DOCSLIB.ORG

Commutative Algebra) Notes, 2Nd Half

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Commutative Algebra) Notes, 2Nd Half Math 221 (Commutative Algebra) Notes, 2nd Half Live [Locally] Free or Die. This is Math 221: Commutative Algebra as taught by Prof. Gaitsgory at Harvard during the Fall Term of 2008, and as understood by yours truly. There are probably typos and mistakes (which are all mine); use at your own risk. If you find the aforementioned mistakes or typos, and are feeling in a generous mood, then you may consider sharing this information with me.1 -S. Gong [email protected] 1 CONTENTS Contents 1 Divisors 2 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 . .2 1.2 Dedekind Domains . .6 1.3 Divisors and Fractional Ideals . .8 1.4 Dedekind Domains and Integral Closedness . 11 1.5 The Picard Group . 16 2 Dimension via Hilbert Functions 21 2.1 Filtrations and Gradations . 21 2.2 Dimension of Modules over a Polynomial Algebra . 24 2.3 For an arbitrary finitely generated algebra . 27 3 Dimension via Transcendency Degree 31 3.1 Definitions . 31 3.2 Agreement with Hilbert dimension . 32 4 Dimension via Chains of Primes 34 4.1 Definition via chains of primes and Noether Normalization . 34 4.2 Agreement with Hilbert dimension . 35 5 Kahler Differentials 37 6 Completions 42 6.1 Introduction to inverse limits and completions . 42 6.2 Artin Rees Pattern . 46 7 Local Rings and Other Notions of Dimension 50 7.1 Dimension theory for Local Noetherian Rings . 50 8 Smoothness and Regularity 53 8.1 Regular Local Rings . 53 8.2 Smoothness/Regularity for algebraically closed fields . 55 8.3 A taste of smoothness and regularity for non-algebraically closed fields . 60 9 Appendix 62 1 1 Divisors 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 Let A be a Noetherian local domain with maximal ideal m. Definition 1.1. We say that A is a discrete valuation ring, or DVR, if m = (π), that is, if the maximal ideal is principal. 2 Lemma 1.2. If A is a DVR, then dimA=m(m=m ) = 1. Proof. Note that the image of π generates m=m2 over A=m, so the dimension is at most 1. However, by Nakayama's Lemma, m=m2 is not the zero module, so the dimension is 1. ≥0 n Lemma 1.3. For A a DVR, for any a 2 A, there is a unique n 2 Z such that a = π a0 for a0 invertible. We call this n the order of a. Proof. Uniqueness is clear by cancellation, so we seek to establish existence. Let n be the k n maximal power of π that divides a. This is finite because \m = f0g. Then if a = π a0, then a0 62 m, so a0 is invertible. Corollary 1.4. Any DVR is a PID (principal ideal domain). n Proof. For any ideal I, choose an element a 2 I of minimal order. Then if a = π a0 for a0 invertible, then πn divides all elements of I, but πn 2 I, so I = (πn), as desired. The above corollary leads directly to the following one: Corollary 1.5. If A is a DVR with maximal ideal m, then the only prime ideals in A are m and (0). Now why are they called DVRs? Here is the apparent reason: Proposition 1.6. DVRs are in bijection with fields K, equipped with a map ν : K ! Z [ f1g satisfying the following properties: 1. ν(a + b) ≥ min(ν(a); ν(b)) for any a; b 2 K 2. ν(ab) = ν(a) + ν(b) for any a; b 2 K∗. 3. ν(x) = 1 if and only if x = 0 This map is called a valuation. n Proof. Given a discrete valuation ring A, take K = Frac(A) and take ν(π a0) = n where a0 is invertible. Conversely, for K take A to be its ring of integers, i.e, A = fa 2 Kjν(a) ≥ 0g. One may check that this is a DVR with m = fa 2 Kjν(a) > 0g, because it is clear that any element with valuation 0 is invertible in A and therefore any element of valuation 1 generates m. 2 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 Definition 1.7. Let M be a module over a domain A. We say that M is torsion-free, if for any nonzero a 2 A, a : M ! M is injective. We say that M is torsion if for any m 2 M, there is nonzero a 2 A such that am = 0. Lemma 1.8. For any module finitely generated module M over a Noetherian domain A, there is a short exact sequence - - - - 0 Mtors M Mtors−free 0 where Mtors is killed by an element of A and Mtors−free is torsion-free. Proof. This is because we may take Mtors to be all the elements which are killed by a non- zero element of A. Then this is clearly a sub-module. Since A is Noetherian, it is finitely generated, which means that it can be killed by one element of A (take the product of the elements that kill the generators). Then it is easy to check that the quotient M=Mtors is torsion-free. Lemma 1.9. For R a PID, a module M over R is flat if and only if it is torsion-free. Proof. This is the content of Problem 2 on the Midterm. Using this, we will classify modules over DVRs. Proposition 1.10. let M be a finitely generated module over a DVR R. Then ⊕n M = Mtors ⊕ R ; n i.e, where Mtors can be annihilated by π for some n. Proof. Set Mtors ⊂ M be as in Lemma 1.8 so that M=Mtors is torsion-free. Therefore, by Corollary 1.4 and Lemma 1.9 we see that it is flat. But it is over a local ring, so that means ⊕n ⊕n that it is free. So we have M=Mtors = R for some n. Furthermore, since R is free, it is additionally projective, so the above sequence splits, so ⊕n M = Mtors ⊕ R as desired. There is nothing more to say about the free part, so let us discuss the torsion part in more detail. Lemma 1.11. Any finitely generated torsion module over a DVR is M R/πnR: Before we prove this, let us give two examples: 3 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 1. Take R = k[[t]], which is a DVR with maximal ideal (t). Thus, by the lemma, for a finitely generated torsion module M, t : M ! M is a nilpotent operator. However, k[[t]]=tn is a Jordan block so we are exactly saying that linear transformations can be written in Jordan block form. 2. Let R = Zp. Here the lemma implies that finitely generated torsion modules over Zp can be written as a direct sum of p-groups. Miaaaoow?? Now let us proceed with the proof of the lemma. Proof of Lemma 1.11. Let n be the minimal integer such that πn kills M. This means that n M is a module over Rn = R/π R, and also there is an element m 2 M, and an injective n−1 map Rn ,! M, because we may choose m to be an element which is not annihilated by π , and then take the map to be 1 7! m. Proceeding by induction, it suffices to show that the above map Rn ,! M splits. But for this it suffices that Rn is an injective module over itself. This property of rings is called the Frobenius property, and it is very rare. We will write this as a lemma. Lemma 1.12. Rn is injective as a module over itself. Proof of Lemma 1.12. Note that a module M over a ring R is injective if and only if for any ideal I ⊂ R, Ext1(R=I; M) = 0. This was shown on Problem Set 8, Problem 2a. 1 Thus we wish to show that for any ideal I, ExtRn (Rn=I; Rn) = 0. Note that since R is a k DVR, we know that it is a PID, and also any element has the form r = π r0 for some k ≥ 0 k and some r0 invertible. Then all ideals in R are of the form (π ) for some k, so all ideals in Rn are also of this form. Therefore, Rn=I = Rm for some m ≤ n, so it suffices to show that 1 for m ≤ n, ExtRn (Rm;Rn) = 0. But note that we have short exact sequence m - π -· - - 0 Rn−m Rn Rm 0 which gives a corresponding long exact sequence of Exts - - ~- 0 HomRn (Rm;Rn) HomRn (Rn;Rn) HomRn (Rn−m;Rn) - 1 - 1 - ExtRn (Rm;Rn) ExtRn (Rn;Rn) ··· But note that any map of Rn modules, Rn−m ! Rn, must map 1 2 Rn−m to an element which is killed by πn−m, which means it must be a multiple of πm, so say is is πma. Then the map is r 7! πmar; which is the image of the map [r 7! ar] 2 HomRn (Rn;Rn): 1 Thus, ~ is surjective. Also note that Rn is projective over itself, so ExtRn (Rn;Rn) = 0. This, along with the surjectivity of ~ shows that 1 ExtRn (Rm;Rn) = 0 as desired. 4 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 As mentioned earlier, this lemma concludes our proof of Lemma 1.11 as well. Definition 1.13. We say that a Noetherian domain A is of Krull dimension 1, if any nonzero prime ideal is maximal.
Load more

Recommended publications
  • Dimension Theory and Systems of Parameters
    Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite.
    [Show full text]
  • An Introduction to Operad Theory
    AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification.
    [Show full text]
  • Math 210B. Finite-Dimensional Commutative Algebras Over a Field Let a Be a Nonzero finite-Dimensional Commutative Algebra Over a field K
    Math 210B. Finite-dimensional commutative algebras over a field Let A be a nonzero finite-dimensional commutative algebra over a field k. Here is a general structure theorem for such A: Theorem 0.1. The set Max(A) of maximal ideals of A is finite, all primes of A are maximal and minimal, and the natural map Y A ! Am m is an isomorphism, with each Am having nilpotent maximal ideal. Qn In particular, if A is reduced then A ' i=1 ki for fields ki, with the maximal ideals given by the kernels of the projections A ! ki. The assertion in the reduced case follows from the rest since if A is reduced then so is each Am (and hence its nilpotent unique maximal ideal vanishes, implying Am must be a field). Note also that the nilpotence of the maximal ideal mAm implies that for some large n we have n n n Am = Am=m Am = (A=m )m = A=m (final equality since m is maximal in A), so the isomorphism in the Theorem can also be expressed Q n as saying A ' m A=m for large n. Most of the proof of this result is worked out in HW1 Exercise 7, and here we just address one point: the nilpotence of the maximal ideal of the local ring Am at each maximal ideal m of A. That is, we claim that the maximal ideal M := mAm is nilpotent. To establish such nilpotence, note that M is finitely generated as an Am-module since Am is noetherian (as A is obviously noetherian!).
    [Show full text]
  • Irreducible Representations of Finite Monoids
    U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately.
    [Show full text]
  • 23. Dimension Dimension Is Intuitively Obvious but Surprisingly Hard to Define Rigorously and to Work With
    58 RICHARD BORCHERDS 23. Dimension Dimension is intuitively obvious but surprisingly hard to define rigorously and to work with. There are several different concepts of dimension • It was at first assumed that the dimension was the number or parameters something depended on. This fell apart when Cantor showed that there is a bijective map from R ! R2. The Peano curve is a continuous surjective map from R ! R2. • The Lebesgue covering dimension: a space has Lebesgue covering dimension at most n if every open cover has a refinement such that each point is in at most n + 1 sets. This does not work well for the spectrums of rings. Example: dimension 2 (DIAGRAM) no point in more than 3 sets. Not trivial to prove that n-dim space has dimension n. No good for commutative algebra as A1 has infinite Lebesgue covering dimension, as any finite number of non-empty open sets intersect. • The "classical" definition. Definition 23.1. (Brouwer, Menger, Urysohn) A topological space has dimension ≤ n (n ≥ −1) if all points have arbitrarily small neighborhoods with boundary of dimension < n. The empty set is the only space of dimension −1. This definition is mostly used for separable metric spaces. Rather amazingly it also works for the spectra of Noetherian rings, which are about as far as one can get from separable metric spaces. • Definition 23.2. The Krull dimension of a topological space is the supre- mum of the numbers n for which there is a chain Z0 ⊂ Z1 ⊂ ::: ⊂ Zn of n + 1 irreducible subsets. DIAGRAM pt ⊂ curve ⊂ A2 For Noetherian topological spaces the Krull dimension is the same as the Menger definition, but for non-Noetherian spaces it behaves badly.
    [Show full text]
  • UC Berkeley UC Berkeley Previously Published Works
    UC Berkeley UC Berkeley Previously Published Works Title Operator bases, S-matrices, and their partition functions Permalink https://escholarship.org/uc/item/31n0p4j4 Journal Journal of High Energy Physics, 2017(10) ISSN 1126-6708 Authors Henning, B Lu, X Melia, T et al. Publication Date 2017-10-01 DOI 10.1007/JHEP10(2017)199 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Published for SISSA by Springer Received: July 7, 2017 Accepted: October 6, 2017 Published: October 27, 2017 Operator bases, S-matrices, and their partition functions JHEP10(2017)199 Brian Henning,a Xiaochuan Lu,b Tom Meliac;d;e and Hitoshi Murayamac;d;e aDepartment of Physics, Yale University, New Haven, Connecticut 06511, U.S.A. bDepartment of Physics, University of California, Davis, California 95616, U.S.A. cDepartment of Physics, University of California, Berkeley, California 94720, U.S.A. dTheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. eKavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Relativistic quantum systems that admit scattering experiments are quan- titatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary de- scriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) mo- mentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal.
    [Show full text]
  • Elements of Minimal Prime Ideals in General Rings
    Elements of minimal prime ideals in general rings W.D. Burgess, A. Lashgari and A. Mojiri Dedicated to S.K. Jain on his seventieth birthday Abstract. Let R be any ring; a 2 R is called a weak zero-divisor if there are r; s 2 R with ras = 0 and rs 6= 0. It is shown that, in any ring R, the elements of a minimal prime ideal are weak zero-divisors. Examples show that a minimal prime ideal may have elements which are neither left nor right zero-divisors. However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors. The union of the minimal prime ideals is studied in 2-primal rings and the union of the minimal strongly prime ideals (in the sense of Rowen) in NI-rings. Mathematics Subject Classification (2000). Primary: 16D25; Secondary: 16N40, 16U99. Keywords. minimal prime ideal, zero-divisors, 2-primal ring, NI-ring. Introduction. E. Armendariz asked, during a conference lecture, if, in any ring, the elements of a minimal prime ideal were zero-divisors of some sort. In what follows this question will be answered in the positive with an appropriate interpretation of \zero-divisor". Two very basic statements about minimal prime ideals hold in a commutative ring R: (I) If P is a minimal prime ideal then the elements of P are zero-divisors, and (II) the union of the minimal prime ideals is M = fa 2 R j 9 r 2 R with ar 2 N∗(R) but r2 = N∗(R)g, where N∗(R) is the prime radical.
    [Show full text]
  • Simpleness of Leavitt Path Algebras with Coefficients in a Commutative
    Simpleness of Leavitt Path Algebras with Coefficients in a Commutative Semiring Y. Katsov1, T.G. Nam2, J. Zumbr¨agel3 [email protected]; [email protected]; [email protected] 1 Department of Mathematics Hanover College, Hanover, IN 47243–0890, USA 2 Institute of Mathematics, VAST 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam 3 Institute of Algebra Dresden University of Technology, Germany Abstract In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commu- tative semiring S, as well as establish some fundamental properties of those algebras. We provide a complete characterization of ideal- simple Leavitt path algebras with coefficients in a semifield S that extends the well-known characterizations when the ground semir- ing S is a field. Also, extending the well-known characterizations when S is a field or commutative ring, we present a complete charac- terization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S. Mathematics Subject Classifications: 16Y60, 16D99, 16G99, 06A12; 16S10, 16S34. Key words: Congruence-simple and ideal-simple semirings, Leav- itt path algebra. arXiv:1507.02913v1 [math.RA] 10 Jul 2015 1 Introduction In some way, “prehistorical” beginning of Leavitt path algebras started with Leavitt algebras ([18] and [19]), Bergman algebras ([7]), and graph C∗- algebras ([9]), considering rings with the Invariant Basis Number property, The second author is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED). The third author is supported by the Irish Research Council under Research Grant ELEVATEPD/2013/82.
    [Show full text]
  • Arxiv:1012.0864V3 [Math.AG] 14 Oct 2013 Nbt Emtyadpr Ler.Ltu Mhsz H Olwn Th Following the Emphasize Us H Let Results Cit.: His Algebra
    ORLOV SPECTRA: BOUNDS AND GAPS MATTHEW BALLARD, DAVID FAVERO, AND LUDMIL KATZARKOV Abstract. The Orlov spectrum is a new invariant of a triangulated category. It was intro- duced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singular- ities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. 1. Introduction The spectrum of a triangulated category was introduced by D. Orlov in [39], building on work of A. Bondal, R. Rouquier, and M. van den Bergh, [44] [11]. This categorical invariant, which we shall call the Orlov spectrum, is simply a list of non-negative numbers. Each number is the generation time of an object in the triangulated category. Roughly, the generation time of an object is the necessary number of exact triangles it takes to build the category using this object.
    [Show full text]
  • A Course in Commutative Algebra
    Graduate Texts in Mathematics 256 A Course in Commutative Algebra Bearbeitet von Gregor Kemper 1. Auflage 2010. Buch. xii, 248 S. Hardcover ISBN 978 3 642 03544 9 Format (B x L): 0 x 0 cm Gewicht: 1200 g Weitere Fachgebiete > Mathematik > Algebra Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Chapter 1 Hilbert’s Nullstellensatz Hilbert’s Nullstellensatz may be seen as the starting point of algebraic geom- etry. It provides a bijective correspondence between affine varieties, which are geometric objects, and radical ideals in a polynomial ring, which are algebraic objects. In this chapter, we give proofs of two versions of the Nullstellensatz. We exhibit some further correspondences between geometric and algebraic objects. Most notably, the coordinate ring is an affine algebra assigned to an affine variety, and points of the variety correspond to maximal ideals in the coordinate ring. Before we get started, let us fix some conventions and notation that will be used throughout the book. By a ring we will always mean a commutative ring with an identity element 1. In particular, there is a ring R = {0},the zero ring,inwhich1=0.AringR is called an integral domain if R has no zero divisors (other than 0 itself) and R ={0}.Asubring of a ring R must contain the identity element of R, and a homomorphism R → S of rings must send the identity element of R to the identity element of S.
    [Show full text]
  • Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes
    COMMUTATIVE ALGEBRA II, SPRING 2019, A. KUSTIN, CLASS NOTES 1. REGULAR SEQUENCES This section loosely follows sections 16 and 17 of [6]. Definition 1.1. Let R be a ring and M be a non-zero R-module. (a) The element r of R is regular on M if rm = 0 =) m = 0, for m 2 M. (b) The elements r1; : : : ; rs (of R) form a regular sequence on M, if (i) (r1; : : : ; rs)M 6= M, (ii) r1 is regular on M, r2 is regular on M=(r1)M, ::: , and rs is regular on M=(r1; : : : ; rs−1)M. Example 1.2. The elements x1; : : : ; xn in the polynomial ring R = k[x1; : : : ; xn] form a regular sequence on R. Example 1.3. In general, order matters. Let R = k[x; y; z]. The elements x; y(1 − x); z(1 − x) of R form a regular sequence on R. But the elements y(1 − x); z(1 − x); x do not form a regular sequence on R. Lemma 1.4. If M is a finitely generated module over a Noetherian local ring R, then every regular sequence on M is a regular sequence in any order. Proof. It suffices to show that if x1; x2 is a regular sequence on M, then x2; x1 is a regular sequence on M. Assume x1; x2 is a regular sequence on M. We first show that x2 is regular on M. If x2m = 0, then the hypothesis that x1; x2 is a regular sequence on M guarantees that m 2 x1M; thus m = x1m1 for some m1.
    [Show full text]
  • MTH 619 MINIMAL PRIMES and IRREDUCIBLE COMPONENTS Let R Be a Commutative Ring and P I, I ∈ I. the Claim Made in Class Was Prop
    MTH 619 Fall 2018 MINIMAL PRIMES AND IRREDUCIBLE COMPONENTS Let R be a commutative ring and pi, i 2 I. The claim made in class was Proposition 1. The sets V (pi) are exactly the irreducible components of Spec(R). We'll need some preparation. First, we have Lemma 2. Let I E R be a radical ideal. If V (I) is irreducible then I is prime. Proof. We prove the contrapositive: assuming I is not irreducible (i.e. it is reducible), we argue that I cannot be prime. Suppose I is not prime. Then, we can find a; b 62 I such that ab 2 I. This means that every prime that contains I also contains ab, and hence contains a or b. We thus have a decomposition V (I) = (V (I) \ V ((a))) [ (V (I) \ V ((b))) of V (I) as a union of closed subsets. It remains to argue that these are both proper subsets in order to conclude that V (I) is not irreducible. By symmetry, it suffices to show that V (I) \ V ((a)) 6= V (I). Note that V (I) \ V ((a)) = V (I + (a)); and the equality V (I) = V (I + (a)) would entail I ⊇ I + (a) because I is radical. But this means that a 2 I, contradicting our choice of a; b 62 I. This finishes the proof of the lemma. Proof of Proposition 1. We have already seen that sets of the form V (p) for prime ideals p are irreducible. It remains to show that if p is a minimal prime ideal then the irreducible set V (p) is maximal (among irreducible subsets of Spec(R)).
    [Show full text]