DOCSLIB.ORG

Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples May 31, 2020

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples May 31, 2020 Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples May 31, 2020 William Heinzer Christel Rotthaus Sylvia Wiegand Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 E-mail address: heinzer@math.purdue.edu Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027 E-mail address: rotthaus@math.msu.edu Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130 E-mail address: swiegand@math.unl.edu 1991 Mathematics Subject Classification. Primary 13A15, 13B35, 13F25; Secondary 13B02, 13B22, 13B40, 13C15, 13H05 Key words and phrases. power series, Noetherian integral domain, completion, generic fiber, flatness, prime spectra. Abstract. A major theme of this monograph is the creation of examples that are appropriate intersections of a field with a homomorphic image of a power series ring over a Noetherian domain. Classical examples of Noetherian integral domains with interesting properties are constructed by Akizuki, Schmidt, and Nagata. This work is continued by Brodmann-Rotthaus, Ferrand-Raynaud, Heitmann, Lequain, McAdam, Nishimura, Ogoma, Ratliff, Rotthaus, Weston and others. In certain circumstances, the intersection examples may be realized as a directed union, and the Noetherian property for the associated directed union is equivalent to a flatness condition. This flatness criterion simplifies the anal- ysis of several classical examples and yields other examples such as • A catenary Noetherian local integral domain of any specified dimension of at least two that has geometrically regular formal fibers and is not universally catenary. • A three-dimensional non-Noetherian unique factorization domain B such that the unique maximal ideal of B has two generators; B has precisely n prime ideals of height two, where n is an arbitrary positive integer; and each prime ideal of B of height two is not finitely generated but all the other prime ideals of B are finitely generated. • A two-dimensional Noetherian local domain that is a birational extension of a polynomial ring in three variables over a field yet fails to have Cohen- Macaulay formal fibers. This example also demonstrates that Serre's condition S1 need not lift to the completion; the example is related to an example of Ogoma. Another theme is an analysis of extensions of integral domains R,! S having trivial generic fiber, that is, every nonzero prime ideal of S has a nonzero intersection with R. Motivated by a question of Hochster and Yao, we present results about • The height of prime ideals maximal in the generic fiber of certain exten- sions involving mixed polynomial-power series rings. • The prime ideal spectrum of a power series ring in one variable over a one-dimensional Noetherian domain. • The dimension of S if R,! S is a local map of complete local domains having trivial generic fiber. A third theme relates to the questions: • What properties of a Noetherian domain extend to a completion? • What properties of an ideal pass to its extension in a completion? • What properties extend for a more general multi-adic completion? We give an example of a three-dimensional regular local domain R having a prime ideal P of height two with the property that the extension of P to the completion of R is not integrally closed. All of these themes are relevant to the study of prime spectra of Noether- ian rings and of the induced spectral maps associated with various extensions of Noetherian rings. We describe the prime spectra of various extensions in- volving power series. Contents Preface ix Chapter 1. Introduction 1 Chapter 2. Tools 9 2.1. Conventions and terminology 9 2.2. Basic theorems 16 2.3. Flatness 18 Exercises 21 Chapter 3. More tools 25 3.1. Introduction to ideal-adic completions 25 3.2. Uncountable transcendence degree for a completion 30 3.3. Power series pitfalls 31 3.4. Basic results about completions 32 3.5. Chains of prime ideals, fibers of maps 34 3.6. Henselian rings 36 3.7. Regularity and excellence 38 Exercises 41 Chapter 4. First examples of the construction 45 4.1. Universality 45 4.2. Elementary examples 47 4.3. Historical examples 48 4.4. Prototypes 51 Exercises 55 Chapter 5. The Inclusion Construction 59 5.1. The Inclusion Construction and a picture 59 5.2. Approximations for the Inclusion Construction 60 5.3. Basic properties of the constructed domains 63 Exercises 71 Chapter 6. Flatness and the Noetherian property 73 6.1. The Noetherian Flatness Theorem 73 6.2. Introduction to Insider Construction 10.7 81 6.3. Nagata's example 82 6.4. Christel's Example 83 6.5. Further implications of the Noetherian Flatness Theorem 84 Exercise 85 v vi CONTENTS Chapter 7. The flat locus of an extension of polynomial rings 87 7.1. Flatness criteria 87 7.2. The Jacobian ideal and the smooth and flat loci 91 7.3. Applications to polynomial extensions 98 Exercises 101 Chapter 8. Excellent rings and formal fibers 103 8.1. Background for excellent rings 103 8.2. Nagata rings and excellence 109 8.3. Henselian rings 111 Exercises 113 Chapter 9. Height-one prime ideals and weak flatness 115 9.1. The limit-intersecting condition 115 9.2. Height-one primes in extensions of Krull domains 119 9.3. Flatness for Inclusion Construction 5.3 122 Exercises 124 Chapter 10. Insider Construction details 127 10.1. Localized polynomial rings over special DVRs 128 10.2. Describing the construction 130 10.3. The non-flat locus of Insider Construction 10.7 131 10.4. Preserving excellence with the Insider Construction 136 Chapter 11. Integral closure under extension to the completion (intclsec) 139 11.1. Integral closure under ring extension 139 11.2. Extending ideals to the completion 142 11.3. Comments and Questions 145 Exercises 147 Chapter 12. Iterative examples 149 12.1. Iterative examples and their properties 149 12.2. Residual algebraic independence 154 Exercises 162 Chapter 13. Approximating discrete valuation rings by regular local rings 165 13.1. Local quadratic transforms and local uniformization 165 13.2. Expressing a DVR as a directed union of regular local rings 167 13.3. Approximating other rank-one valuation rings 171 Exercises 171 Chapter 14. Non-Noetherian examples of dimension 3 173 14.1. A family of examples in dimension 3 173 14.2. Two cases of Examples 14.1 and their spectra 177 Exercises 183 Chapter 15. Noetherian properties of non-Noetherian rings 185 15.1. A general question and the setting for this chapter 185 15.2. More properties of Insider Constructions 187 15.3. A four-dimensional non-Noetherian domain 189 Exercises 196 CONTENTS vii Chapter 16. Non-Noetherian examples in higher dimension 197 16.1. Higher dimensional non-Noetherian examples 197 16.2. A 4-dimensional prime spectrum 200 16.3. D + M constructions, coherence, finite conductor domains 204 Exercises 207 Chapter 17. The Homomorphic Image Construction 209 17.1. Two construction methods and a picture 209 17.2. Approximations for Homomorphic Image Construction 211 17.3. Basic properties of the Approximation Domains 213 17.4. Noetherian flatness for homomorphic images 215 17.5. Weak Flatness for Homomorphic Image Construction 17.2 218 17.6. Inclusion Constructions are Homomorphic Image Constructions 219 17.7. Connecting the two constructions 224 Exercises 229 Chapter 18. Catenary local rings with geometrically normal formal fibers 231 18.1. History, terminology and summary 231 18.2. Geometrically normal formal fibers and the catenary property 233 18.3. Flatness for the intersection of finitely many ideals 237 18.4. Regular maps and geometrically regular formal fibers 238 18.5. Examples that are not universally catenary 240 18.6. The depth of the constructed rings 244 Exercises 244 Chapter 19. An Ogoma-like example 245 19.1. Cohen-Macaulay formal fibers 245 19.2. Constructing the Ogoma-like example 252 Chapter 20. Multi-ideal-adic completions of Noetherian rings 257 20.1. Ideal filtrations and completions 257 20.2. Basic mechanics for the multi-adic completion 258 20.3. Preserving Noetherian under multi-adic completion 262 20.4. Preserving excellence or Henselian under multi-adic completion 265 Exercises 267 Chapter 21. Noetherian flatness and multi{adic constructions 269 21.1. Flatness lemmas 269 21.2. Multi-adic inclusion constructions 270 Chapter 22. Idealwise algebraic independence, 275 22.1. Idealwise independence, weakly flat and PDE extensions 275 22.2. Primarily independent elements 280 22.3. Residually algebraically independent elements 285 Exercises 293 Chapter 23. Idealwise algebraic independence II 295 23.1. Primary independence and flatness 295 23.2. Composition, base change and polynomial extensions 297 23.3. Passing to the Henselization 302 23.4. Summary diagram for the independence concepts 306 viii CONTENTS Chapter 24. Krull domains with Noetherian x-adic completions 309 24.1. Applying Inclusion Construction 5.3 310 24.2. Limit-intersecting elements 311 24.3. Example where the intersection domain equals the approximation domain and is non-Noetherian 317 24.4. A birational connection and additional examples 324 Exercises 327 Chapter 25. Inclusion Constructions over excellent normal local domains 329 25.1. Primarily limit-intersecting extensions and flatness 329 25.2. Existence of primarily limit-intersecting extensions 332 Exercise 339 Chapter 26. Weierstrass techniques for generic fiber rings 341 26.1. Terminology, background and results 341 26.2. Variations on a theme by Weierstrass 345 Exercises 355 Chapter 27. Generic fiber rings of mixed polynomial-power series rings 357 27.1. Weierstrass implications for the ring B = k[[X]] [Y ](X;Y ) 357 27.2. Weierstrass implications for the ring C = k[Y ](Y )[[X]] 360 27.3. Subrings of the power series ring k[[z; t]] 363 27.4. Weierstrass implications for the localized polynomial ring A 366 27.5.
Load more
Loading...
Loading...
Loading...
Loading...
Loading...

429 pages remaining, click to 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]
  • Varieties of Quasigroups Determined by Short Strictly Balanced Identities
    Czechoslovak Mathematical Journal Jaroslav Ježek; Tomáš Kepka Varieties of quasigroups determined by short strictly balanced identities Czechoslovak Mathematical Journal, Vol. 29 (1979), No. 1, 84–96 Persistent URL: http://dml.cz/dmlcz/101580 Terms of use: © Institute of Mathematics AS CR, 1979 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 29 (104) 1979, Praha VARIETIES OF QUASIGROUPS DETERMINED BY SHORT STRICTLY BALANCED IDENTITIES JAROSLAV JEZEK and TOMAS KEPKA, Praha (Received March 11, 1977) In this paper we find all varieties of quasigroups determined by a set of strictly balanced identities of length ^ 6 and study their properties. There are eleven such varieties: the variety of all quasigroups, the variety of commutative quasigroups, the variety of groups, the variety of abelian groups and, moreover, seven varieties which have not been studied in much detail until now. In Section 1 we describe these varieties. A survey of some significant properties of arbitrary varieties is given in Section 2; in Sections 3, 4 and 5 we assign these properties to the eleven varieties mentioned above and in Section 6 we give a table summarizing the results. 1. STRICTLY BALANCED QUASIGROUP IDENTITIES OF LENGTH £ 6 Quasigroups are considered as universal algebras with three binary operations •, /, \ (the class of all quasigroups is thus a variety).
    [Show full text]
  • Commutative Algebra, Lecture Notes
    COMMUTATIVE ALGEBRA, LECTURE NOTES P. SOSNA Contents 1. Very brief introduction 2 2. Rings and Ideals 2 3. Modules 10 3.1. Tensor product of modules 15 3.2. Flatness 18 3.3. Algebras 21 4. Localisation 22 4.1. Local properties 25 5. Chain conditions, Noetherian and Artin rings 29 5.1. Noetherian rings and modules 31 5.2. Artin rings 36 6. Primary decomposition 38 6.1. Primary decompositions in Noetherian rings 42 6.2. Application to Artin rings 43 6.3. Some geometry 44 6.4. Associated primes 45 7. Ring extensions 49 7.1. More geometry 56 8. Dimension theory 57 8.1. Regular rings 66 9. Homological methods 68 9.1. Recollections 68 9.2. Global dimension 71 9.3. Regular sequences, global dimension and regular rings 76 10. Differentials 83 10.1. Construction and some properties 83 10.2. Connection to regularity 88 11. Appendix: Exercises 91 References 100 1 2 P. SOSNA 1. Very brief introduction These are notes for a lecture (14 weeks, 2×90 minutes per week) held at the University of Hamburg in the winter semester 2014/2015. The goal is to introduce and study some basic concepts from commutative algebra which are indispensable in, for instance, algebraic geometry. There are many references for the subject, some of them are in the bibliography. In Sections 2-8 I mostly closely follow [2], sometimes rearranging the order in which the results are presented, sometimes omitting results and sometimes giving statements which are missing in [2]. In Section 9 I mostly rely on [9], while most of the material in Section 10 closely follows [4].
    [Show full text]
  • The Structure Theory of Complete Local Rings
    The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings.
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • MAXIMAL and NON-MAXIMAL ORDERS 1. Introduction Let K Be A
    MAXIMAL AND NON-MAXIMAL ORDERS LUKE WOLCOTT Abstract. In this paper we compare and contrast various prop- erties of maximal and non-maximal orders in the ring of integers of a number field. 1. Introduction Let K be a number field, and let OK be its ring of integers. An order in K is a subring R ⊆ OK such that OK /R is finite as a quotient of abelian groups. From this definition, it’s clear that there is a unique maximal order, namely OK . There are many properties that all orders in K share, but there are also differences. The main difference between a non-maximal order and the maximal order is that OK is integrally closed, but every non-maximal order is not. The result is that many of the nice properties of Dedekind domains do not hold in arbitrary orders. First we describe properties common to both maximal and non-maximal orders. In doing so, some useful results and constructions given in class for OK are generalized to arbitrary orders. Then we de- scribe a few important differences between maximal and non-maximal orders, and give proofs or counterexamples. Several proofs covered in lecture or the text will not be reproduced here. Throughout, all rings are commutative with unity. 2. Common Properties Proposition 2.1. The ring of integers OK of a number field K is a Noetherian ring, a finitely generated Z-algebra, and a free abelian group of rank n = [K : Q]. Proof: In class we showed that OK is a free abelian group of rank n = [K : Q], and is a ring.
    [Show full text]
  • Math 412. Quotient Rings of Polynomial Rings Over a Field
    (c)Karen E. Smith 2018 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 412. Quotient Rings of Polynomial Rings over a Field. For this worksheet, F always denotes a fixed field, such as Q; R; C; or Zp for p prime. DEFINITION. Fix a polynomial f(x) 2 F[x]. Define two polynomials g; h 2 F[x] to be congruent modulo f if fj(g − h). We write [g]f for the set fg + kf j k 2 F[x]g of all polynomials congruent to g modulo f. We write F[x]=(f) for the set of all congruence classes of F[x] modulo f. The set F[x]=(f) has a natural ring structure called the Quotient Ring of F[x] by the ideal (f). CAUTION: The elements of F[x]=(f) are sets so the addition and multiplication, while induced by those in F[x], is a bit subtle. A. WARM UP. Fix a polynomial f(x) 2 F[x] of degree d > 0. (1) Discuss/review what it means that congruence modulo f is an equivalence relation on F[x]. (2) Discuss/review with your group why every congruence class [g]f contains a unique polynomial of degree less than d. What is the analogous fact for congruence classes in Z modulo n? 2 (3) Show that there are exactly four distinct congruence classes for Z2[x] modulo x + x. List them out, in set-builder notation. For each of the four, write it in the form [g]x2+x for two different g.
    [Show full text]
  • Multiplication Domains, Nagata Rings, and Kronecker Function Rings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 319 (2008) 309–319 www.elsevier.com/locate/jalgebra Prüfer ∗-multiplication domains, Nagata rings, and Kronecker function rings Gyu Whan Chang Department of Mathematics, University of Incheon, Incheon 402-749, Republic of Korea Received 31 January 2007 Available online 30 October 2007 Communicated by Steven Dale Cutkosky Abstract Let D be an integrally closed domain, ∗ a star-operation on D, X an indeterminate over D,andN∗ = ∗ {f ∈ D[X]|(Af ) = D}.Forane.a.b. star-operation ∗1 on D,letKr(D, ∗1) be the Kronecker function ring of D with respect to ∗1. In this paper, we use ∗ todefineanewe.a.b. star-operation ∗c on D.Then ∗ [ ] = ∗ we prove that D is a Prüfer -multiplication domain if and only if D X N∗ Kr(D, c), if and only if Kr(D, ∗c) is a quotient ring of D[X], if and only if Kr(D, ∗c) is a flat D[X]-module, if and only if each ∗-linked overring of D is a Prüfer v-multiplication domain. This is a generalization of the following well- known fact that if D is a v-domain, then D is a Prüfer v-multiplication domain if and only if Kr(D, v) = [ ] [ ] [ ] D X Nv , if and only if Kr(D, v) is a quotient ring of D X , if and only if Kr(D, v) is a flat D X -module. © 2007 Elsevier Inc. All rights reserved. Keywords: (e.a.b.) ∗-operation; Prüfer ∗-multiplication domain; Nagata ring; Kronecker function ring 1.
    [Show full text]
  • Quotient-Rings.Pdf
    4-21-2018 Quotient Rings Let R be a ring, and let I be a (two-sided) ideal. Considering just the operation of addition, R is a group and I is a subgroup. In fact, since R is an abelian group under addition, I is a normal subgroup, and R the quotient group is defined. Addition of cosets is defined by adding coset representatives: I (a + I)+(b + I)=(a + b)+ I. The zero coset is 0+ I = I, and the additive inverse of a coset is given by −(a + I)=(−a)+ I. R However, R also comes with a multiplication, and it’s natural to ask whether you can turn into a I ring by multiplying coset representatives: (a + I) · (b + I)= ab + I. I need to check that that this operation is well-defined, and that the ring axioms are satisfied. In fact, everything works, and you’ll see in the proof that it depends on the fact that I is an ideal. Specifically, it depends on the fact that I is closed under multiplication by elements of R. R By the way, I’ll sometimes write “ ” and sometimes “R/I”; they mean the same thing. I Theorem. If I is a two-sided ideal in a ring R, then R/I has the structure of a ring under coset addition and multiplication. Proof. Suppose that I is a two-sided ideal in R. Let r, s ∈ I. Coset addition is well-defined, because R is an abelian group and I a normal subgroup under addition. I proved that coset addition was well-defined when I constructed quotient groups.
    [Show full text]
  • RINGS of MINIMAL MULTIPLICITY 1. Introduction We Shall Discuss Two Theorems of S. S. Abhyankar About Rings of Minimal Multiplici
    RINGS OF MINIMAL MULTIPLICITY J. K. VERMA Abstract. In this exposition, we discuss two theorems of S. S. Abhyankar about Cohen-Macaulay local rings of minimal multiplicity and graded rings of minimal multiplicity. 1. Introduction We shall discuss two theorems of S. S. Abhyankar about rings of minimal multiplicity which he proved in his 1967 paper \Local rings of high embedding dimension" [1] which appeared in the American Journal of Mathematics. The first result gives a lower bound on the multiplicity of the maximal ideal in a Cohen-Macaulay local ring and the second result gives a lower bound on the multiplicity of a standard graded domain over an algebraically closed field. The origins of these results lie in projective geometry. Let k be an algebraically closed field and X be a projective variety. We say that it is non-degenerate if it is not contained in a hyperplane. Let I(X) be the ideal of X. It is a homogeneous ideal of the polynomial ring S = k[x0; x1; : : : ; xr]: The homogeneous coordinate ring of X is defined as R = S(X) = S=I(X): Then R is a graded ring and 1 we write it as R = ⊕n=0Rn: Here Rn = Sn=(I(X) \ Sn): The Hilbert function of R is the function HR(n) = dimk Rn: Theorem 1.1 (Hilbert-Serre). There exists a polynomial PR(x) 2 Q[x] so that for all large n; HR(n) = PR(n): The degree of PR(x) is the dimension d of X and we can write x + d x + d − 1 P (x) = e(R) − e (R) + ··· + (−1)de (R): R d 1 d − 1 d Definition 1.2.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • CATENARITY in MODULE-FINITE ALGEBRAS 1. Introduction Let R Be
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3495{3502 S 0002-9939(99)04962-X Article electronically published on May 13, 1999 CATENARITY IN MODULE-FINITE ALGEBRAS SHIRO GOTO AND KENJI NISHIDA (Communicated by Ken Goodearl) Abstract. The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dim Λ=Q +htΛQfor any Q Spec Λ, and the equality ∈ n =htΛQ htΛP holds true for any pair P Q of prime ideals in Λ and for − ⊆ any saturated chain P = P0 P1 Pn =Qof prime ideals between P ⊂ ⊂···⊂ and Q. 1. Introduction Let R be a commutative Noetherian ring and let Λ be an R-algebra which is finitely generated as an R-module. Here we don’t assume Λ to be a commutative ring. The purpose of this article is to prove the following Theorem 1.1. Any module-finite R-algebra Λ is catenary if R is universally cate- nary. Before going ahead, let us recall the definition of catenarity and universal cate- narity ([Ma, p. 84]). We say that our ring Λ is catenary if for any pair P Q of ⊆ prime ideals in Λ and for any saturated chain P = P0 P1 Pn = Q of prime ideals between P and Q, the length n of the chain⊂ is independent⊂ ··· ⊂ of its par- ticular choice and hence equal to htΛ=P Q=P ,wherehtΛ=P Q=P denotes the height of the prime ideal Q=P in Λ=P .
    [Show full text]