DOCSLIB.ORG

Chapter 4 the Algebra–Geometry Dictionary

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Cox et al., Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-16721-3_4 176 Chapter 4 The Algebra–Geometry Dictionary The Hilbert Basis Theorem assures us thatV(I) is actually an affine variety, for it tells us that there exists afinite set of polynomialsf 1,...,f s I such thatI= ∈ f 1,...,f s , and we proved in Proposition 9 of Chapter 2,§5 thatV(I) is the set of common⟨ roots⟩ of these polynomials. Thus, we have a map ideals affine varieties I −→ V(I). These two maps give us a correspondence between ideals and varieties. In this chap- ter, we will explore the nature of this correspondence. Thefirst thing to note is that this correspondence (more precisely, the mapV) is not one-to-one: different ideals can give the same variety. For example, x and x 2 are different ideals ink[x] which have the same varietyV(x) =V(x 2) =⟨ ⟩0 . More⟨ ⟩ serious problems can arise if thefieldk is not algebraically closed. For{ example,} consider the three polynomials 1,1+x 2, and 1+x 2 +x 4 inR[x]. These generate different ideals 2 2 4 I1 = 1 =R[x],I 2 = 1+x ,I 3 = 1+x +x , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ but each polynomial has no real roots, so that the corresponding varieties are all empty: V(I1) =V(I 2) =V(I 3) = . ∅ Examples of polynomials in two variables without real roots include 1+x 2 +y 2 and 1+x 2 +y 4. These give different ideals inR[x,y] which correspond to the empty variety. Does this problem of having different ideals represent the empty variety go away if thefieldk is algebraically closed? It does in the one-variable case when the ring is k[x]. To see this, recall from §5 of Chapter 1 that any idealI ink[x] can be generated by a single polynomial becausek[x] is a principal ideal domain. So we can write I= f for some polynomialf k[x]. ThenV(I) is the set of roots off ; i.e., the⟨ set⟩ ofa k such thatf(a)∈ = 0. But sincek is algebraically closed, every nonconstant∈ polynomial ink[x] has a root. Hence, the only way that we could have V(I) = would be to havef be a nonzero constant. In this case, 1/f k. Thus, 1=(1/∅f) f I, which means thatg=g 1 I for allg k[x]. This shows ∈ thatI=k[x·] ∈is the only ideal ofk[x] that· ∈ represents∈ the empty variety whenk is algebraically closed. A wonderful thing now happens: the same property holds when there is more than one variable. In any polynomial ring, algebraic closure is enough to guarantee that the only ideal which represents the empty variety is the entire polynomial ring itself. This is the Weak Nullstellensatz, which is the basis of (and is equivalent to) one of the most celebrated mathematical results of the late nineteenth century, Hilbert’s Nullstellensatz. Such is its impact that, even today, one customarily uses the original German name Nullstellensatz: a word formed, in typical German fashion, from three simpler words: Null (= Zero), Stellen (= Places), Satz (= Theorem). §1 Hilbert’s Nullstellensatz 177 Theorem 1 (The Weak Nullstellensatz). Let k be an algebraically closedfield and let I k[x 1,...,x n] be an ideal satisfyingV(I)= . Then I=k[x 1,...,x n]. ⊆ ∅ Proof. Our proof is inspired by GLEBSKY (2012). We will prove the theorem in contrapositive form: I!k[x 1,...,x n] = V(I)= . ⇒ ̸ ∅ We will make frequent use of the standard equivalenceI=k[x 1,...,x n] 1 I. This is part (a) of Exercise 16 from Chapter 1,§4. ⇔ ∈ Givena k andf k[x 1,...,x n], let ¯f=f(x 1,...,x n 1,a) k[x 1,...,x n 1]. Similar to∈ Exercise∈2 of Chapter 3,§5 and Exercise 15 of Chapter− ∈ 3,§6, the set − Ix a = ¯f f I n= { | ∈ } is an ideal ofk[x 1,...,x n 1]. The key step in the proof is the following claim. − Claim. Ifk is algebraically closed andI!k[x 1,...,x n] is a proper ideal, then there isa k such thatI xn=a !k[x 1,...,x n 1]. ∈ − Once we prove the claim, an easy induction gives elementsa 1,...,a n k such ∈ thatI xn=an,...,x1=a1 !k. But the only ideals ofk are 0 andk (Exercise 3), so that { } Ixn=an,...,x1=a1 = 0 . This implies(a 1,...,a n) V(I). We conclude thatV(I)= , and the theorem{ will} follow. ∈ ̸ ∅ To prove the claim, there are two cases, depending on the size ofI k[x n]. ∩ Case 1.I k[x n]= 0 . Letf I k[x n] be nonzero,and note thatf is nonconstant, ∩ ̸ { } ∈ ∩ since otherwise 1 I k[x n] I, contradicting? I=k[x 1,...,x n]. ∈ ∩ ⊆ r ̸ mi Sincek is algebraically closed,f=c (xn b i) wherec,b 1,...,b r k and i=1 − ∈ c= 0. Suppose thatI xn=bi =k[x 1,...,x n 1] for alli. Then for alli there isB i I ̸ − ∈ withB i(x1,...,x n 1,b i) = 1. This implies that − 1=B i(x1,...,x n 1,b i) =B i(x1,...,x n 1,x n (x n b i)) =B i +A i(xn b i) − − − − − for someA i k[x 1,...,x n]. Since this holds fori=1,...,r, we obtain ∈ @r @r mi mi 1= (Ai(xn b i) +B i) =A (xn b i) +B, − − i=1 i=1 ? ? r mi r mi 1 whereA= A andB I. This and (xn b i) =c − f I imply that i=1 i ∈ i=1 − ∈ 1 I, which contradictsI=k[x 1,...,x n]. ThusI xn=bi =k[x 1,...,x n 1] for somei. ∈ ̸ ̸ − Thisb i is the desireda. Case 2.I k[x n] = 0 . Let g 1,...,g t be a Gröbner basis ofI for lex order with ∩ { } { } x1 > >x n and write ··· αi αi (1)g i =c i(xn)x + terms<x , αi wherec i(xn) k[x n] is nonzero andx is a monomial inx 1,...,x n 1. ∈ − 178 Chapter 4 The Algebra–Geometry Dictionary Now picka k such thatc i(a)= 0 for alli. This is possible since algebraically closedfields are∈ infinite by Exercise̸ 4. It is easy to see that the polynomials ¯gi =g i(x1,...,x n 1,a) − form a basis ofI xn=a (Exercise 5). Substitutingx n =a into equation ( 1), one easily αi αi sees that LT(¯gi) =c i(a)x sincec i(a)= 0. Also note thatx = 1, since otherwise ̸ ̸ gi =c i I k[x n] = 0 , yetc i = 0. This shows that LT(¯gi) is nonconstant for alli. ∈ ∩ { } ̸ We claim that the¯gi form a Gröbner basis ofI xn=a. Assuming the claim, it follows that 1/ I xn=a since no LT(¯gi) can divide 1. ThusI xn=a =k[x 1,...,x n 1], which is what we∈ want to show. ̸ − To prove the claim, takeg i,g j G and consider the polynomial ∈ xγ xγ S=c j(xn) gi c i(xn) gj, xαi − xαj wherex γ = lcm(xαi ,x αj ). By construction,x γ > LT("S) (be sure you understand t this). SinceS I, it has a standard representationS= Al gl. Then evaluating ∈ l=1 atx n =a gives γ γ " x x t ¯ cj(a) ¯gi c i(a) ¯gj = S= l Al ¯gl. xαi − xαj =1 αi Since LT(¯gi) =c i(a)x , we see that S is theS-polynomialS(¯g i,¯gj) up to the nonzero constantc i(a)cj(a). Then γ x > LT(S) LT(Algl),A lgl =0 ≥ ̸ implies that γ x > LT(A¯l¯gl), A¯l¯gl =0 ̸ γ (Exercise 6). Sincex = lcm(LM(¯gi), LM(¯gj)), it follows thatS(¯gi,¯gj) has an lcm representation for alli,j and hence is a Gröbner basis by Theorem 6 of Chapter 2, §9.
Load more

Recommended publications
  • 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]
  • A Primer of Commutative Algebra
    A Primer of Commutative Algebra James S. Milne April 5, 2009, v2.00 Abstract These notes prove the basic theorems in commutative algebra required for alge- braic geometry and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. Available at www.jmilne.org/math/. Contents 1 Rings and algebras . 2 2 Ideals . 3 3 Noetherian rings . 7 4 Unique factorization . 12 5 Integrality . 14 6 Rings of fractions . 19 7 Direct limits. 23 8 Tensor Products . 24 9 Flatness . 28 10 The Hilbert Nullstellensatz . 32 11 The max spectrum of a ring . 34 12 Dimension theory for finitely generated k-algebras . 42 13 Primary decompositions . 45 14 Artinian rings . 49 15 Dimension theory for noetherian rings . 50 16 Regular local rings . 54 17 Connections with geometry . 56 NOTATIONS AND CONVENTIONS Our convention is that rings have identity elements,1 and homomorphisms of rings respect the identity elements. A unit of a ring is an element admitting an inverse. The units of a c 2009 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. 1An element e of a ring A is an identity element if ea a ae for all elements a of the ring. It is usually D D denoted 1A or just 1. Some authors call this a unit element, but then an element can be a unit without being a unit element. Worse, a unit need not be the unit. 1 1 RINGS AND ALGEBRAS 2 2 ring A form a group, which we denote A.
    [Show full text]