Injective Modules and Torsion Functors 10
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Injective Modules: Preparatory Material for the Snowbird Summer School on Commutative Algebra
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them. Let R be a commutative Noetherian ring with an identity element. An R- module E is injective if HomR( ;E) is an exact functor. The main messages of these notes are − Every R-module M has an injective hull or injective envelope, de- • noted by ER(M), which is an injective module containing M, and has the property that any injective module containing M contains an isomorphic copy of ER(M). A nonzero injective module is indecomposable if it is not the direct • sum of nonzero injective modules. Every injective R-module is a direct sum of indecomposable injective R-modules. Indecomposable injective R-modules are in bijective correspondence • with the prime ideals of R; in fact every indecomposable injective R-module is isomorphic to an injective hull ER(R=p), for some prime ideal p of R. The number of isomorphic copies of ER(R=p) occurring in any direct • sum decomposition of a given injective module into indecomposable injectives is independent of the decomposition. Let (R; m) be a complete local ring and E = ER(R=m) be the injec- • tive hull of the residue field of R. The functor ( )_ = HomR( ;E) has the following properties, known as Matlis duality− : − (1) If M is an R-module which is Noetherian or Artinian, then M __ ∼= M. -
Unofficial Math 501 Problems (PDF)
Uno±cial Math 501 Problems The following document consists of problems collected from previous classes and comprehensive exams given at the University of Illinois at Urbana-Champaign. In the exercises below, all rings contain identity and all modules are unitary unless otherwise stated. Please report typos, sug- gestions, gripes, complaints, and criticisms to: [email protected] 1) Let R be a commutative ring with identity and let I and J be ideals of R. If the R-modules R=I and R=J are isomorphic, prove I = J. Note, we really do mean equals! 2) Let R and S be rings where we assume that R has an identity element. If M is any (R; S)-bimodule, prove that S HomR(RR; M) ' M: 3) Let 0 ! A ! B ! C ! 0 be an exact sequence of R-modules where R is a ring with identity. If A and C are projective, show that B is projective. 4) Let R be a ring with identity. (a) Prove that every ¯nitely generated right R-module is noetherian if and only if R is a right noetherian ring. (b) Give an example of a ¯nitely generated module which is not noetherian. ® ¯ 5) Let 0 ! A ! B ! C ! 0 be a short exact sequence of R-modules where R is any ring. Assume there exists an R-module homomorphism ± : B ! A such that ±® = idA. Prove that there exists an R-module homomorphism : C ! B such that ¯ = idC . 6) If R is any ring and RM is an R-module, prove that the functor ¡ R M is right exact. -
Change of Rings Theorems
CHANG E OF RINGS THEOREMS CHANGE OF RINGS THEOREMS By PHILIP MURRAY ROBINSON, B.SC. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master of Science McMaster University (July) 1971 MASTER OF SCIENCE (1971) l'v1cHASTER UNIVERSITY {Mathematics) Hamilton, Ontario TITLE: Change of Rings Theorems AUTHOR: Philip Murray Robinson, B.Sc. (Carleton University) SUPERVISOR: Professor B.J.Mueller NUMBER OF PAGES: v, 38 SCOPE AND CONTENTS: The intention of this thesis is to gather together the results of various papers concerning the three change of rings theorems, generalizing them where possible, and to determine if the various results, although under different hypotheses, are in fact, distinct. {ii) PREFACE Classically, there exist three theorems which relate the two homological dimensions of a module over two rings. We deal with the first and last of these theorems. J. R. Strecker and L. W. Small have significantly generalized the "Third Change of Rings Theorem" and we have simply re organized their results as Chapter 2. J. M. Cohen and C. u. Jensen have generalized the "First Change of Rings Theorem", each with hypotheses seemingly distinct from the other. However, as Chapter 3 we show that by developing new proofs for their theorems we can, indeed, generalize their results and by so doing show that their hypotheses coincide. Some examples due to Small and Cohen make up Chapter 4 as a completion to the work. (iii) ACKNOW-EDGMENTS The author expresses his sincere appreciation to his supervisor, Dr. B. J. Mueller, whose guidance and helpful criticisms were of greatest value in the preparation of this thesis. -
Topics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. -
Injective Modules and Divisible Groups
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 5-2015 Injective Modules And Divisible Groups Ryan Neil Campbell University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Algebra Commons Recommended Citation Campbell, Ryan Neil, "Injective Modules And Divisible Groups. " Master's Thesis, University of Tennessee, 2015. https://trace.tennessee.edu/utk_gradthes/3350 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Ryan Neil Campbell entitled "Injective Modules And Divisible Groups." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. David Anderson, Major Professor We have read this thesis and recommend its acceptance: Shashikant Mulay, Luis Finotti Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Injective Modules And Divisible Groups AThesisPresentedforthe Master of Science Degree The University of Tennessee, Knoxville Ryan Neil Campbell May 2015 c by Ryan Neil Campbell, 2015 All Rights Reserved. ii Acknowledgements I would like to thank Dr. -
Gille Beamer Part 1
Torsors over affine curves Philippe Gille Institut Camille Jordan, Lyon PCMI (Utah), July 12, 2021 This is the first part The Swan-Serre correspondence I This is the correspondence between projective finite modules of finite rank and vector bundles, it arises from the case of a paracompact topological space [37]. We explicit it in the setting of affine schemes following the book of G¨ortz-Wedhorn [18, ch. 11] up to slightly different conventions. I Vector group schemes. Let R be a ring (commutative, unital). (a) Let M be an R{module. We denote by V(M) the affine R{scheme defined by V(M) = SpecSym•(M) ; it is affine over R and represents the R{functor S 7! HomS−mod (M ⊗R S; S) = HomR−mod (M; S) [11, 9.4.9]. Vector group schemes • I V(M) = Spec Sym (M) ; I It is called the vector group scheme attached to M, this construction commutes with arbitrary base change of rings R ! R0. I Proposition [32, I.4.6.1] The functor M ! V(M) induces an antiequivalence of categories between the category of R{modules and that of vector group schemes over R. An inverse functor is G 7! G(R). Vector group schemes II I (b) We assume now that M is locally free of finite rank and denote by M_ its dual. In this case Sym•(M) is of finite presentation (ibid, 9.4.11). Also the R{functor S 7! M ⊗R S is representable by the affine R{scheme V(M_) which is also denoted by W(M) [32, I.4.6]. -
Arxiv:Math/0301347V1
Contemporary Mathematics Morita Contexts, Idempotents, and Hochschild Cohomology — with Applications to Invariant Rings — Ragnar-Olaf Buchweitz Dedicated to the memory of Peter Slodowy Abstract. We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As applications of our approach we explore to what extent Hochschild cohomology of an invariant ring coincides with the invariants of the Hochschild cohomology. Contents Introduction 1 1. Morita Contexts 3 2. The Grade of the Stable Endomorphism Ring 6 3. Equivalences to the Generalized Nakayama Conjecture 12 4. The Comparison Homomorphism for Hochschild Cohomology 14 5. Hochschild Cohomology of Auslander Contexts 17 6. Invariant Rings: The General Case 20 7. Invariant Rings: The Commutative Case 23 References 27 arXiv:math/0301347v1 [math.AC] 29 Jan 2003 Introduction One of the basic features of Hochschild (co-)homology is its invariance under derived equivalence, see [25], in particular, it is invariant under Morita equivalence, a result originally established by Dennis-Igusa [18], see also [26]. This raises the question how Hochschild cohomology compares in general for algebras that are just ingredients of a Morita context. -
Galois Extensions of Structured Ring Spectra 3
GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA John Rognes August 31st 2005 Abstract. We introduce the notion of a Galois extension of commutative S-algebras (E∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological K-theory, Lubin–Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this general- ity. Its proof involves the notions of separable and ´etale extensions of commutative S-algebras, and the Goerss–Hopkins–Miller theory for E∞ mapping spaces. We show that the global sphere spectrum S is separably closed, using Minkowski’s discrimi- nant theorem, and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf–Galois extensions of com- mutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions. Contents 1. Introduction 2. Galois extensions in algebra 2.1. Galois extensions of fields 2.2. Regular covering spaces 2.3. Galois extensions of commutative rings 3. Closed categories of structured module spectra 3.1. Structured spectra arXiv:math/0502183v2 [math.AT] 6 Dec 2005 3.2. Localized categories 3.3. Dualizable spectra 3.4. Stably dualizable groups 3.5. The dualizing spectrum 3.6. The norm map 4. Galois extensions in topology 4.1. Galois extensions of E-local commutative S-algebras 4.2. The Eilenberg–Mac Lane embedding 4.3. Faithful extensions 5. -
Complex Oriented Cohomology Theories and the Language of Stacks
COMPLEX ORIENTED COHOMOLOGY THEORIES AND THE LANGUAGE OF STACKS COURSE NOTES FOR 18.917, TAUGHT BY MIKE HOPKINS Contents Introduction 1 1. Complex Oriented Cohomology Theories 2 2. Formal Group Laws 4 3. Proof of the Symmetric Cocycle Lemma 7 4. Complex Cobordism and MU 11 5. The Adams spectral sequence 14 6. The Hopf Algebroid (MU∗; MU∗MU) and formal groups 18 7. More on isomorphisms, strict isomorphisms, and π∗E ^ E: 22 8. STACKS 25 9. Stacks and Associated Stacks 29 10. More on Stacks and associated stacks 32 11. Sheaves on stacks 36 12. A calculation and the link to topology 37 13. Formal groups in prime characteristic 40 14. The automorphism group of the Lubin-Tate formal group laws 46 15. Formal Groups 48 16. Witt Vectors 50 17. Classifying Lifts | The Lubin-Tate Space 53 18. Cohomology of stacks, with applications 59 19. p-typical Formal Group Laws. 63 20. Stacks: what's up with that? 67 21. The Landweber exact functor theorem 71 References 74 Introduction This text contains the notes from a course taught at MIT in the spring of 1999, whose topics revolved around the use of stacks in studying complex oriented cohomology theories. The notes were compiled by the graduate students attending the class, and it should perhaps be acknowledged (with regret) that we recorded only the mathematics and not the frequent jokes and amusing sideshows which accompanied it. Please be wary of the fact that what you have in your hands is the `alpha- version' of the text, which is only slightly more than our direct transcription of the stream-of- consciousness lectures. -
Lectures on Local Cohomology
Contemporary Mathematics Lectures on Local Cohomology Craig Huneke and Appendix 1 by Amelia Taylor Abstract. This article is based on five lectures the author gave during the summer school, In- teractions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concern- ing the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals. Contents 1. Introduction 1 2. Local Cohomology 3 3. Injective Modules over Noetherian Rings and Matlis Duality 10 4. Cohen-Macaulay and Gorenstein rings 16 d 5. Vanishing Theorems and the Structure of Hm(R) 22 6. Vanishing Theorems II 26 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 32 8. Appendix 2: Bass numbers and Gorenstein Rings 37 References 41 1. Introduction Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization 2000 Mathematics Subject Classification. Primary 13C11, 13D45, 13H10. Key words and phrases. local cohomology, Gorenstein ring, initial ideal. The first author was supported in part by a grant from the National Science Foundation, DMS-0244405. -
Characteristic Elements in Noncommutative Iwasawa Theory
Characteristic Elements in Noncommutative Iwasawa Theory Meinen Eltern Contents Introduction 1 General Notation and Conventions 9 1. p-adic Lie groups 11 2. The Iwasawa algebra - a review of Lazard’s work 13 3. Iwasawa-modules 18 4. Virtual objects and some requisites from K-theory 24 5. The relative situation 28 5.1. Non-commutative power series rings 29 5.2. The Weierstrass preparation theorem 32 5.3. Faithful modules and non-principal reflexive ideals 33 6. Ore sets associated with group extensions 36 6.1. The direct product case 40 6.2. The semi-direct product case 42 6.3. The GL2 case 43 7. Twisting 45 7.1. Twisting of Λ-modules 45 7.2. Evaluating at representations 50 7.3. Equivariant Euler-characteristics 52 8. Descent of K-theory 57 9. Characteristic elements of Selmer groups 65 9.1. Local Euler factors 66 9.2. The characteristic element of an elliptic curve over k∞ 67 9.3. The false Tate curve case 68 9.4. The GL2-case 71 10. Towards a main conjecture 71 Appendix A. Filtered rings 74 Appendix B. Induction 76 References 79 Introduction Let p be a prime number, which, for simplicity, we shall always assume odd. The goal of non-commutative Iwasawa theory is to extend Iwasawa theory over Zp- extensions to the case of p-adic Lie extensions. Thus it might be useful to recall briefly some main aspects of the classical theory: Let k be a finite extension of Q, and write kcyc for the cyclotomic Zp-extension of k. -
9 the Injective Envelope
9 The Injective Envelope. In Corollaries 3.6 and 3.11 we saw that every module RM is a factor of a projective and the submodule of an injective. So we would like to learn whether there exist such projective and injective modules that are in some sense minimal and universal for M, sort of closures for M among the projective and injective modules. It turns out, strangely, that here we discover that our usual duality for projectives and injectives breaks down; there exist minimal injective extensions for all modules, but not minimal projectives for all modules. Although we shall discuss the projective case briey, in this section we focus on the important injective case. The rst big issue is to decide what we mean by minimal projective and injective modules for M. Fortunately, the notions of of essential and superuous submodules will serve nice as criteria. Let R be a ring — initially we place no further restriction. n epimorphism superuous Ker f M A(n) f : M → N is essential in case monomorphism Im f –/ N So for example, if M is a module of nite length, then the usual map M → M/Rad M is a superuous epimorphism and the inclusion map Soc M → M is an essential monomorphism. (See Proposition 8.9.) 9.1. Lemma. (1) An epimorphism f : M → N is superuous i for every homomorphism g : K → M,iff g is epic, then g is epic. (2) A monomorphism f : M → N is essential i for every homomorphism g : N → K,ifg f is monic, then g is monic.