DOCSLIB.ORG

Characterization of Artinian Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Characterization of Artinian Rings MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 43 8.3. characterization of Artin rings. Recall 7.14 (7.7): In a Noe- therian ring, every ideal a contains a power of its radical r(a). Taking a = 0 and r(0) = nilrad A we get: Corollary 8.13 (A-M 7.15). A Noetherian nilrad A is nilpotent. ⇒ Lemma 8.14. Suppose m A is a maximal ideal (A/m field) and A is Artinian or Noetherian. ⊂Then m/m2 is a fin dim vector space over A/m. Proof. A/m2 is Noetherian or Artinian and its proper ideals are exactly the vector subspaces of m/m2. ! Lemma 8.15 (A-M 6.11). Suppose that 0 A is a finite product of ⊆ (not necessarily distinct) maximal ideals 0 = m1m2 mn. Then the following are equivalent: · · · (1) A is Noetherian. (2) A is Artinian. (3) Each quotient m m /m m is finite dimensional as a 1 · · · i 1 · · · i+1 vector space over the field A/mi+1. The proof is by induction on n using the exact sequence 0 m1 mn 1 A A/m1 mn 1 0 → · · · − → → · · · − → Theorem 8.16 (A-M 8.5). A Artinian A Noetherian and dim A = 0. ⇐⇒ Proof. Both sides imply dim A = 0. So, assuming dim A = 0, we need to show A Artinian iff A Noetherian. By the lemma, it suffices to show that 0 is a finite product of maximal ideals. k ( ) A Artinian n = mi nilpotent (n = 0). So, mi mi = 0. So,⇒A is Noetherian.⇒ ∩ ⊆ ∩ ( ) A Noetherian 0 has primary decomposition.! So A has only finitely⇐ many minimal⇒primes. But dim A = 0 implies all primes are maximal. So, A has only finitely many maximal ideals mi. By 7.15 (8.13) this implies n = m is nilpotent. By 6.11 (8.15), A is Artinian. ∩ i ! Proposition 8.17. Suppose A is Noetherian local with max ideal m. Then either n (1) m = 0 for some n ( 6.11 A is Artin local) or (2) mn = mn+1 = ⇒ ' ' · · · Proof. By Nakayama (ma = a a = 0), ⇐⇒ mn = mn+1 mn = 0 ⇐⇒ ! 44 MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA Theorem 8.18 (A-M 8.7: Structure thm for Artin rings). n A Artin A = A A : Artin local ⇐⇒ ∼ i i i=1 " k n Proof. ( ) Suppose n = 0, n = i=1 mi. ⇒ ϕ Claim: A n A/mk is an isomorphism. i=1 i # Proof: (Using−→ properties of coprime ideals from 1.3.3) ! ϕ onto and m coprime mk coprime i ⇒ i ⇒ mk = mk $ i ∩ i ker ϕ = mk = mk = 0 !(ϕ is 1-1) ⇒ ∩ i i ( ) This follows from " Claim⇐ : A, B Artin A B Artin. ⇒ × Proof of claim: A 0 " A B # 0 B is a short exact sequence of A B modules where× the first× and third× terms satisfy the DCC for submo×dules since the submodules of A 0 are the ideals in A and the submodules of 0 B are the ideals of B×. So, the middle satisfies the × DCC for submodules = ideals. ! Proposition 8.19 (A-M 8.8). If A is Artin local then TFAE: (1) Every ideal is principal. (2) m is principal. (3) dim m/m2 1 (k = A/m). k ≤ Proof. (1) (2) (3) is obvious. So, suppose (3). ⇒ ⇒ 2 Case 1. dimk m/m = 0 m = m2 m = 0 A = A/m = k is a field ⇒ ⇒NAK ⇒ and 0 = (0) is the only ideal. 2 Case 2. dimk m/m = 1 m = (x) mr = (xr) ⇒ ⇒ m = rad A is nilpotent (mn = 0) Claim: These are all the ideals of A. Proof: Let a A be a nontrivial proper ideal. ⊂ a = A a m a = 0 a mn = 0 ' ⇒ ⊂ ' ⇒ '⊂ So, a mr, a mr+1 for some r ⊂ '⊂ r+1 r+1 r r+1 r r m ! a + m m a + m = m NAK a = m ⇒ ⊂ ⇒ ⇒ which shows (1). The book proof of the last line is: y = axr a, y / mr+1 a / m a : unit xr a a = (xr) = mr ∃ ∈ ∈ ⇒ ∈ ⇒ ⇒ ∈ ⇒ MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 45 ! 8.4. examples, problems. Example 8.20. A = k[x , x , ] is generated, a = (x , x2, x3, ). 1 2 · · · ∞ 1 2 3 · · · Then B = A/a is local with unique maximal ideal m = (x1, x2, ). Also m is the unique prime ideal. So, dim B = 0. But m is· ·not· nilpotent. So, A is not Artinian and not Noetherian. In fact, m is not finitely generated. Example 8.21. A = k[x2, x3]/(x4) k[x]/(x4). Then A is Artin local, m = (x2, x3), m2 = 0 and dim m = dim⊂ m/m2 = 2. Problem. Show that a local ring A with maximal ideal m is Artinian iff m is nilpotent and m/m2 is a finite dimension vector space over A/m. Use the fact that there is an epimorphism from the n-fold tensor power of m/m2 onto mn/mn+1. Problem #6, p.92 A Noetherian, q a p-primary ideal. Consider chains of primary ideals from q to p. Show that all such chains are bounded and all maximal chains have the same length. Step 1. We may assume q = 0. Proof: For simplicity suppose the maximal length is 3: q q q p in A ⊂ 1 ⊂ 2 ⊂ 0 q /q q /q p/q in A/q ⊂ 1 ⊂ 2 ⊂ p/q-primary prime p q q q q p q / is prime in A/ . i%/ are&' / (-primary%&'( and distinct. So, there is a 1-1 correspondence between p-primary ideals between q and p and p/q-primary ideals between 0 and p/q. So, we may replace A by A/q and assume that q = 0. Step 2. 0 is p-primary A p has no zero divisors. Proof: If s A p is a⇒z.d. −then st = 0 for some nonzero t. But 0 is p-primary. So,∈ either− t = 0 or sn = 0. But both are impossible. ı 1 Step 3. S = A p has no z.d. a " S− A = Ap and p-primary q: − ⇒ → ∀ 1 1 (1) S− q is S− p-primary (2) qec = q There is a 1-1 correspondence between p-primary ideals in A and ⇒ 1 S− p-primary ideals in the local ring Ap. 1 A Ap = S− A 1 e 1 p-primary q S− q = q S− p-primary c 0→ 1 p-primary a a S− p-primary ← 46 MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA Step 4. Ap is Artin local. Proof: dim Ap = 0. p is a minimal prime (It is the only prime associated to 0.) Step 5 In an Artin local ring all ideals are m-primary. 2 dimk m/m = d1 i i+1 dimk m /m = di m m+1 dimk m /m = dr r+1 m = 0 dr+1 = 0 Claim. All maximal chains of proper ideals have length d1 + d2 + + dk · · ·Proof: For each ideal a define a mi + mi+1 a = dim ∩ i k mi+1 (1) If a ! b then 0 ai < bi di ≤ ≤ (2) If bi ai 2 then c s.t. a ! c ! b [Let i be minimal so − ≥ ) ∃)i ) i i+1 that bi > ai. Take b b m , b / a m + m . Let c = a + (b). Then) c =)a + 1 and∈c ∩= a for∈ j ∩= i.] i i j j ' Therefore, the numbers ai take all values from 0 to di for any maximal chain of ideals. ) ).
Load more

Recommended publications
  • 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]
  • 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]
  • Arxiv:Math/0005288V1 [Math.QA] 31 May 2000 Ilb Setal H Rjciecodnt Igo H Variety
    Mannheimer Manuskripte 254 math/0005288 SINGULAR PROJECTIVE VARIETIES AND QUANTIZATION MARTIN SCHLICHENMAIER Abstract. By the quantization condition compact quantizable K¨ahler mani- folds can be embedded into projective space. In this way they become projec- tive varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular vari- eties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, sin- gularities, and quotients appearing in geometric invariant theory are recalled. Contents Introduction 1 1. From quantizable compact K¨ahler manifolds to projective varieties 3 2. Projective varieties 6 2.1. The definition of a projective variety 6 2.2. Embeddings into Projective Space 9 2.3. The projective coordinate ring 12 3. Singularities 14 4. Quotients 17 4.1. Quotients in algebraic geometry 17 4.2. The relation with the symplectic quotient 19 References 20 Introduction arXiv:math/0005288v1 [math.QA] 31 May 2000 Compact K¨ahler manifolds which are quantizable, i.e. which admit a holomor- phic line bundle with curvature form equal to the K¨ahler form (a so called quantum line bundle) are projective algebraic manifolds. This means that with the help of the global holomorphic sections of a suitable tensor power of the quantum line bundle they can be embedded into a projective space of certain dimension.
    [Show full text]
  • Rings of Quotients and Localization a Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Ar
    RINGS OF QUOTIENTS AND LOCALIZATION A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREME N TS FOR THE DEGREE OF MASTER OF ARTS IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE TEXAS WOMAN'S UNIVERSITY COLLEGE OF ARTS AND SCIENCE S BY LILLIAN WANDA WRIGHT, B. A. DENTON, TEXAS MAY, l974 TABLE OF CONTENTS INTRODUCTION . J Chapter I. PRIME IDEALS AND MULTIPLICATIVE SETS 4 II. RINGS OF QUOTIENTS . e III. CLASSICAL RINGS OF QUOTIENTS •• 25 IV. PROPERTIES PRESERVED UNDER LOCALIZATION 30 BIBLIOGRAPHY • • • • • • • • • • • • • • • • ••••••• 36 iii INTRODUCTION The concept of a ring of quotients was apparently first introduced in 192 7 by a German m a thematician Heinrich Grell i n his paper "Bezeihungen zwischen !deale verschievener Ringe" [ 7 ] . I n his work Grelt observed that it is possible to associate a ring of quotients with the set S of non -zero divisors in a r ing. The elements of this ring of quotients a r e fractions whose denominators b elong to Sand whose numerators belong to the commutative ring. Grell's ring of quotients is now called the classical ring of quotients. 1 GrelL 1 s concept of a ring of quotients remained virtually unchanged until 1944 when the Frenchman Claude C hevalley presented his paper, "On the notion oi the Ring of Quotients of a Prime Ideal" [ 5 J. C hevalley extended Gre ll's notion to the case wher e Sis the compte- ment of a primt: ideal. (Note that the set of all non - zer o divisors and the set- theoretic c om?lement of a prime ideal are both instances of .multiplicative sets -- sets tha t are closed under multiplicatio n.) 1According to V.
    [Show full text]
  • Arxiv:1710.09830V1 [Math.AC] 26 Oct 2017
    COMPUTATIONS OVER LOCAL RINGS IN MACAULAY2 MAHRUD SAYRAFI Thesis Advisor: David Eisenbud Abstract. Local rings are ubiquitous in algebraic geometry. Not only are they naturally meaningful in a geometric sense, but also they are extremely useful as many problems can be attacked by first reducing to the local case and taking advantage of their nice properties. Any localization of a ring R, for instance, is flat over R. Similarly, when studying finitely generated modules over local rings, projectivity, flatness, and freeness are all equivalent. We introduce the packages PruneComplex, Localization and LocalRings for Macaulay2. The first package consists of methods for pruning chain complexes over polynomial rings and their localization at prime ideals. The second package contains the implementation of such local rings. Lastly, the third package implements various computations for local rings, including syzygies, minimal free resolutions, length, minimal generators and presentation, and the Hilbert–Samuel function. The main tools and procedures in this paper involve homological methods. In particular, many results depend on computing the minimal free resolution of modules over local rings. Contents I. Introduction 2 I.1. Definitions 2 I.2. Preliminaries 4 I.3. Artinian Local Rings 6 II. Elementary Computations 7 II.1. Is the Smooth Rational Quartic a Cohen-Macaulay Curve? 12 III. Other Computations 13 III.1. Computing Syzygy Modules 13 arXiv:1710.09830v1 [math.AC] 26 Oct 2017 III.2. Computing Minimal Generators and Minimal Presentation 16 III.3. Computing Length and the Hilbert-Samuel Function 18 IV. Examples and Applications in Intersection Theory 20 V. Other Examples from Literature 22 VI.
    [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]
  • CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and Properties
    page 1 of Chapter 2 CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and Properties 2.1.1 Definitions and Comments A ring R is an abelian group with a multiplication operation (a, b) → ab that is associative and satisfies the distributive laws: a(b+c)=ab+ac and (a + b)c = ab + ac for all a, b, c ∈ R. We will always assume that R has at least two elements,including a multiplicative identity 1 R satisfying a1R =1Ra = a for all a in R. The multiplicative identity is often written simply as 1,and the additive identity as 0. If a, b,and c are arbitrary elements of R,the following properties are derived quickly from the definition of a ring; we sketch the technique in each case. (1) a0=0a =0 [a0+a0=a(0+0)=a0; 0a +0a =(0+0)a =0a] (2) (−a)b = a(−b)=−(ab)[0=0b =(a+(−a))b = ab+(−a)b,so ( −a)b = −(ab); similarly, 0=a0=a(b +(−b)) = ab + a(−b), so a(−b)=−(ab)] (3) (−1)(−1) = 1 [take a =1,b= −1 in (2)] (4) (−a)(−b)=ab [replace b by −b in (2)] (5) a(b − c)=ab − ac [a(b +(−c)) = ab + a(−c)=ab +(−(ac)) = ab − ac] (6) (a − b)c = ac − bc [(a +(−b))c = ac +(−b)c)=ac − (bc)=ac − bc] (7) 1 = 0 [If 1 = 0 then for all a we have a = a1=a0 = 0,so R = {0},contradicting the assumption that R has at least two elements] (8) The multiplicative identity is unique [If 1 is another multiplicative identity then 1=11 =1] 2.1.2 Definitions and Comments If a and b are nonzero but ab = 0,we say that a and b are zero divisors;ifa ∈ R and for some b ∈ R we have ab = ba = 1,we say that a is a unit or that a is invertible.
    [Show full text]
  • A Guide to Cohen's Structure Theorem for Complete
    A GUIDE TO COHEN'S STRUCTURE THEOREM FOR COMPLETE LOCAL RINGS D. Katz The purpose of this note is to provide for my algebra class a guide to the proof of Cohen's structure theorem for complete local rings as given by Cohen himself, in the original 1946 paper [C]. In spite of more modern and concise treatments by the likes of Nagata and Grothendieck, the original proof is a model of clarity and the entire paper is a commutative algebra masterpiece. First, a few words on the original paper. The paper is divided into three parts. Part I deals with the general theory of completions of local rings and what Cohen calls generalized local rings. The former being Noetherian commutative rings with a unique maximal ideal, the latter being quasi-local rings (R; m) with m finitely generated and satisfying Krull's n intersection theorem, \n≥1m = 0. Generalized local rings are needed in Part II when (in modern jargon) certain faithfully flat extensions of local rings are constructed in order to pass to the case where the residue field of the new ring is perfect. Using the associated graded ring, Cohen shows in Part I that the completion of a generalized local ring is a local ring (see [C; Theorem 3]). Incidently, Cohen asks whether or not generalized local rings are always local. The answer is no, but I believe that the generalized local rings he considers are local rings. Part II of the paper presents the structure theorem for complete local rings. It is interesting to note that some of the easier arguments that serve as `base cases' for his proof either appeared in or are based on earlier work in the 1930's on valuations rings by Hasse and Schmidt, Maclane, and Teichmuller.
    [Show full text]
  • Local Cohomology
    LOCAL COHOMOLOGY by Mel Hochster These are lecture notes based on seminars and courses given by the author at the Univer- sity of Michigan over a period of years. This particular version is intended for Mathematics 615, Winter 2011. The objective is to give a treatment of local cohomology that is quite elementary, assuming, for the most part, only a modest knowledge of commutative alge- bra. There are some sections where further prerequisites, usually from algebraic geometry, are assumed, but these may be omitted by the reader who does not have the necessary background. This version contains the first twenty sections, and is tentative. There will be extensive revisions and additions. The final version will also have Appendices dealing with some prerequisites. ||||||||| Throughout, all given rings are assumed to be commutative, associative, with identity, and all modules are assumed to be unital. By a local ring (R; m; K) we mean a Noetherian ring R with a unique maximal ideal m and residue field K = R=m. Topics to be covered include the study of injective modules over Noetherian rings, Matlis duality (over a complete local ring, the category of modules with ACC is anti- equivalent to the category of modules with DCC), the notion of depth, Cohen-Macaulay and Gorenstein rings, canonical modules and local duality, the Hartshorne-Lichtenbaum vanishing theorem, and the applications of D-modules (in equal characteristic 0) and F- modules (in characteristic p > 0) to study local cohomology of regular rings. The tie-in with injective modules arises, in part, because the local cohomology of a regular local rings with support in the maximal ideal is the same as the injective huk of the residue class field.
    [Show full text]
  • Some Remarks on the Formal Power Series Ring Bulletin De La S
    BULLETIN DE LA S. M. F. MATTHEW J. O’MALLEY Some remarks on the formal power series ring Bulletin de la S. M. F., tome 99 (1971), p. 247-258 <http://www.numdam.org/item?id=BSMF_1971__99__247_0> © Bulletin de la S. M. F., 1971, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 99, 1971, p. 247 a a58. SOME REMARKS ON THE FORMAL POWER SERIES RING BY MATTHEW J. (TMALLEY [Houston, Texas.] Let R be an integral domain with identity, let X be an indeterminate over jR, let S be the formal power series ring jR[[X]], and let G be a finite group of .R-automorphisms of S. If J? is a local ring (that is, a noetherian ring with unique maximal ideal M), and if R is complete in the M-adic topology, then P. SAMUEL shows the existence of f^S such that the ring S° of invariants of G is R[[f]] [9]. This paper was motivated by an attempt to generalize this result. Specifically, we will prove that the same conclusion holds if R is any noetherian integral domain with identity whose integral closure is a finite jR-module.
    [Show full text]
  • 1 Ringed and Locally Ringed Spaces 2 the Prime Spectrum of a Ring
    18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009) Schemes We next introduce locally ringed spaces, affine schemes, and general schemes. References: Hartshorne II.2, Eisenbud-Harris I.1, EGA 1.1. 1 Ringed and locally ringed spaces A ringed space is a topological space X equipped with a sheaf OX on X with values in Ring (called the structure sheaf ). This definition isn't so useful because it doesn't force the topology to have much to do with the ring structure; for instance, any ring can be viewed as a ringed space on a one-element topological space. A more useful notion is that of a locally ringed space. This is a ringed space in which for each x 2 X, the stalk OX;x of OX at x is a local ring, i.e., a ring with a unique maximal ideal mX;x. (The zero ring is not a local ring!) For example, suppose X is a manifold and let OX be the sheaf of real-valued continuous functions. We check that (X; OX ) forms a locally ringed space. Given x 2 X, let mX;x be the ideal of OX;x consisting of germs of functions taking the value 0 at x. This is clearly an ideal, and the quotient OX;x=mX;x is certainly contained in R. Since X is a manifold, the quotient is nonzero, so mX;x is indeed a maximal ideal of OX;x. To check that it is the unique maximal ideal, it suffices to check that any f 2 OX;x not contained in mX;x is a unit in OX;x.
    [Show full text]