Rings, Modules, and the Total, by Friedrich Kasch and Adolf Mader
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Sketch Notes — Rings and Fields
Sketch Notes — Rings and Fields Neil Donaldson Fall 2018 Text • An Introduction to Abstract Algebra, John Fraleigh, 7th Ed 2003, Adison–Wesley (optional). Brief reminder of groups You should be familiar with the majority of what follows though there is a lot of time to remind yourself of the harder material! Try to complete the proofs of any results yourself. Definition. A binary structure (G, ·) is a set G together with a function · : G × G ! G. We say that G is closed under · and typically write · as juxtaposition.1 A semigroup is an associative binary structure: 8x, y, z 2 G, x(yz) = (xy)z A monoid is a semigroup with an identity element: 9e 2 G such that 8x 2 G, ex = xe = x A group is a monoid in which every element has an inverse: 8x 2 G, 9x−1 2 G such that xx−1 = x−1x = e A binary structure is commutative if 8x, y 2 G, xy = yx. A group with a commutative structure is termed abelian. A subgroup2 is a non-empty subset H ⊆ G which remains a group under the same binary operation. We write H ≤ G. Lemma. H is a subgroup of G if and only if it is a non-empty subset of G closed under multiplication and inverses in G. Standard examples of groups: sets of numbers under addition (Z, Q, R, nZ, etc.), matrix groups. Standard families: cyclic, symmetric, alternating, dihedral. 1You should be comfortable with both multiplicative and additive notation. 2More generally any substructure. 1 Cosets and Factor Groups Definition. -
Morita Equivalence and Generalized Kähler Geometry by Francis
Morita Equivalence and Generalized Kahler¨ Geometry by Francis Bischoff A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2019 by Francis Bischoff Abstract Morita Equivalence and Generalized K¨ahlerGeometry Francis Bischoff Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2019 Generalized K¨ahler(GK) geometry is a generalization of K¨ahlergeometry, which arises in the study of super-symmetric sigma models in physics. In this thesis, we solve the problem of determining the underlying degrees of freedom for the class of GK structures of symplectic type. This is achieved by giving a reformulation of the geometry whereby it is represented by a pair of holomorphic Poisson structures, a holomorphic symplectic Morita equivalence relating them, and a Lagrangian brane inside of the Morita equivalence. We apply this reformulation to solve the longstanding problem of representing the metric of a GK structure in terms of a real-valued potential function. This generalizes the situation in K¨ahlergeometry, where the metric can be expressed in terms of the partial derivatives of a function. This result relies on the fact that the metric of a GK structure corresponds to a Lagrangian brane, which can be represented via the method of generating functions. We then apply this result to give new constructions of GK structures, including examples on toric surfaces. Next, we study the Picard group of a holomorphic Poisson structure, and explore its relationship to GK geometry. We then apply our results to the deformation theory of GK structures, and explain how a GK metric can be deformed by flowing the Lagrangian brane along a Hamiltonian vector field. -
Classification of Finite Abelian Groups
Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k. -
Arxiv:2001.09075V1 [Math.AG] 24 Jan 2020
A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address: [email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi. -
BLECHER • All Numbers Below Are Integers, Usually Positive. • Read Up
MATH 4389{ALGEBRA `CHEATSHEET'{BLECHER • All numbers below are integers, usually positive. • Read up on e.g. wikipedia on the Euclidean algorithm. • Diophantine equation ax + by = c has solutions iff gcd(a; b) doesn't divide c. One may find the solutions using the Euclidean algorithm (google this). • Euclid's lemma: prime pjab implies pja or pjb. • gcd(m; n) · lcm(m; n) = mn. • a ≡ b (mod m) means mj(a − b). An equivalence relation (reflexive, symmetric, transitive) on the integers. • ax ≡ b (mod m) has a solution iff gcd(m; a)jb. • Division Algorithm: there exist unique integers q and r with a = bq + r and 0 ≤ r < b; so a ≡ r (mod b). • Fermat's theorem: If p is prime then ap ≡ a (mod p) Also, ap−1 ≡ 1 (mod p) if p does not divide a. • For definitions for groups, rings, fields, subgroups, order of element, etc see Algebra fact sheet" below. • Zm = f0; 1; ··· ; m−1g with addition and multiplication mod m, is a group and commutative ring (= Z =m Z) • Order of n in Zm (i.e. smallest k s.t. kn ≡ 0 (mod m)) is m=gcd(m; n). • hai denotes the subgroup generated by a (the smallest subgroup containing a). For example h2i = f0; 2; 2+2 = 4; 2 + 2 + 2 = 6g in Z8. (In ring theory it may be the subring generated by a). • A cyclic group is a group that can be generated by a single element (the group generator). They are abelian. • A finite group is cyclic iff G = fg; g + g; ··· ; g + g + ··· + g = 0g; so G = hgi. -
Cohomology Theory of Lie Groups and Lie Algebras
COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf. -
Lecture 1.3: Direct Products and Sums
Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise. -
Problems from Ring Theory
Home Page Problems From Ring Theory In the problems below, Z, Q, R, and C denote respectively the rings of integers, JJ II rational numbers, real numbers, and complex numbers. R generally denotes a ring, and I and J usually denote ideals. R[x],R[x, y],... denote rings of polynomials. rad R is defined to be {r ∈ R : rn = 0 for some positive integer n} , where R is a ring; rad R is called the nil radical or just the radical of R . J I Problem 0. Let b be a nilpotent element of the ring R . Prove that 1 + b is an invertible element Page 1 of 14 of R . Problem 1. Let R be a ring with more than one element such that aR = R for every nonzero Go Back element a ∈ R. Prove that R is a division ring. Problem 2. If (m, n) = 1 , show that the ring Z/(mn) contains at least two idempotents other than Full Screen the zero and the unit. Problem 3. If a and b are elements of a commutative ring with identity such that a is invertible Print and b is nilpotent, then a + b is invertible. Problem 4. Let R be a ring which has no nonzero nilpotent elements. Prove that every idempotent Close element of R commutes with every element of R. Quit Home Page Problem 5. Let A be a division ring, B be a proper subring of A such that a−1Ba ⊆ B for all a 6= 0 . Prove that B is contained in the center of A . -
On Semilattice Structure of Mizar Types
FORMALIZED MATHEMATICS Volume 11, Number 4, 2003 University of Białystok On Semilattice Structure of Mizar Types Grzegorz Bancerek Białystok Technical University Summary. The aim of this paper is to develop a formal theory of Mizar types. The presented theory is an approach to the structure of Mizar types as a sup-semilattice with widening (subtyping) relation as the order. It is an abstrac- tion from the existing implementation of the Mizar verifier and formalization of the ideas from [9]. MML Identifier: ABCMIZ 0. The articles [20], [14], [24], [26], [23], [25], [3], [21], [1], [11], [12], [16], [10], [13], [18], [15], [4], [2], [19], [22], [5], [6], [7], [8], and [17] provide the terminology and notation for this paper. 1. Semilattice of Widening Let us mention that every non empty relational structure which is trivial and reflexive is also complete. Let T be a relational structure. A type of T is an element of T . Let T be a relational structure. We say that T is Noetherian if and only if: (Def. 1) The internal relation of T is reversely well founded. Let us observe that every non empty relational structure which is trivial is also Noetherian. Let T be a non empty relational structure. Let us observe that T is Noethe- rian if and only if the condition (Def. 2) is satisfied. (Def. 2) Let A be a non empty subset of T . Then there exists an element a of T such that a ∈ A and for every element b of T such that b ∈ A holds a 6< b. -
Morita Equivalence and Continuous-Trace C*-Algebras, 1998 59 Paul Howard and Jean E
http://dx.doi.org/10.1090/surv/060 Selected Titles in This Series 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R.