K n. Exercise 5. Show that a morphism of rings f : R 1 R 2 induces a map f : Spec R 2 Spec R 1, by proving that a pullback of a prime ideal is prime. Show that the induced map is continuous with respect to the Zariski topology. Exercise 6. Describe V(I) K 1 for I = (x 2 + 1) when K = C and K = R. Exercise 7. Realize the set of n n nilpotent matrices as an affine variety. What is its dimension? 6
7 Exercise Consider a polynomial f K[x] (e.g. f = x). Let D be the (open) set D f = {p K n : f(p) 0}. Construct an affine variety V and a polynomial map inducing a bijection V D. 2. Realize nondegenerate n n matrices as an affine variety. Exercise Use (or not) your favorite computer algebra system to determine the ideal of the image of the map given by formula (2). What is the meaning of the lowest degree polynomial in this ideal? 2. Describe the ideal of the image of the map given by formula (1). 3. Generalize the previous point to more (independent) variables possibly with different (but finite) number of states. Exercise 10. Determine for which prime numbers p, the ideal I 2 = x 2 2y 2 F p [x, y] is prime. Exercise 11. For a polynomial f = a c ax a we call the degree k part of f the homogeneous polynomial a: a =k c ax a. 1. Provide an example of a homogeneous ideal generated by nonhomogeneous polynomials. 2. Prove that an ideal I = f 1,..., f j is homogeneous if and only if for any f i and any k the degree k part of f i belongs to I. 3. Propose an algorithm that, given a set of generators of I K[x], decides if I is a homogeneous ideal. References [1] J. Bochnak M. Coste and M.F. Roy. Real algebraic geometry. Vol. 36. Springer Science & Business Media, [2] D. Cox, J. Little and D. O Shea: Ideals, Varieties, and Algorithms. An introduction to computational algebraic geometry and commutative algebra, Third edition, Undergraduate Texts in Mathematics, Springer, New York, [3] I. Shafarevich: Basic algebraic geometry. 1. Varieties in projective space. Translated from the 1988 Russian edition and with notes by Miles Reid. (1994). [4] R. Vakil: Foundations of algebraic geometry. 7
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
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 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 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 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 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 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 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 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 informationPolynomials, 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
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 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 informationCHEVALLEY 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
More information12. 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
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 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 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 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 informationAlgebraic 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
More informationRings 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...
More informationThe GeometryAlgebra Dictionary
Chapter 1 The GeometryAlgebra Dictionary This chapter is an introduction to affine algebraic geometry. Working over a field k, we will write A n (k) for the affine nspace over k and k[x 1,..., x n ]
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 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 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 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 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 informationMIT Algebraic techniques and semidefinite optimization February 16, Lecture 4
MIT 6.972 Algebraic techniques and semidefinite optimization February 16, 2006 Lecture 4 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture we will review some basic elements of abstract
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 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 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 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 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 informationPOLYNOMIAL 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
More informationExtended Index. 89f depth (of a prime ideal) 121f ArtinRees Lemma. 107f descending chain condition 74f Artinian module
Extended Index cokernel 19f for Atiyah and MacDonald's Introduction to Commutative Algebra colon operator 8f Key: comaximal ideals 7f  listings ending in f give the page where the term is defined commutative
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 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 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 informationVector 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 79 December 2010 1 The notion of vector bundle In affine geometry,
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 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 information214A HOMEWORK KIM, SUNGJIN
214A HOMEWORK KIM, SUNGJIN 1.1 Let A = k[[t ]] be the ring of formal power series with coefficients in a field k. Determine SpecA. Proof. We begin with a claim that A = { a i T i A : a i k, and a 0 k }.
More informationBasic facts and definitions
Synopsis Thursday, September 27 Basic facts and definitions We have one one hand ideals I in the polynomial ring k[x 1,... x n ] and subsets V of k n. There is a natural correspondence. I V (I) = {(k 1,
More informationLECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n
More informationBinomial Exercises A = 1 1 and 1
Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.
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 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 informationINTERSECTION THEORY CLASS 2
INTERSECTION THEORY CLASS 2 RAVI VAKIL CONTENTS 1. Last day 1 2. Zeros and poles 2 3. The Chow group 4 4. Proper pushforwards 4 The webpage http://math.stanford.edu/ vakil/245/ is up, and has last day
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More informationCommutative 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.....................
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 informationBEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
More informationChapter 1. Affine algebraic geometry. 1.1 The Zariski topology on A n
Chapter 1 Affine algebraic geometry We shall restrict our attention to affine algebraic geometry, meaning that the algebraic varieties we consider are precisely the closed subvarieties of affine n space
More informationBoolean 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
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 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 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 informationCOMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES
COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting dtuples of commuting n n matrices that
More informationALGEBRA 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)
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 informationLecture 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
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 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 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 information1. 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 quasiprojective varieties over a field k Affine Varieties 1.
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 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 informationThe Zariski Spectrum of a ring
Thierry Coquand September 2010 Use of prime ideals Let R be a ring. We say that a 0,..., a n is unimodular iff a 0,..., a n = 1 We say that Σa i X i is primitive iff a 0,..., a n is unimodular Theorem:
More informationAlgebraic geometry of the ring of continuous functions
Algebraic geometry of the ring of continuous functions Nicolas Addington October 27 Abstract Maximal ideals of the ring of continuous functions on a compact space correspond to points of the space. For
More informationRings With Topologies Induced by Spaces of Functions
Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,
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 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 informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
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 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 information8. 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
More information14 Lecture 14: Basic generallities on adic spaces
14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions
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 informationwhere 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
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 informationTHE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS. Contents
THE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS ALICE MARK Abstract. This paper is a simple summary of the first most basic definitions in Algebraic Geometry as they are presented in Dummit and Foote ([1]),
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 (Math 6130)
Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,
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 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 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 informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
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 informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
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 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 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 informationORAL QUALIFYING EXAM QUESTIONS. 1. Algebra
ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)
More informationCRing Project, Chapter 7
Contents 7 Integrality and valuation rings 3 1 Integrality......................................... 3 1.1 Fundamentals................................... 3 1.2 Le sorite for integral extensions.........................
More informationA GLIMPSE OF ALGEBRAIC KTHEORY: Eric M. Friedlander
A GLIMPSE OF ALGEBRAIC KTHEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic
More informationFOUNDATIONS 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,
More informationABSTRACT. Department of Mathematics. interesting results. A graph on n vertices is represented by a polynomial in n
ABSTRACT Title of Thesis: GRÖBNER BASES WITH APPLICATIONS IN GRAPH THEORY Degree candidate: Angela M. Hennessy Degree and year: Master of Arts, 2006 Thesis directed by: Professor Lawrence C. Washington
More informationGEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS
GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and
More informationALGEBRAIC 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
More informationMath 137 Lecture Notes
Math 137 Lecture Notes Evan Chen Spring 2015 This is Harvard College s Math 137, instructed by Yaim Cooper. The formal name for this class is Algebraic Geometry ; we will be studying complex varieties.
More informationTensors. Notes by Mateusz Michalek and Bernd Sturmfels for the lecture on June 5, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Tensors Notes by Mateusz Michalek and Bernd Sturmfels for the lecture on June 5, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra This lecture is divided into two parts. The first part,
More information