Injective Modules and Divisible Groups
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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. -
18 Divisible Groups
18 Divisible groups Every group splits as a direct sum G = D © R where D is divisible and R is reduced. However, the divisible subgroup is natural whereas the reduced subgroup is not. [Recall that all groups are abelian in this chapter.] Definition 18.1. A group G is called divisible if for every x 2 G and every positive integer n there is a y 2 G so that ny = x, i.e., every element of G is divisible by every positive integer. For example the additive group of rational numbers Q and real numbers R and any vector space over Q or R are divisible but Z is not divisible. [In fact there are no nontrivial finitely generated divisible groups.] Another example of a divisible group is Z=p1. This is usually defined as a direct limit: Z=p1 = lim Z=pk ! where Z=pk maps to Z=pk+1 by multiplication by p. This is the quotient of the direct sum ©Z=pk by the relation that n 2 Z=pk is identified with pjn 2 Z=pj+k. Another way to say this is that Z=p1 is generated by elements x0; x1; x2; x3; ¢ ¢ ¢ which are related by xk = pxk+1 and x0 = 0. Since xk has order pk it is divisible by any number relatively prime to p and is divisible j by any power of p since xk = p xj+k. Yet another description of Z=p1 is the multiplicative group of all p-power roots of unity. These are complex numbers of the form e2¼in=pk where n is an integer. -
HKR CHARACTERS, P-DIVISIBLE GROUPS and the GENERALIZED CHERN CHARACTER
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 11, November 2010, Pages 6159–6181 S 0002-9947(2010)05194-3 Article electronically published on June 11, 2010 HKR CHARACTERS, p-DIVISIBLE GROUPS AND THE GENERALIZED CHERN CHARACTER TAKESHI TORII Abstract. In this paper we describe the generalized Chern character of clas- sifying spaces of finite groups in terms of Hopkins-Kuhn-Ravenel generalized group characters. For this purpose we study the p-divisible group and its level structures associated with the K(n)-localization of the (n +1)stMorava E-theory. 1. Introduction For a finite group G,weletBG be its classifying space. A generalized cohomol- ogy theory h∗(−) defines a contravariant functor from the category of finite groups to the category of graded modules by assigning to a finite group G a graded module h∗(BG). When h∗(−) has a graded commutative multiplicative structure, this func- tor takes its values in the category of graded commutative h∗-algebras. If we have a description of this functor in terms of finite groups G, then it is considered that we have some intrinsic information for the cohomology theory h∗(−). For example, when h∗(−) is the complex K-theory K∗(−), there is a natural ring homomorphism from the complex representation ring R(G)toK∗(BG), and Atiyah [4] has shown that this induces a natural isomorphism after completion at the augmentation ideal I of R(G), ∗ ∼ ∧ K (BG) = R(G)I , where we regard K∗(−)asaZ/2Z-graded cohomology theory. This isomorphism is considered to be intimately related to the definition of K-theory by vector bundles. -
Moduli of P-Divisible Groups
Cambridge Journal of Mathematics Volume 1, Number 2, 145–237, 2013 Moduli of p-divisible groups Peter Scholze and Jared Weinstein We prove several results about moduli spaces of p-divisible groups such as Rapoport–Zink spaces. Our main goal is to prove that Rapoport–Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of p- adic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport–Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings, [Fal10]. For this, we give a general classification of p-divisible groups over OC ,whereC is an algebraically closed complete extension of Qp, in the spirit of Riemann’s classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonn´e module functor for p-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective). 1 Introduction 146 2 Adic spaces 157 2.1 Adic spaces 157 2.2 Formal schemes and their generic fibres 161 2.3 Perfectoid spaces 164 2.4 Inverse limits in the category of adic spaces 167 3 Preparations on p-divisible groups 170 3.1 The universal cover of a p-divisible group 170 3.2 The universal vector extension, and the universal cover 173 3.3 The Tate module of a p-divisible group 175 3.4 The logarithm 177 145 146 Peter Scholze and Jared Weinstein 3.5 Explicit Dieudonn´e theory 179 4 Dieudonn´e theory over semiperfect rings -
October 17, 2014 P-DIVISIBLE GROUPS Let's Set Some Conventions. Let S = Spec R Where R Is a Complete Local Noetherian Ring. Le
October 17, 2014 p-DIVISIBLE GROUPS SPEAKER: JOSEPH STAHL Let's set some conventions. Let S = Spec R where R is a complete local noetherian ring. Let K = Frac R and k be the residue field of R. Recall that S is connected. 1. Reminder on finite flat group schemes Last time, we defined a group scheme over S to be a group object in S − Sch. If G is a finite flat group scheme then we have a connected-´etaleexact sequence j 0 ! G0 −!i G −! G´et ! 0: If we write G = Spec A then G0 = Spec A0 where A0 is the local quotient through which the co-unit of the Hopf algebra structure " : A ! R factors. And G´et = Spec(A´et) where A´et ⊂ A is the maximal ´etale subalgebra of A. Proposition 1. The two functors G 7! G0 and G 7! G´et are exact. Before we do this, let's recall something about ´etaleschemes. Let α : Spec k ! S be a geometric point. If X is an S-scheme then let X(α) = HomSch =S(Spec k; X) sep so if X = Spec A then X(α) = HomR(A; k). Let π1(S; α) = Gal(k =k). Then π1(S; α) acts on X(α) and X 7! X(α) defines an equivalence of categories between finite ´etale S-schemes and finite continuous π1(S; α)-sets. Proof. Let 0 ! G0 ! G ! G00 ! 0 be a short exact sequence in the category of group schemes over S. The easy directions are showing \connected parts" is left exact and \´etaleparts" is right exact. -
Monomorphism - Wikipedia, the Free Encyclopedia
Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G. -
SUSTAINED P-DIVISIBLE GROUPS in This Chapter We Develop And
SUSTAINED p-DIVISIBLE GROUPS CHING-LI CHAI & FRANS OORT preliminary draft, version 9/10/2017 for a chapter in a monograph Hecke Orbits In this chapter we develop and explain the notion of sustained p-divisible groups, which provides a scheme-theoretic definition of the notion of central leaves intro- duced in [21]. 1. Introduction: what is a sustained p-divisible group 1.1. In a nutshell, a p-divisible group X ! S over a base scheme S in equi- characteristic p is said to be sustained if its pn-torsion subgroup X[pn] ! S is constant locally in the flat topology of S, for every natural number n; see 1.3 for the precise definition. In music a sustained (sostenuto in Italian) tone is constant, but as the underlying harmony may change it does not feel constant at all. Hence we use the adjective \sustained" to describe a familiy of \the same objects" whereas the whole family need not be constant. This concept enriches the notion of geometrically fiberwise constant family of p-divisible groups. Proofs of statements in this introduction are given in later sections. Careful readers are likely to have zeroed in on an important point which was glossed over in the previous paragraph: being \constant" over a base scheme is fundatmentally a relative concept. There is no satisfactory definition of a \constant family Y ! S of algebraic varieties" over a general base scheme S, while for a scheme S over a field κ, a κ-constant family over S is a morphism Y ! S which is S-isomorphic to Y0 ×Spec(κ) S for some κ-scheme Y0. -
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. -
THE MODULE of PSEUDO-RATIONAL RELATIONS of a QUOTIENT DIVISIBLE GROUP Introduction in the Most General Case, an Abelian Group G
Algebra i analiz St. Petersburg Math. J. Tom 22 (2010), 1 Vol. 22 (2011), No. 1, Pages 163–174 S 1061-0022(2010)01136-1 Article electronically published on November 17, 2010 THE MODULE OF PSEUDO-RATIONAL RELATIONS OF A QUOTIENT DIVISIBLE GROUP A. V. TSAREV Abstract. It is proved that the module of pseudo-rational relations determines a quotient divisible group up to quasi-isomorphism. Introduction In the most general case, an Abelian group G may be called quotient divisible if it contains a free subgroup F such that G/F is a divisible periodic group. In the present paper, we consider only quotient divisible torsion-free groups of finite rank that do not contain divisible periodic subgroups. Fomin and Wickless introduced such groups in [1]; moreover, they constructed the category of quotient divisible groups with quasihomomor- phisms as morphisms and showed that this category is dual to the well-known category of torsion-free groups of finite rank with quasihomomorphisms as morphisms. Even earlier, in [2], Beaumont and Pierce considered quotient divisible torsion-free groups of finite rank (the term itself “quotient divisible group” first appeared in that paper), for which they constructed a system of invariants that determine such a group up to quasi-isomorphism. It was an initial example of a description of Abelian groups up to quasi-isomorphism rather than up to isomorphism, which was quite timely in view of the papers [3] by J´onsson and [4] by Yakovlev. In the present paper, we develop an idea of Fomin from [5] about the invocation of modules over the ring R of pseudo-rational numbers for the study of quotient divisible groups. -
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. -
Notes on P-Divisible Groups
Notes on p-Divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete noetherian local ring, m its maximal ideal. We keep the assumption that the residue ¯eld k = R=m is of characteristic p > 0. (Indeed, most of the de¯nitions and propositions are true for arbitrary noetherian schemes; however, we will work mostly with the described case later on.) By a ¯nite group scheme G over R, we mean an a±ne commutative group scheme G = Spec A, where A is a ¯nite R-algebra and it is free of rank m as an R-module. m is called the order of the group. It can be shown that if G has order m, then G ¡!£m G is the trivial map. In other words, 8x 2 G(S), mx = e for any R-scheme S. (This is only known as Deligne theorem when G is commutative. It is still a conjecture for noncommutative case.) The augmentation ideal I is the kernel of the unity map ² : A ! R. The di®erential 2 » of G is given by G=R = I=I R A. (Using the map à : G £ G ! G £ G, given by Ã(x; y) = (xy; y)). For (commutative) ¯nite group scheme G = Spec A, one can de¯ne its Cartier Dual _ _ _ G = Spec A , where A = HomR¡mod(A; R) and its multiplication is given by the dual of the comultiplication m¤ : A ! A A.