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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. -
Two Constructions in Monoidal Categories Equivariantly Extended Drinfel’D Centers and Partially Dualized Hopf Algebras
Two constructions in monoidal categories Equivariantly extended Drinfel'd Centers and Partially dualized Hopf Algebras Dissertation zur Erlangung des Doktorgrades an der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften Fachbereich Mathematik der Universit¨atHamburg vorgelegt von Alexander Barvels Hamburg, 2014 Tag der Disputation: 02.07.2014 Folgende Gutachter empfehlen die Annahme der Dissertation: Prof. Dr. Christoph Schweigert und Prof. Dr. Sonia Natale Contents Introduction iii Topological field theories and generalizations . iii Extending braided categories . vii Algebraic structures and monoidal categories . ix Outline . .x 1. Algebra in monoidal categories 1 1.1. Conventions and notations . .1 1.2. Categories of modules . .3 1.3. Bialgebras and Hopf algebras . 12 2. Yetter-Drinfel'd modules 25 2.1. Definitions . 25 2.2. Equivalences of Yetter-Drinfel'd categories . 31 3. Graded categories and group actions 39 3.1. Graded categories and (co)graded bialgebras . 39 3.2. Weak group actions . 41 3.3. Equivariant categories and braidings . 48 4. Equivariant Drinfel'd center 51 4.1. Half-braidings . 51 4.2. The main construction . 55 4.3. The Hopf algebra case . 61 5. Partial dualization of Hopf algebras 71 5.1. Radford biproduct and projection theorem . 71 5.2. The partial dual . 73 5.3. Examples . 75 A. Category theory 89 A.1. Basic notions . 89 A.2. Adjunctions and monads . 91 i ii Contents A.3. Monoidal categories . 92 A.4. Modular categories . 97 References 99 Introduction The fruitful interplay between topology and algebra has a long tradi- tion. On one hand, invariants of topological spaces, such as the homotopy groups, homology groups, etc. -
Categories of Modules with Differentials
JOURNAL OF ALGEBRA 185, 50]73Ž. 1996 ARTICLE NO. 0312 Categories of Modules with Differentials Paul Popescu Department of Mathematics, Uni¨ersity of Craio¨a, 13, A.I. Cuza st., Craio¨a, 1100, Romania Communicated by Walter Feit CORE Metadata, citation and similar papers at core.ac.uk Received August 1, 1994 Provided by Elsevier - Publisher Connector 1. INTRODUCTION The definitions of the module with arrow Ž.module fleche , the infinitesi- mal module, and the Lie pseudoalgebra, as used here, are considered inwx 7 . Moreover, we define the preinfinitesimal module, called inwx 1 ``un pre- espace d'Elie Cartan regulier.'' Inwx 7 the Lie functor is constructed from the category of differentiable groupoids in the category of Lie algebroids, but it is inwx 2 that the first general and abstract treatment of the algebraic properties of Lie alge- broids is made, giving a clear construction of the morphisms of Lie algebroids. We shall use it fully in this paper to define completely the categories M W A Ž.Ž.modules and arrows , P I M preinfinitesimal modules , IMŽ.Ž.infinitesimal modules , and L P A Lie pseudoalgebras ; we call these categories the categories of modules with differentials. We notice that in wx7 and wx 1 the objects of these categories and the subcategories of modules over a fixed algebra are considered. Inwx 3 the category of L P A of Lie pseudoalgebras is defined and some aspects related to the functional covariance or contravariance correspondence with categories constructed with additional structures on vector bundles are studied. In Section 2 we give a brief description of the category MA , of modules over commutative k-algebrasŽ considered also inwx 3. -
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. -
The Elliptic Drinfeld Center of a Premodular Category Arxiv
The Elliptic Drinfeld Center of a Premodular Category Ying Hong Tham Abstract Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X; λ), where X ∈ Obj(C) and λ is a el half-braiding. For a premodular category C, we construct a new category Z (C) which 1 2 i we call the Elliptic Drinfeld Center, which has objects (X; λ ; λ ), where the λ 's are half-braidings that satisfy some compatibility conditions. We discuss an SL2(Z)-action el on Z (C) that is related to the anomaly appearing in Reshetikhin-Turaev theory. This construction is motivated from the study of the extended Crane-Yetter TQFT, in particular the category associated to the once punctured torus. 1 Introduction and Preliminaries In [CY1993], Crane and Yetter define a 4d TQFT using a state-sum involving 15j symbols, based on a sketch by Ooguri [Oog1992]. The state-sum begins with a color- ing of the 2- and 3-simplices of a triangulation of the four manifold by integers from 0; 1; : : : ; r. These 15j symbols then arise as the evaluation of a ribbon graph living on the boundary of a 4-simplex. The labels 0; 1; : : : ; r correspond to simple objects of the Verlinde modular category, the semi-simple subquotient of the category of finite dimen- πi r 2 sional representations of the quantum group Uqsl2 at q = e as defined in [AP1995]. Later Crane, Kauffman, and Yetter [CKY1997] extend this~ definition+ to colorings with objects from a premodular category (i.e. -
Stable Model Categories Are Categories of Modules 1
STABLE MODEL CATEGORIES ARE CATEGORIES OF MODULES STEFAN SCHWEDE AND BROOKE SHIPLEY Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1. Introduction The recent discovery of highly structured categories of spectra has opened the way for a new wholesale use of algebra in stable homotopy theory. In this paper we use this new algebra of spectra to characterize stable model categories, the settings for doing stable homotopy theory, as categories of highly structured modules. This characterization also leads to a Morita theory for equivalences between categories of highly structured modules. -
Mel Hochster's Lecture Notes
MATH 614 LECTURE NOTES, FALL, 2017 by Mel Hochster Lecture of September 6 We assume familiarity with the notions of ring, ideal, module, and with the polynomial ring in one or finitely many variables over a commutative ring, as well as with homomor- phisms of rings and homomorphisms of R-modules over the ring R. As a matter of notation, N ⊆ Z ⊆ Q ⊆ R ⊆ C are the non-negative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively, throughout this course. Unless otherwise specified, all rings are commutative, associative, and have a multiplica- tive identity 1 (when precision is needed we write 1R for the identity in the ring R). It is possible that 1 = 0, in which case the ring is f0g, since for every r 2 R, r = r ·1 = r ·0 = 0. We shall assume that a homomorphism h of rings R ! S preserves the identity, i.e., that h(1R) = 1S. We shall also assume that all given modules M over a ring R are unital, i.e., that 1R · m = m for all m 2 M. When R and S are rings we write S = R[θ1; : : : ; θn] to mean that S is generated as a ring over its subring R by the elements θ1; : : : ; θn. This means that S contains R and the elements θ1; : : : ; θn, and that no strictly smaller subring of S contains R and the θ1; : : : ; θn. It also means that every element of S can be written (not necessarily k1 kn uniquely) as an R-linear combination of the monomials θ1 ··· θn . -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
Lifting Defects for Nonstable K 0-Theory of Exchange Rings And
LIFTING DEFECTS FOR NONSTABLE K0-THEORY OF EXCHANGE RINGS AND C*-ALGEBRAS FRIEDRICH WEHRUNG Abstract. The assignment (nonstable K0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (i) There is no functor Γ, from simplicial monoids with order-unit with nor- malized positive homomorphisms to exchange rings, such that V ◦ Γ =∼ id. (ii) There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V ◦ Γ =∼ id. 3 (iii) There is a {0, 1} -indexed commutative diagram D~ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality ℵ3 (resp., an ℵ3-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group but it is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring. -
Arxiv:Math/0310146V1 [Math.AT] 10 Oct 2003 Usinis: Question Fr Most Aut the the of Algebra”
MORITA THEORY IN ABELIAN, DERIVED AND STABLE MODEL CATEGORIES STEFAN SCHWEDE These notes are based on lectures given at the Workshop on Structured ring spectra and their applications. This workshop took place January 21-25, 2002, at the University of Glasgow and was organized by Andy Baker and Birgit Richter. Contents 1. Introduction 1 2. Morita theory in abelian categories 2 3. Morita theory in derived categories 6 3.1. The derived category 6 3.2. Derived equivalences after Rickard and Keller 14 3.3. Examples 19 4. Morita theory in stable model categories 21 4.1. Stable model categories 22 4.2. Symmetric ring and module spectra 25 4.3. Characterizing module categories over ring spectra 32 4.4. Morita context for ring spectra 35 4.5. Examples 38 References 42 1. Introduction The paper [Mo58] by Kiiti Morita seems to be the first systematic study of equivalences between module categories. Morita treats both contravariant equivalences (which he calls arXiv:math/0310146v1 [math.AT] 10 Oct 2003 dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’). The term ‘Morita theory’ is now used for results concerning equivalences of various kinds of module categories. The authors of the obituary article [AGH] consider Morita’s theorem “probably one of the most frequently used single results in modern algebra”. In this survey article, we focus on the covariant form of Morita theory, so our basic question is: When do two ‘rings’ have ‘equivalent’ module categories ? We discuss this question in different contexts: • (Classical) When are the module categories of two rings equivalent as categories ? Date: February 1, 2008. -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
Derived Categories of Modules and Coherent Sheaves
Derived Categories of Modules and Coherent Sheaves Yuriy A. Drozd Abstract We present recent results on derived categories of modules and coherent sheaves, namely, tame{wild dichotomy and semi-continuity theorem for derived categories over finite dimensional algebras, as well as explicit calculations for derived categories of modules over nodal rings and of coherent sheaves over projective configurations of types A and A~. This paper is a survey of some recent results on the structure of derived categories obtained by the author in collaboration with Viktor Bekkert and Igor Burban [6, 11, 12]. The origin of this research was the study of Cohen{ Macaulay modules and vector bundles by Gert-Martin Greuel and myself [27, 28, 29, 30] and some ideas from the work of Huisgen-Zimmermann and Saor´ın [42]. Namely, I understood that the technique of \matrix problems," briefly explained below in subsection 2.3, could be successfully applied to the calculations in derived categories, almost in the same way as it was used in the representation theory of finite-dimensional algebras, in study of Cohen{ Macaulay modules, etc. The first step in this direction was the semi-continuity theorem for derived categories [26] presented in subsection 2.1. Then Bekkert and I proved the tame{wild dichotomy for derived categories over finite di- mensional algebras (see subsection 2.2). At the same time, Burban and I described the indecomposable objects in the derived categories over nodal rings (see Section 3) and projective configurations of types A and A~ (see Sec- tion 4). Note that it follows from [23, 29] that these are the only cases, where such a classification is possible; for all other pure noetherian rings (or projec- tive curves) even the categories of modules (respectively, of vector bundles) are wild.