Linear Source Invertible Bimodules and Green Correspondence
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Modules and Vector Spaces
Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module. -
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. -
QUIVER BIALGEBRAS and MONOIDAL CATEGORIES 3 K N ≥ 0
QUIVER BIALGEBRAS AND MONOIDAL CATEGORIES HUA-LIN HUANG (JINAN) AND BLAS TORRECILLAS (ALMER´IA) Abstract. We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories. 1. Introduction This paper is devoted to the study of natural bialgebra structures on the path coalgebra of an arbitrary quiver and monoidal structures on the category of its locally nilpotent and locally finite representations. A further purpose is to establish a quiver setting for general pointed bialgebras and pointed monoidal categories. Our original motivation is to extend the Hopf quiver theory [4, 7, 8, 12, 13, 25, 31] to the setting of generalized Hopf structures. As bialgebras are a fundamental generalization of Hopf algebras, we naturally initiate our study from this case. The basic problem is to determine what kind of quivers can give rise to bialgebra structures on their associated path algebras or coalgebras. It turns out that the path coalgebra of an arbitrary quiver admits natural bial- gebra structures, see Theorem 3.2. This seems a bit surprising at first sight by comparison with the Hopf case given in [8], where Cibils and Rosso showed that the path coalgebra of a quiver Q admits a Hopf algebra structure if and only if Q is a Hopf quiver which is very special. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
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. -
11. Finitely-Generated Modules
11. Finitely-generated modules 11.1 Free modules 11.2 Finitely-generated modules over domains 11.3 PIDs are UFDs 11.4 Structure theorem, again 11.5 Recovering the earlier structure theorem 11.6 Submodules of free modules 1. Free modules The following definition is an example of defining things by mapping properties, that is, by the way the object relates to other objects, rather than by internal structure. The first proposition, which says that there is at most one such thing, is typical, as is its proof. Let R be a commutative ring with 1. Let S be a set. A free R-module M on generators S is an R-module M and a set map i : S −! M such that, for any R-module N and any set map f : S −! N, there is a unique R-module homomorphism f~ : M −! N such that f~◦ i = f : S −! N The elements of i(S) in M are an R-basis for M. [1.0.1] Proposition: If a free R-module M on generators S exists, it is unique up to unique isomorphism. Proof: First, we claim that the only R-module homomorphism F : M −! M such that F ◦ i = i is the identity map. Indeed, by definition, [1] given i : S −! M there is a unique ~i : M −! M such that ~i ◦ i = i. The identity map on M certainly meets this requirement, so, by uniqueness, ~i can only be the identity. Now let M 0 be another free module on generators S, with i0 : S −! M 0 as in the definition. -
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 . -
NOTES in COMMUTATIVE ALGEBRA: PART 1 1. Results/Definitions Of
NOTES IN COMMUTATIVE ALGEBRA: PART 1 KELLER VANDEBOGERT 1. Results/Definitions of Ring Theory It is in this section that a collection of standard results and definitions in commutative ring theory will be presented. For the rest of this paper, any ring R will be assumed commutative with identity. We shall also use "=" and "∼=" (isomorphism) interchangeably, where the context should make the meaning clear. 1.1. The Basics. Definition 1.1. A maximal ideal is any proper ideal that is not con- tained in any strictly larger proper ideal. The set of maximal ideals of a ring R is denoted m-Spec(R). Definition 1.2. A prime ideal p is such that for any a, b 2 R, ab 2 p implies that a or b 2 p. The set of prime ideals of R is denoted Spec(R). p Definition 1.3. The radical of an ideal I, denoted I, is the set of a 2 R such that an 2 I for some positive integer n. Definition 1.4. A primary ideal p is an ideal such that if ab 2 p and a2 = p, then bn 2 p for some positive integer n. In particular, any maximal ideal is prime, and the radical of a pri- mary ideal is prime. Date: September 3, 2017. 1 2 KELLER VANDEBOGERT Definition 1.5. The notation (R; m; k) shall denote the local ring R which has unique maximal ideal m and residue field k := R=m. Example 1.6. Consider the set of smooth functions on a manifold M. -
Deformations of Bimodule Problems
FUNDAMENTA MATHEMATICAE 150 (1996) Deformations of bimodule problems by Christof G e i ß (M´exico,D.F.) Abstract. We prove that deformations of tame Krull–Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems. 1. Introduction. Let k be an algebraically closed field. Consider the variety algV (k) of associative unitary k-algebra structures on a vector space V together with the operation of GlV (k) by transport of structure. In this context we say that an algebra Λ1 is a deformation of the algebra Λ0 if the corresponding structures λ1, λ0 are elements of algV (k) and λ0 lies in the closure of the GlV (k)-orbit of λ1. In [11] it was shown, us- ing Drozd’s tame-wild theorem, that deformations of tame algebras are tame. Similar results may be expected for other classes of problems where Drozd’s theorem is valid. In this paper we address the case of bimodule problems with trivial differential (in the sense of [4]); here we interpret defor- mations via the structure constants of the respective projective realizations. Note that the bimodule problems include as special cases the representa- tion theory of finite-dimensional algebras ([4, 2.2]), subspace problems (4.1) and prinjective modules ([16, 1]). We also discuss some examples concerning subspace problems. -
MATH 210A, FALL 2017 Question 1. Consider a Short Exact Sequence 0
MATH 210A, FALL 2017 HW 3 SOLUTIONS WRITTEN BY DAN DORE, EDITS BY PROF.CHURCH (If you find any errors, please email [email protected]) α β Question 1. Consider a short exact sequence 0 ! A −! B −! C ! 0. Prove that the following are equivalent. (A) There exists a homomorphism σ : C ! B such that β ◦ σ = idC . (B) There exists a homomorphism τ : B ! A such that τ ◦ α = idA. (C) There exists an isomorphism ': B ! A ⊕ C under which α corresponds to the inclusion A,! A ⊕ C and β corresponds to the projection A ⊕ C C. α β When these equivalent conditions hold, we say the short exact sequence 0 ! A −! B −! C ! 0 splits. We can also equivalently say “β : B ! C splits” (since by (i) this only depends on β) or “α: A ! B splits” (by (ii)). Solution. (A) =) (B): Let σ : C ! B be such that β ◦ σ = idC . Then we can define a homomorphism P : B ! B by P = idB −σ ◦ β. We should think of this as a projection onto A, in that P maps B into the submodule A and it is the identity on A (which is exactly what we’re trying to prove). Equivalently, we must show P ◦ P = P and im(P ) = A. It’s often useful to recognize the fact that a submodule A ⊆ B is a direct summand iff there is such a projection. Now, let’s prove that P is a projection. We have β ◦ P = β − β ◦ σ ◦ β = β − idC ◦β = 0. Thus, by the universal property of the kernel (as developed in HW2), P factors through the kernel α: A ! B. -
Bimodule Categories and Monoidal 2-Structure Justin Greenough University of New Hampshire, Durham
University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Fall 2010 Bimodule categories and monoidal 2-structure Justin Greenough University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Greenough, Justin, "Bimodule categories and monoidal 2-structure" (2010). Doctoral Dissertations. 532. https://scholars.unh.edu/dissertation/532 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. BIMODULE CATEGORIES AND MONOIDAL 2-STRUCTURE BY JUSTIN GREENOUGH B.S., University of Alaska, Anchorage, 2003 M.S., University of New Hampshire, 2008 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics September, 2010 UMI Number: 3430786 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMT Dissertation Publishing UMI 3430786 Copyright 2010 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 This thesis has been examined and approved. -
Extensions of a Ring by a Ring with a Bimodule Structure
EXTENSIONS OF A RING BY A RING WITH A BIMODULE STRUCTURE CARL W. KOHLS Abstract. A type of ring extension is considered that was intro- duced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings. This paper is a sequel to [2 ], in which a class of examples was given of a type of ring extension defined in terms of two functions. Here we exhibit some other functions that may be used to construct such extensions, and show that in certain cases (in particular, when the first ring is an integral domain and the second is a commutative ring with identity), the functions must have a prescribed form. We also characterize this type of ring extension, which Szendrei defined di- rectly by the ring operations, in terms of the manner in which the two given rings are embedded in the extension. The reader is referred to [2] for background. Only the basic defini- tion is repeated here. Let A and B be rings. We define the ring A*B to be the direct sum of A and B as additive groups, with multiplication given by (1) (a,b)(c,d) = (ac+ {b,d},aad + b<rc+ bd), where a is a homomorphism from A onto a ring of permutable bimul- tiplications of B, and { •, • } is a biadditive function from BXB into A satisfying the equations (2) bo-{c,d) = o-[b.c)d, (3) {b,cd\ = {bc,d\, (4) {b,aac\ = {baa,c\, (5) \o-ab,c} = a{b,c}, and {&,c<ra} = \b,c\a, for all a£^4 and all b, c, dEB.