Multiplicative Theory of Ideals This Is Volume 43 in PURE and APPLIED MATHEMATICS a Series of Monographs and Textbooks Editors: PAULA
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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]. -
Open Problems
Open Problems 1. Does the lower radical construction for `-rings stop at the first infinite ordinal (Theorems 3.2.5 and 3.2.8)? This is the case for rings; see [ADS]. 2. If P is a radical class of `-rings and A is an `-ideal of R, is P(A) an `-ideal of R (Theorem 3.2.16 and Exercises 3.2.24 and 3.2.25)? 3. (Diem [DI]) Is an sp-`-prime `-ring an `-domain (Theorem 3.7.3)? 4. If R is a torsion-free `-prime `-algebra over the totally ordered domain C and p(x) 2C[x]nC with p0(1) > 0 in C and p(a) ¸ 0 for every a 2 R, is R an `-domain (Theorems 3.8.3 and 3.8.4)? 5. Does each f -algebra over a commutative directed po-unital po-ring C whose left and right annihilator ideals vanish have a unique tight C-unital cover (Theorems 3.4.5, 3.4.11, and 3.4.12 and Exercises 3.4.16 and 3.4.18)? 6. Can an f -ring contain a principal `-idempotent `-ideal that is not generated by an upperpotent element (Theorem 3.4.4 and Exercise 3.4.15)? 7. Is each C-unital cover of a C-unitable (totally ordered) f -algebra (totally or- dered) tight (Theorem 3.4.10)? 8. Is an archimedean `-group a nonsingular module over its f -ring of f -endomorphisms (Exercise 4.3.56)? 9. If an archimedean `-group is a finitely generated module over its ring of f -endomorphisms, is it a cyclic module (Exercise 3.6.20)? 10. -
Quantizations of Regular Functions on Nilpotent Orbits 3
QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS IVAN LOSEV Abstract. We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa. 1. Introduction 1.1. Nilpotent orbits and their quantizations. Let G be a connected semisimple algebraic group over C and let g be its Lie algebra. Pick a nilpotent orbit O ⊂ g. This orbit is a symplectic algebraic variety with respect to the Kirillov-Kostant form. So the algebra C[O] of regular functions on O acquires a Poisson bracket. This algebra is also naturally graded and the Poisson bracket has degree −1. So one can ask about quantizations of O, i.e., filtered algebras A equipped with an isomorphism gr A −→∼ C[O] of graded Poisson algebras. We are actually interested in quantizations that have some additional structures mir- roring those of O. Namely, the group G acts on O and the inclusion O ֒→ g is a moment map for this action. We want the G-action on C[O] to lift to a filtration preserving action on A. -
Algebraic Numbers
FORMALIZED MATHEMATICS Vol. 24, No. 4, Pages 291–299, 2016 DOI: 10.1515/forma-2016-0025 degruyter.com/view/j/forma Algebraic Numbers Yasushige Watase Suginami-ku Matsunoki 6 3-21 Tokyo, Japan Summary. This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field Q induced by substitution of an algebraic number to the polynomial ring of Q[x] turns to be a field. MSC: 11R04 13B21 03B35 Keywords: algebraic number; integral dependency MML identifier: ALGNUM 1, version: 8.1.05 5.39.1282 1. Preliminaries From now on i, j denote natural numbers and A, B denote rings. Now we state the propositions: (1) Let us consider rings L1, L2, L3. Suppose L1 is a subring of L2 and L2 is a subring of L3. Then L1 is a subring of L3. (2) FQ is a subfield of CF. (3) FQ is a subring of CF. R (4) Z is a subring of CF. Let us consider elements x, y of B and elements x1, y1 of A. Now we state the propositions: (5) If A is a subring of B and x = x1 and y = y1, then x + y = x1 + y1. (6) If A is a subring of B and x = x1 and y = y1, then x · y = x1 · y1. -
7902127 Fordt DAVID JAKES Oni the COMPUTATION of THE
7902127 FORDt DAVID JAKES ON i THE COMPUTATION OF THE MAXIMAL ORDER IK A DEDEKIND DOMAIN. THE OHIO STATE UNIVERSITY, PH.D*, 1978 University. Microfilms International 3q o n . z e e b r o a o . a n n a r b o r , mi 48ios ON THE COMPUTATION OF THE MAXIMAL ORDER IN A DEDEKIND DOMAIN DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By David James Ford, B.S., M.S. # * * K * The Ohio State University 1978 Reading Committee: Approved By Paul Ponomarev Jerome Rothstein Hans Zassenhaus Hans Zassenhaus Department of Mathematics ACKNOWLDEGMENTS I must first acknowledge the help of my Adviser, Professor Hans Zassenhaus, who was a fountain of inspiration whenever I was dry. I must also acknowlegde the help, financial and otherwise, that my parents gave me. Without it I could not have finished this work. All of the computing in this project, amounting to between one and two thousand hours of computer time, was done on a PDP-11 belonging to Drake and Ford Engineers, through the generosity of my father. VITA October 28, 1946.......... Born - Columbus, Ohio 1967...................... B.S. in Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 1967-196 8 ................. Teaching Assistant Department of Mathematics The Ohio State University Columbus, Ohio 1968-197 1................. Programmer Children's Hospital Medical Center Boston, Massachusetts 1971-1975................. Teaching Associate Department of Mathematics The Ohio State University Columbus, Ohio 1975-1978................. Programmer Prindle and Patrick, Architects Columbus, Ohio PUBLICATIONS "Relations in Q (Zh X ZQ) and Q (ZD X ZD)" (with Michael Singer) n 4 o n o o Communications in Algebra, Vol. -
Norms and Traces MAT4250 — Høst 2013 Norms and Traces
Norms and traces MAT4250 — Høst 2013 Norms and traces Version 0.2 — 10. september 2013 klokken 09:55 We start by recalling a few facts from linear algebra that will be very useful for us, and we shall do it over a general base field K (of any characteristic). Assume that V is afinitedimensionalvectorspaceoverK.Toanylinearoperatorφ: V V there are ! associated two polynomials, both in a canonical way. Firstly, there is the characteristic polynomial Pφ(t), which is, as we know, defined by Pφ(t)=det(t id φ). It is a monic polynomial whose degree n equals the dimension · − dimK V .TwoofthecoefficientsofPφ(t) play a special prominent role, the trace and the determinant. They are up to sign, respectively the coefficient of the subdominant term and the constant term of the characteristic polynomial Pφ(t): n n 1 n P (t)=t tr φt − + +( 1) det φ φ − ··· − Secondly, there is the minimal polynomial of φ.Itisdefinedasthemonicgenerator of the annihilator of φ,thatisthemonicgeneratoroftheidealIφ = f(t) K[t] { 2 | f(φ)=0 . The two polynomials arise in very different ways, but they are intimately } related. The Cayley-Hamilton theorem states that Pφ(φ)=0,sothecharacteristic polynomial of φ belongs to the ideal Iφ,andtheminimalpolynomialdividesthecha- racteristic polynomial. In general they are different. For example, the degree of the characteristic poly is always equal to dimK V ,butthedegreeoftheminimalpolycan be anything between one and dimK V (degree one occurring for scalar multiples of the identity map). We often will work over extensions ⌦ of the base field K.AvectorspaceV and alinearoperatorφ may accordingly be extended as the tensor products V K ⌦ and ⌦ φ idK .Ifv1,...,vn is a basis for V ,thenv1 1,...,vn 1 will be a basis for V K ⌦— ⌦ ⌦ ⌦ ⌦ the only thing that changes, is that we are allowed to use elements from ⌦ as coefficients. -
Notes on Ring Theory
Notes on Ring Theory by Avinash Sathaye, Professor of Mathematics February 1, 2007 Contents 1 1 Ring axioms and definitions. Definition: Ring We define a ring to be a non empty set R together with two binary operations f,g : R × R ⇒ R such that: 1. R is an abelian group under the operation f. 2. The operation g is associative, i.e. g(g(x, y),z)=g(x, g(y,z)) for all x, y, z ∈ R. 3. The operation g is distributive over f. This means: g(f(x, y),z)=f(g(x, z),g(y,z)) and g(x, f(y,z)) = f(g(x, y),g(x, z)) for all x, y, z ∈ R. Further we define the following natural concepts. 1. Definition: Commutative ring. If the operation g is also commu- tative, then we say that R is a commutative ring. 2. Definition: Ring with identity. If the operation g has a two sided identity then we call it the identity of the ring. If it exists, the ring is said to have an identity. 3. The zero ring. A trivial example of a ring consists of a single element x with both operations trivial. Such a ring leads to pathologies in many of the concepts discussed below and it is prudent to assume that our ring is not such a singleton ring. It is called the “zero ring”, since the unique element is denoted by 0 as per convention below. Warning: We shall always assume that our ring under discussion is not a zero ring. -
ON SUPPORTS and ASSOCIATED PRIMES of MODULES OVER the ENVELOPING ALGEBRAS of NILPOTENT LIE ALGEBRAS Introduction Let G Be a Conn
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 6, Pages 2131{2170 S 0002-9947(01)02741-6 Article electronically published on January 3, 2001 ON SUPPORTS AND ASSOCIATED PRIMES OF MODULES OVER THE ENVELOPING ALGEBRAS OF NILPOTENT LIE ALGEBRAS BORIS SIROLAˇ Abstract. Let n be a nilpotent Lie algebra, over a field of characteristic zero, and U its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of U related to finitely generated U-modules V ,andin particular the set Ass V of associated primes for such V (note that now Ass V is equal to the set Annspec V of annihilator primes for V ); (2) the problem of nontriviality for the modules V=PV when P is a (maximal) prime of U,andin particular when P is the augmentation ideal Un of U. We define the support of V , as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set Ass V . Weprovethefollowing generalization of a stability result given by W. Casselman and M. S. Osborne inthecasewhenN, N as in the theorem, are abelian. We also present some of its interesting consequences. Theorem. Let Q be a finite-dimensional Lie algebra over a field of char- acteristic zero, and N an ideal of Q;denotebyU(N) the universal enveloping algebra of N.LetV be a Q-module which is finitely generated as an N-module. -
P-INTEGRAL BASES of a CUBIC FIELD 1. Introduction Let K = Q(Θ
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 1949{1953 S 0002-9939(98)04422-0 p-INTEGRAL BASES OF A CUBIC FIELD S¸ABAN ALACA (Communicated by William W. Adams) Abstract. A p-integral basis of a cubic field K is determined for each rational prime p, and then an integral basis of K and its discriminant d(K)areobtained from its p-integral bases. 1. Introduction Let K = Q(θ) be an algebraic number field of degree n, and let OK denote the ring of integral elements of K.IfOK=α1Z+α2Z+ +αnZ,then α1,α2,...,αn is said to be an integral basis of K. For each prime··· ideal P and each{ nonzero ideal} A of K, νP (A) denotes the exponent of P in the prime ideal decomposition of A. Let P be a prime ideal of K,letpbe a rational prime, and let α K.If ∈ νP(α) 0, then α is called a P -integral element of K.Ifαis P -integral for each prime≥ ideal P of K such that P pO ,thenαis called a p-integral element | K of K.Let!1;!2;:::;!n be a basis of K over Q,whereeach!i (1 i n)isap-integral{ element} of K.Ifeveryp-integral element α of K is given≤ as≤ α = a ! + a ! + +a ! ,wherethea are p-integral elements of Q,then 1 1 2 2 ··· n n i !1;!2;:::;!n is called a p-integral basis of K. { In Theorem} 2.1 a p-integral basis of a cubic field K is determined for every rational prime p, and in Theorem 2.2 an integral basis of K is obtained from its p-integral bases. -
On Finiteness Conjectures for Endomorphism Algebras of Abelian Surfaces
ON FINITENESS CONJECTURES FOR ENDOMORPHISM ALGEBRAS OF ABELIAN SURFACES NILS BRUIN, E. VICTOR FLYNN, JOSEP GONZALEZ,´ AND VICTOR ROTGER Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL2- type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves. 1. Introduction Let A/K be an abelian variety defined over a number field K. For any field extension L/K, let EndL(A) denote the ring of endomorphisms of A defined over 0 L and EndL(A) = EndL(A) ⊗ Q. A conjecture, which may be attributed to Robert Coleman1, asserts the following. Conjecture C(e, g): Let e, g ≥ 1 be positive integers. Then, up to isomorphism, there exist only finitely many rings O over Z such that EndL(A) 'O for some abelian variety A/K of dimension g and a field extension L/K of a number field K of degree [K : Q] ≤ e. The conjecture holds in dimension 1: by the theory of complex multiplication, if E/K is an elliptic curve and L/K is an extension of number√ fields, then EndL(E) is either Z or an order O in an imaginary quadratic√ field Q( −d) such that the ring class field HO attached to O is contained in K( −d). -
Introduction to Algebraic Number Theory Lecture 3
Introduction to Algebraic Number Theory Lecture 3 Andrei Jorza 2014-01-20 Today: more number rings; traces and norms. Textbook here is http://wstein.org/books/ant/ant.pdf 2 Number Rings (continued) (2.1) We have shown that for a number field K the algebraic integers OK form a ring. Definition 1. An order is a subring O ⊂ OK such that OK =O is finite. The ring of integers is said to be the maximal order. Some examples later. (2.2) Having shown that for a number field K the algebraic integers OK form a ring we should answer some natural questions: 1. Is OK torsion-free? Of course, since K is. 2. Is OK a finite Z-module? We know that every Z[α] ⊂ OK is finite over Z and the question is whether OK is generated by finitely many algebraic integers. 3. A finite Z-module is just a finitely generated abelian group and once we show that OK is finite over Z ∼ d and torsion-free we deduce that OK = Z for d = rank(OK ). What is this rank? 4. Can we find generators for OK as a Z-module? (2.3) p p Example 2. If m is a square-free integer not equal to 1 then the ring of integers of ( m) is [ m] when p Q Z 1+ m m ≡ 2; 3 (mod 4) and Z[ 2 ] when m ≡ 1 (mod 4). p 2 2 2 Proof. If a + b m 2 OK then the minimal polynomial X − 2aX + a − b m 2 Z[X] and so 2a = p 2 Z. -
Appendix D: Integrality and Algebraic Integers
Skript zur Vorlesung: Charaktertheorie SS 2020 69 Example 16 ˆ E.g. the tensor product of two 2 2-matrices is of the form » fi b “ P p q „ ⇢ „ ⇢ — ffi 4 — ffi M K – fl Lemma-Definition›Ñ C.4 (Tensor› productÑ of K-endomorphisms) 1 2 b b If : V V and : W W are two endomorphisms of finite-dimensional K -vector spaces V 1 2 1 2 K and W , then the tensorb productbof and›Ñ is theb K -endomorphism of V W defined by 1 2 bK fiÑ p bK qp b q “ p qb p q : V W V W 1 2 1 2 p b q“ p q p q : 1 2 1 2 Furthermore, Tr Tr Tr b. “ t u “t u 1 2 V Proof : It is straightforward to check that is K -linear. Moreover, choosing ordered bases B 1 W 1 b b “t b | § § § § u and B of V and W respectively,K it is straightforward from the definitions 1 2 V W to check that the matrix of w.r.t. the ordered basis B 1 1 is 1 V 2 W the Kronecker product of the matrices of w.r.t. B and of w.r.t. to B . The trace formula follows. D Integrality and Algebraic Integers Z We recall/introduce here some notions of the Commutative Algebra lecture on integrality of ring ele- ments. However, we are essentially interested in the field of complex numbers and its subring . Definition D.1 (integral element, algebraic integer) Let A be a subring ofP a commutative ring B.