DOCSLIB.ORG

Math 572 – Lecture Notes Chapter 15 – Commutative Algebra and Algebraic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 572 – Lecture Notes Chapter 15 – Commutative Algebra and Algebraic Geometry Math 572 { Lecture Notes Chapter 15 { Commutative Algebra and Algebraic Geometry Lecture 32 §15.1 Noetherian Rings and Affine Algebraic Sets In this chapter, R will usually be a commutative ring with 1 6= 0. Definition A commutative ring R is Noetherian or satisfies the ascending chain condi- tion if there are no infinite increasing chains of ideals in R. That is, whenever I1 ⊆ I2 ⊆ I3 ⊆ · · · is an increasing chain of ideals, there exists some m such that Ik = Im for all k ≥ m. Proposition 1 If I is an ideal in a Noetherian ring R, then R=I is Noetherian. Any homomor- phic image of a Noetherian ring is Noetherian. Proof. Any infinite ascending chain of ideals in R=I would correspond to an infinite ascending chain of ideals in R. The homomorphic image is isomorphic to R=I where I is the kernel of the homomorphism. 1 Theorem 2 The following are equivalent: (1) R is Noetherian. (2) Every nonempty set of ideals of R contains a maximal element under in- clusion. (3) Every ideal of R is finitely generated. Proof. (1))(2). Assume R is Noetherian. Let Σ by any nonempty collection of ideals, but assume Σ has no maximal element. Choose I1 2 Σ. Since I1 is not maximal, there exists I2 2 Σ such that I1 ( I2: Since I2 is not maximal, there exists I3 2 Σ such that I1 ( I2 ( I3: Continuing in this fashion produces an infinite ascending chain in Σ: I1 ( I2 ( I3 ( ··· ( Ik ( Ik+1 ( ··· : This contradicts the fact that R is Noetherian. Hence Σ must have a maximal element. (2))(3). Assume (2) holds. Let I be any ideal of R. If I is not finitely generated, there exist elements x1; x2; x3;::: 2 I such that I1 = (x1) ( I2 = (x1; x2) ( I3 = (x1; x2; x3) ( ··· Let Σ be the collection of ideals in the sequence above. By (2), Σ contains a 0 0 maximal element I . But then I = In for some n, which means that I was actually finitely generated, a contradiction. So, every ideal of R is finitely generated. (3))(1). Assume every ideal of R is finitely generated. Let I1 ⊆ I2 ⊆ I3 ⊆ · · · 2 be a chain of ideals, and let 1 [ I = Ii: i=1 Then I is an ideal, which by (3), is finitely generated I = (x1; x2; : : : ; xk) Then xi 2 Imi for some mi. Let m = max(m1; : : : ; mk). Then x1; : : : ; xk 2 In for all n ≥ m. Thus, In = I for all n ≥ m and the ascending chain of ideals is finite. Hence, R is Noetherian. Note In the previous proof, we did not every use the assumption of commutativity or that the ring contained 1. Hence, we could have defined a Noetherian ring as a ring (not necessarily commutative and not necessarily containing 1) satisfying the ascending chain condition. Then the previous theorem still holds. Example Every ideal in a PID is finitely generated (by a single element). Hence, any PID is Noetherian. Z, Z[i], F [x] for F a field are Noetherian. Z[x1; x2; x3;:::] is not Noetherian since the ideal (x1; x2; x3;:::) is not finitely generated. Z[x1; : : : ; xn] is Noetherian (by Hilbert's basis theorem). Example Noetherian ring may having infinitely long descending chains. For example, in Z 2 3 (2) ) (2 ) ) (2 ) ) ··· Theorem 3 (Hilbert's Basis Theorem) If R is a Noetherian ring, then so is R[x]. 3 Proof. Let I be any ideal in R[x]. We'll show that I is finitely generated, and so R[x] will be Noetherian. Let L ⊆ R be the set of all leading coefficients of polynomials in I. Thus, axm + lower order terms 2 I ) a 2 L: We'll show that L is an ideal of R. Since the zero polynomial is in I, 0 2 L. Let f = axd + ··· and g = bxe + ··· be polynomials in I. For any r 2 R, either ar − b = 0 or ar − b is the leading coefficient of rxef − xdg: This shows that L is an ideal in R. Since R is Noetherian, L is finitely generated: L = (a1; : : : ; an) ai 2 R: For each ai choose fi 2 I such that ei fi = aix + ··· Set N = max(e1; : : : ; en). For each d 2 f0; 1;:::;N − 1g, let Ld = \set of leading coefficients of elements of degree d in I along with 0". Let f = axd + ··· and g = bxd + ··· be polynomials of degree d in I. For any r 2 R, either ra − b = 0 or ra − b is the leading coefficient of rf − g Hence, Ld is also an ideal of R, and so is finitely generated. Ld = (bd1; bd2; : : : ; bdnd ): For each bdi choose fdi 2 I such that d fdi = bdix + ··· : 4 Let I0 be the ideal in R[x] defined by 0 I = ff1; : : : ; fng [ ffdi : 0 ≤ d < N; 1 ≤ i ≤ ndg : By construction I0 ⊆ I: We'll show I0 = I and so I is finitely generated. If I 6= I0, there exists some nonzero polynomial f 2 I n I0 of minimal degree m. Write f = axm + ··· where deg(f) = m and a is the leading coefficient. Case 1. Suppose m ≥ N. Since a 2 L a = r1a1 + ··· + rnan for some ri 2 R. Then m−e1 m−en 0 g = r1x f1 + ··· rnx fn 2 I has degree m and leading coefficient a. Then f − g 2 I n I0 has degree < m. By the minimality of m, f − g = 0 and so f = g 2 I0, a contradiction. Case 2. Suppose m = d < N. In this case, a 2 Ld. Write a = r1bd1 + ··· + rnd bdnd ; ri 2 R: Then 0 g = r1fd1 + ··· + rnd fdnd 2 I had degree m = d and has the same leading coefficient as f. As before, we have a contradiction. Hence, I = I0. This proves that R[x] is Noetherian. Corollary If R is Noetherian, R[x1; : : : ; xn] is Noetherian. If k is a field k[x1; : : : ; xn] is Noetherian. 5 Lecture 33 Lecture 34 Definition Let k be a field. A ring R is called a k-algebra if k is contained in the center of R and the identity of k is the identity of R. Definition (1) The ring R is a finitely generated k-algebra if R is generated as a ring by k together with some finite set r1; r2; : : : ; rn of elements of R. (2) Let R and S be k-algebras. A map : R ! S is a k-algebra homomor- phism if is a ring homomorphism that is the identity on k. Corollary 5 The ring R is a finitely generated k-algebra if and only if there is some surjective k-algebra homomorphism φ: k[x1; x2; : : : ; xn] ! R: Hence, any finitely generated k-algebra is Noetherian. Proof. If R is generated as a k-algebra by r1; : : : ; rn, we define the map φ: k[x1; : : : ; xn] ! R by φ(xi) = ri and φ(a) = a for all a 2 k. Then φ extends uniquely to a surjective homomorphism. Conversely, given a surjective homomorphism, the image of x1; : : : ; xn generates R as a k-algebra, proving that R is finitely generated. Then R is isomorphic to the quotients of the Noetherian ring k[x1; : : : ; xn] modulo the kernel, which is again Noetherian. Affine Algebraic Sets 6 An is the set of n-tuples of the field k is called affine n-space. (Apply the forgetful functor to the vector space kn to get affine space.) n Polynomials f 2 k[x1; : : : ; xn] may be regarded as functions f : A ! k by evalua- tion: f :(a1; : : : ; an) 7! f(a1; : : : ; an) 2 k: This gives a ring of functions, denoted k[An], called the coordinate ring of An. Definition If S ⊆ k[An], then n Z(S) := f(a1; : : : ; an) 2 A : f(a1; : : : ; an) = 0 for all f 2 Sg is called an affine algebraic set. V = Z(S) is called the locus of S in An. Examples 1. If n = 1 and f(x) 2 k[x], then Z(f) is the set of roots of f. n 2. One point subsets f(a1; : : : ; an)g of A are Z(x1 − a1; : : : ; xn − an). 3. If f 2 k[x1; : : : ; xn] is a nonconstant polynomial, Z(f) is called a hyper- surface. In R2, Z(y−x2) is the parabola y = x2, Z(xy−1) is the hyperbola y = 1=x. In R3, Z(x2 + y2=4 + z2=9 − 1) is an ellipsoid. 7 Basic Properties of Z Let S; T ⊆ k[An]. (1) If S ⊆ T , then Z(T ) ⊆ Z(S). (2) Z(I) = Z(S) where I = (S) is the ideal generated by S. (3) Z(S) \Z(T ) = Z(S [ T ). (4) For ideals I;J ⊆ k[An], Z(I) [Z(J) = Z(IJ). (5) Z(0) = An and Z(1) = ;. By (2), we may consider the map n n Z : fideals of k[A ]g ! faffine algebraic sets in A g: Since k[x1; : : : ; xn] is Noetherian, an ideal I = (f1; : : : ; fq) is finitely generated, and from (3) Z(I) = Z(f1) \Z(f2) \···\Z(fq): So, each affine algebraic set is the intersection of finitely many hypersurfaces. Definition For any A ⊆ An, I(A) := ff 2 k[x1; : : : ; xn]: f(a1; : : : ; an) = 0 for all (a1; : : : ; an) 2 Ag: I(A) is an ideal and is the largest ideal that vanishes on A. 8 Basic Properties of I Let A; B ⊆ An (6) If A ⊆ B, then I(B) ⊆ I(A).
Load more

Recommended publications
  • Chapter 4 the Algebra–Geometry Dictionary
    Chapter 4 The Algebra–Geometry Dictionary In this chapter, we will explore the correspondence between ideals and varieties. In §§1 and 2, we will prove the Nullstellensatz, a celebrated theorem which iden- tifies exactly which ideals correspond to varieties. This will allow us to construct a “dictionary” between geometry and algebra, whereby any statement about varieties can be translated into a statement about ideals (and conversely). We will pursue this theme in §§3 and 4, where we will define a number of natural algebraic operations on ideals and study their geometric analogues. In keeping with the computational emphasis of the book, we will develop algorithms to carry out the algebraic op- erations. In §§5 and 6, we will study the more important algebraic and geometric concepts arising out of the Hilbert Basis Theorem: notably the possibility of decom- posing a variety into a union of simpler varieties and the corresponding algebraic notion of writing an ideal as an intersection of simpler ideals. In §7, we will prove the Closure Theorem from Chapter 3 using the tools developed in this chapter. §1 Hilbert’s Nullstellensatz In Chapter 1, we saw that a varietyV k n can be studied by passing to the ideal ⊆ I(V)= f k[x 1,...,x n] f(a)= 0 for alla V { ∈ | ∈ } of all polynomials vanishing onV. Hence, we have a map affine varieties ideals V −→ I(V). Conversely, given an idealI k[x 1,...,x n], we can define the set ⊆ V(I)= a k n f(a)= 0 for allf I . { ∈ | ∈ } © Springer International Publishing Switzerland 2015 175 D.A.
    [Show full text]
  • Notes on Irreducible Ideals
    BULL. AUSTRAL. MATH. SOC. I3AI5, I3H05, I 4H2O VOL. 31 (1985), 321-324. NOTES ON IRREDUCIBLE IDEALS DAVID J. SMITH Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an Af-primary ideal of a local ring (if, M) satisfying the condition that Q : M = Q + M where s is the index of Q , then all irreducible components of Q have index s . (Q is "index- unmixed" .) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples. Introduction Let i? be a commutative ring with identity. An ideal J of R is irreducible if it is not a proper intersection of any two ideals of if . A discussion of some elementary properties of irreducible ideals is found in Zariski and Samuel [4] and Grobner [!]• If i? is Noetherian then every ideal of if has an irredundant representation as a finite intersection of irreducible ideals, and every irreducible ideal is primary. It is properties of representations of primary ideals as intersections of irreducible ideals which will be discussed here. We assume henceforth that if is Noetherian. Received 25 October 1981*. Copyright Clearance Centre, Inc. Serial-fee code: OOOl*-9727/85 $A2.00 + 0.00. 32 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 24 Sep 2021 at 06:01:50, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
    [Show full text]
  • Mel Hochster's Lecture Notes
    MATH 614 LECTURE NOTES, FALL, 2017 by Mel Hochster Lecture of September 6 We assume familiarity with the notions of ring, ideal, module, and with the polynomial ring in one or finitely many variables over a commutative ring, as well as with homomor- phisms of rings and homomorphisms of R-modules over the ring R. As a matter of notation, N ⊆ Z ⊆ Q ⊆ R ⊆ C are the non-negative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively, throughout this course. Unless otherwise specified, all rings are commutative, associative, and have a multiplica- tive identity 1 (when precision is needed we write 1R for the identity in the ring R). It is possible that 1 = 0, in which case the ring is f0g, since for every r 2 R, r = r ·1 = r ·0 = 0. We shall assume that a homomorphism h of rings R ! S preserves the identity, i.e., that h(1R) = 1S. We shall also assume that all given modules M over a ring R are unital, i.e., that 1R · m = m for all m 2 M. When R and S are rings we write S = R[θ1; : : : ; θn] to mean that S is generated as a ring over its subring R by the elements θ1; : : : ; θn. This means that S contains R and the elements θ1; : : : ; θn, and that no strictly smaller subring of S contains R and the θ1; : : : ; θn. It also means that every element of S can be written (not necessarily k1 kn uniquely) as an R-linear combination of the monomials θ1 ··· θn .
    [Show full text]
  • Commutative Algebra
    Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .
    [Show full text]
  • 2-Irreducible and Strongly 2-Irreducible Ideals of Commutative Rings
    2-IRREDUCIBLE AND STRONGLY 2-IRREDUCIBLE IDEALS OF COMMUTATIVE RINGS ∗ H. MOSTAFANASAB AND A. YOUSEFIAN DARANI Abstract. An ideal I of a commutative ring R is said to be irreducible if it cannot be written as the intersection of two larger ideals. A proper ideal I of a ring R is said to be strongly irreducible if for each ideals J, K of R, J ∩ K ⊆ I implies that J ⊆ I or K ⊆ I. In this paper, we introduce the concepts of 2-irreducible and strongly 2-irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. We say that a proper ideal I of a ring R is 2-irreducible if for each ideals J, K and L of R, I = J ∩ K ∩ L implies that ei- ther I = J ∩ K or I = J ∩ L or I = K ∩ L. A proper ideal I of a ring R is called strongly 2-irreducible if for each ideals J, K and L of R, J ∩K ∩L ⊆ I implies that either J ∩K ⊆ I or J ∩L ⊆ I or K ∩L ⊆ I. Keywords: Irreducible ideals, 2-irreducible ideals, strongly 2-irreducible ideals. MSC(2010): Primary: 13A15, 13C05; Secondary: 13F05, 13G05. 1. Introduction Throughout this paper all rings are commutative with a nonzero identity. Recall that an ideal I of a commutative ring R is irreducible if I = J K for ideals J and K of R implies that either I = J or I = K. A proper∩ ideal I of a ring R is said to be strongly irreducible if for each ideals J, K of R, J K I implies that J I or K I (see [3], [8]).
    [Show full text]
  • Commutative Ideal Theory Without Finiteness Conditions: Completely Irreducible Ideals
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 7, Pages 3113–3131 S 0002-9947(06)03815-3 Article electronically published on March 1, 2006 COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS: COMPLETELY IRREDUCIBLE IDEALS LASZLO FUCHS, WILLIAM HEINZER, AND BRUCE OLBERDING Abstract. An ideal of a ring is completely irreducible if it is not the intersec- tion of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those com- mutative rings in which every ideal is an irredundant intersection of completely irreducible ideals. Introduction Let R denote throughout a commutative ring with 1. An ideal of R is called irreducible if it is not the intersection of two proper overideals; it is called completely irreducible if it is not the intersection of any set of proper overideals. Our goal in this paper is to examine the structure of completely irreducible ideals of a commutative ring on which there are imposed no finiteness conditions. Other recent papers that address the structure and ideal theory of rings without finiteness conditions include [3], [4], [8], [10], [14], [15], [16], [19], [25], [26]. AproperidealA of R is completely irreducible if and only if there is an element x ∈ R such that A is maximal with respect to not containing x.
    [Show full text]
  • Algebraic Geometry Over Groups I: Algebraic Sets and Ideal Theory
    Algebraic geometry over groups I: Algebraic sets and ideal theory Gilbert Baumslag Alexei Myasnikov Vladimir Remeslennikov Abstract The object of this paper, which is the ¯rst in a series of three, is to lay the foundations of the theory of ideals and algebraic sets over groups. TABLE OF CONTENTS 1. INTRODUCTION 1.1 Some general comments 1.2 The category of G-groups 1.3 Notions from commutative algebra 1.4 Separation and discrimination 1.5 Ideals 1.6 The a±ne geometry of G-groups 1.7 Ideals of algebraic sets 1.8 The Zariski topology of equationally Noetherian groups 1.9 Decomposition theorems 1.10 The Nullstellensatz 1.11 Connections with representation theory 1.12 Related work 2. NOTIONS FROM COMMUTATIVE ALGEBRA 2.1 Domains 2.2 Equationally Noetherian groups 2.3 Separation and discrimination 2.4 Universal groups 1 3. AFFINE ALGEBRAIC SETS 3.1 Elementary properties of algebraic sets and the Zariski topology 3.2 Ideals of algebraic sets 3.3 Morphisms of algebraic sets 3.4 Coordinate groups 3.5 Equivalence of the categories of a±ne algebraic sets and coordinate groups 3.6 The Zariski topology of equationally Noetherian groups 4. IDEALS 4.1 Maximal ideals 4.2 Radicals 4.3 Irreducible and prime ideals 4.4 Decomposition theorems for ideals 5. COORDINATE GROUPS 5.1 Abstract characterization of coordinate groups 5.2 Coordinate groups of irreducible algebraic sets 5.3 Decomposition theorems for coordinate groups 6. THE NULLSTELLENSATZ 7. BIBLIOGRAPHY 2 1 Introduction 1.1 Some general comments The beginning of classical algebraic geometry is concerned with the geometry of curves and their higher dimensional analogues.
    [Show full text]
  • Primary Decomposition
    + Primary Decomposition The decomposition of an ideal into primary ideals is a traditional pillar of ideal theory. It provides the algebraic foundation for decomposing an algebraic variety into its irreducible components-although it is only fair to point out that the algebraic picture is more complicated than naive geometry would suggest. From another point of view primary decomposition provides a gen- eralization of the factorization of an integer as a product of prime-powers. In the modern treatment, with its emphasis on localization, primary decomposition is no longer such a central tool in the theory. It is still, however, of interest in itself and in this chapter we establish the classical uniqueness theorems. The prototypes of commutative rings are z and the ring of polynomials kfxr,..., x,] where k is a field; both these areunique factorization domains. This is not true of arbitrary commutative rings, even if they are integral domains (the classical example is the ring Z[\/=1, in which the element 6 has two essentially distinct factorizations, 2.3 and it + r/-S;(l - /=)). However, there is a generalized form of "unique factorization" of ideals (not of elements) in a wide class of rings (the Noetherian rings). A prime ideal in a ring A is in some sense a generalization of a prime num- ber. The corresponding generalization of a power of a prime number is a primary ideal. An ideal q in a ring A is primary if q * A and. if xy€q => eitherxeqory" eqforsomen > 0. In other words, q is primary o Alq * 0 and every zero-divisor in l/q is nilpotent.
    [Show full text]
  • Primary Decomposition of Divisorial Ideals in Mori Domains
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 117, 327-342 (1988) Primary Decomposition of Divisorial Ideals in Mori Domains EVAN G. HOUSTON AND THOMAS G. LUCAS Department of Mathematics, Lfniversity of Xorth Carolina at Charlotte, Charlotte, North Carolina 28223 AND T. M. VISWANATHAN* Universidade Estadual de Campinas. Instituto de Matenuitica e Estatistica, Cai.va Postal 6165, 13081 Campinas, Sao Paula. Brazil Communicated by Melvin Hochster Received July 17. 1986 DEDICATED 'I-O THE MEMORY OF SRINIVASA RAMANUJAN ON THE OCCASION OF THE 100~~ ANNIVERSARYOF HIS BIRTH 1. INTR~DUC-H~N Mori domains have received a good deal of attention in the literature recently [l-5, 10, 131. A Mori domain is an integral domain which satisfies the ascending chain condition on divisorial ideals. The best-known exam- ples are (arbitrary) Noetherian domains and Krull domains, but other examples do exist [ 1, 2, 41. As one would expect from the definition, many results on Noetherian domains have analogues in the Mori case. The goal of this paper is to study primary decomposition of divisorial ideals in Mori domains. In the next section we prove that a divisorial ideal in a Mori domain has only finitely many associated primes. We also introduce and study the notion of a d-irreducible ideal in a Mori domain R: the ideal I of R is d-irreducible if it is divisorial and cannot be written as the intersection of two properly larger divisorial ideals of R. We show that every d-irreducible ideal is of the form (a) :b and has a unique maximal associated prime.
    [Show full text]
  • Introduction to Projective Varieties by Enrique Arrondo(*)
    Introduction to projective varieties by Enrique Arrondo(*) Version of November 26, 2017 This is still probably far from being a final version, especially since I had no time yet to complete the second part (which is so far not well connected with the first one). Anyway, any kind of comments are very welcome, in particular those concerning the general structure, or suggestions for shorter and/or correct proofs. 0. Algebraic background 1. Projective sets and their ideals; Weak Nullstellensatz 2. Irreducible components 3. Hilbert polynomial. Nullstellensatz 4. Graded modules; resolutions and primary decomposition 5. Dimension, degree and arithmetic genus 6. Product of varieties 7. Regular maps 8. Properties of morphisms 9. Resolutions and dimension 10. Ruled varieties 11. Tangent spaces and cones; smoothness 12. Transversality 13. Parameter spaces 14. Affine varieties vs projective varieties; sheaves 15. The local ring at a point 16. Introduction to affine and projective schemes 17. Vector bundles 18. Coherent sheaves 19. Schemes (*) Departamento de Algebra,´ Facultad de Ciencias Matem´aticas, Universidad Com- plutense de Madrid, 28040 Madrid, Spain, Enrique [email protected] 1 The scope of these notes is to present a soft and practical introduction to algebraic geometry, i.e. with very few algebraic requirements but arriving soon to deep results and concrete examples that can be obtained \by hand". The notes are based on some basic PhD courses (Milan 1998 and Florence 2000) and a summer course (Perugia 1998) that I taught. I decided to produce these notes while preparing new similar courses (Milan and Perugia 2001). My approach consists of avoiding all the algebraic preliminaries that a standard al- gebraic geometry course uses for affine varieties and thus start directly with projective varieties (which are the varieties that have good properties).
    [Show full text]
  • Math 210C Lecture 2 Notes
    Math 210C Lecture 2 Notes Daniel Raban April 3, 2019 1 Primary Decomposition of Ideals 1.1 Primary decomposition of ideals in noetherian rings Recall that anp ideal I of a commutative ring R is primary if whenever ab 2 I, either a 2 I or b 2 I.A primary decomposition of an ideal I is a collection fq1;:::; qng of Tn T p p primary ideals of R with i=1 qi = I. It is minimal if qi 6⊇ j6=i qj and qi 6= qj for all i 6= j. Definition 1.1. An ideal is decomposible if it has a primary decomposition. Lemma 1.1. If I is decomposable, then it has a minimal primary decomposition. Definition 1.2. A proper ideal I is irreducible if for any two ideals a and b with I = a\b, either I = a or I = b. Proposition 1.1. Let R be noetherian. Then every irreducible ideal is primary. Proof. Let I ⊆ R be irreudicble. Let a; b 2 I with b2 = I. For each n ≥ 1, let Jn = fr 2 n R : a r 2 Ig be an ideal of R. Notice that Jn ⊆ Jn+1 for all n. By the ascending chain condition, there exists N such that Jn = Jn+1 for all n ≥ N. Let a = (aN ) + I, b = (b) + I. We claim that a \ b = I. Let c 2 a \ b. Then c = anr + q with r 2 R and q 2 I since c 2 a, and ac 2 (ab) + U ⊆ (ab) + I, since c 2 b.
    [Show full text]
  • Algebraic Geometry Notes
    Algebraic Geometry Notes Sean Sather-Wagstaff Department of Mathematics, 412 Minard Hall, North Dakota State University, Fargo, North Dakota 58105-5075, USA E-mail address: Sean.Sather-Wagstaff@ndsu.edu URL: www.ndsu.edu/pubweb/~ssatherw 1 March 26, 2012 Contents Contents i Preface 1 1 Affine Space 3 1.1 Algebraic Subsets . 3 1.2 Zariski Topology . 6 1.3 Geometric Ideals . 8 1.4 Hilbert’s Nullstellensatz . 10 1.5 Irreducible Closed Subsets . 11 1.6 Finding Irreducible Components . 16 2 Projective Space 23 2.1 Motivation . 23 n 2.2 Projective Space Pk ................................... 24 2.3 Homogeneous Polynomials . 25 n 2.4 The Zariski Topology on Pk ............................... 28 n 2.5 Geometric Ideals in Pk .................................. 32 2.6 Projective Nullstellensatz . 33 2.7 Irreducible Closed Subsets . 35 2.8 Regular Functions . 37 2.9 Finding Irreducible Components . 38 3 Sheaves 41 3.1 Presheaves . 41 n 3.2 Regular Functions on Ak ................................. 42 Index 45 i Introduction What is Algebraic Geometry? The study of geometric objects determined by algebraic “data”, i.e. polynomials. Some examples are lines in R2, conics in R2, planes in R3, spheres, ellipsoids, etc. in R3. Geometric objects of interest include: solution sets to systems of polynomial equations, study them using algebraic techniques. For example: is the solution set finite or infinite? Example 0.0.1 Let f; g 2 C[x; y; z] and let V = f(x; y; z) 2 C3 j f(x; y; z) = 0 = g(x; y; z)g. Assume that V 6= ; (Hilbert’s Nullstellensatz says that this is equivalent to (f; g) 6= C[x; y; z]).
    [Show full text]