DOCSLIB.ORG

Appendices a Contramodules Over Coalgebras Over Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendices a Contramodules Over Coalgebras Over Fields Appendices A Contramodules over Coalgebras over Fields Let C be a coassociative coalgebra with counit over a field k.Itiswellknown (see [82, Theorems 2.1.3 and 2.2.1] or [68, 5.1.1]) that C is the union of its finite- dimensional subcoalgebras and any C-comodule is a union of finite-dimensional comodules over finite-dimensional subcoalgebras of C. The dual assertion for C-contramodules is not true: for the most common of noncosemisimple infinite- dimensional coalgebras C there exist C-contramodules P such that the intersection of the images of Homk(C/U, P)inP over all finite-dimensional subcoalgebras U ⊂ C is nonzero. A weaker statement holds, however: if the contraaction map Homk(C/U, P) −−→ P is surjective for every finite-dimensional subcoalgebra U of C,thenP = 0. Besides, even though adic filtrations of contramodules are not in general separated, they are always complete. Using the related techniques we show that any contraflat C-con- tramodule is projective, generalizing the well-known result that any flat module over a finite-dimensional algebra is projective [7]. A.1 Counterexamples A.1.1 Let C be the coalgebra for which the dual algebra C∗ is isomorphic to the algebra of formal power series k[[x]]. Then a C-contramodule P can be equivalently defined as a k-vector space endowed with the following operation of summation of n sequences of vectors with formal coefficients x : for any elements p0, p1, ... in P, ∞ n an element of P denoted by n=0 x pn is defined. This operation should satisfy the following equations: ∞ ∞ ∞ n n n x (apn + bqn)=a x pn + b x qn n=0 n=0 n=0 for a, b ∈ k, pn, qn ∈ P (linearity); ∞ n x pn = p0 when p1 = p2 = ···=0 n=0 (counity); and ∞ ∞ ∞ i j n x x pij = x pij i=0 j=0 n=0 i+j=n for any pij ∈ P, i, j =0, 1,... (contraassociativity). In the latter equation, the interior summation sign in the right-hand side denotes the conventional finite sum 230 A. Contramodules over Coalgebras over Fields of elements of a vector space, while the three other summation signs refer to the contramodule infinite summation operation. The following examples of C-contramodules are revealing. Let E denote the free C-contramodule generated by the sequence of symbols e0, e1, . ; its elements ∞ can be represented as formal sums i=0 ai(x)ei,whereai(x) are formal power series in x such that ai(x) → 0 in the topology of k[[x]] as i →∞.LetF denote the free C-contramodule generated by the sequence of symbols f1, f2, ...; then C-contramodule homomorphisms from F to E correspond bijectively to sequences of elements of E that are images of the elements fi. We are interested in the map −→ i − ∞ g : F E sending fi to x ei e0; in other words, an element i=1 bi(x)fi of F ∞ i − ∞ is mapped to the element i=1 x bi(x)ei i=1 bi(x) e0. It is clear from this formula that the element e0 ∈ E does not belong to the image of g.LetP denote the cokernel of the morphism g and pi denote the images of the elements ei in P. n Then one has p0 = x pn in P; in other words, the element p0 belongs to the image of Homk(C/U, P) under the contraaction map Homk(C, P) −→ P for any finite-dimensional subcoalgebra U =(k[[x]]/xn)∗ of C. Now let E be the free C-contramodule generated by the symbols e1, e2, ..., −→ P denote the cokernel of the map g : F E sending fi to xiei,andpi denote the images of ei in P . Then the result of the contramodule infinite summation ∞ n n n=1 x pn is nonzero in P , even though every element x pn is equal to zero. Therefore, the contramodule summation operation cannot be understood as any kind of limit of finite partial sums. Actually, the C-contramodule P is just the n direct sum of the contramodules k[[x]]/x k[[ x]] over n = 1, 2, . in the category C ∞ n of -contramodules. Notice that the element n=1 x pn also belongs to the image of Homk(C/U, P )inP for any finite-dimensional subcoalgebra U ⊂ C. Remark. In the above notation, a C-contramodule structure on a k-vector space P is uniquely determined by the underlying structure of a module over the algebra of polynomials k[x]; the natural functor C-contra −→ k[x]-mod is fully faithful. ∞ n ∈ Indeed, for any p0, p1, ... in P the sequence qm = n=0 x pm+n P, m =0, 1, . is the unique solution of the system of equations qm = pm + xqm+1.The image of this functor is a full abelian subcategory closed under kernels, coker- nels, extensions, and infinite products; it consists of all k[x]-modules P such that i −1 D Extk[x](k[x, x ],P)=0fori = 0, 1. It follows that if is a coalgebra for which the dual algebra D∗ is isomorphic to a quotient algebra of the algebra of formal power series k[[x1,...,xm]] in a finite number of (commuting) variables by a closed ideal, then the natural functor D-contra −→ k[x1,...,xm]-mod is fully faithful. A.1.2 Now let us give an example of finite-dimensional (namely, two-dimen- sional) contramodule P over a coalgebra C such that the intersection of the images of Homk(C/U, P)inP is nonzero. Notice that for any coalgebra C there are natu- ral left C∗-module structures on any left C-comodule and any left C-contramodule; that is there are natural faithful functors C-comod −−→ C∗-mod and C-contra −−→ C∗-mod A.1. Counterexamples 231 (where C∗ is considered as an abstract algebra without any topology). The functor C-comod −→ C∗-mod is fully faithful, while the functor C-contra −→ C∗-mod is fully faithful on finite-dimensional contramodules. Let V be a vector space and C be the coalgebra such that the dual algebra ∗ ∗ C has the form ki2 ⊕ i2V i1 ⊕ ki1,wherei1 and i2 are idempotent elements with ∗ i1i2 = i2i1 =0andi1 + i2 =1.ThenleftC -modules are essentially pairs of k-vector spaces M1, M2 endowed with a map ∗ V ⊗k M1 −−→ M2, left C-comodules are pairs of vector spaces M1, M2 endowed with a map M1 −−→ V ⊗k M2, and left C-contramodules are pairs of vector spaces P1, P2 endowed with a map Homk(V,P1) −−→ P2. In particular, the functor C-contra −→ C∗-mod is not surjective on morphisms of infinite-dimensional objects, while the functor C-comod −→ C∗-mod is not surjec- tive on the isomorphism classes of finite-dimensional objects. (Neither is in general the functor C-contra −→ C∗-mod, as one can see in the example of an analogous coalgebra with three idempotent linear functions instead of two and three vector spaces instead of one; when k is a finite field and C is the countable direct sum of copies of the coalgebra k, there even exists a one-dimensional C∗-module which comes from no C-comodule or C-contramodule.) Let P be the C-contramodule with P1 = k = P2 corresponding to a linear function V ∗ −→ k coming from no element of V . Then the intersection of the images of Homk(C/U, P)inP over all finite-dimensional subcoalgebras U ⊂ C is equal to P2. More generally, for any coalgebra C any finite-dimensional left C-comodule M has a natural left C-contramodule structure given by the composition ∗ ∗ Homk(C, M) C ⊗k M −−→ C ⊗k C ⊗k M −−→ M of the map induced by the C-coaction in M and the map induced by the pairing ∗ C ⊗k C −→ k. The category of finite-dimensional left C-comodules is isomorphic to a full subcategory of the category of finite-dimensional left C-contramodules; a finite-dimensional C-contramodule comes from a C-comodule if and only if it comes from a contramodule over a finite-dimensional subcoalgebra of C.Wewillseebelow that every irreducible C-contramodule is a finite-dimensional contramodule over a finite-dimensional subcoalgebra of C; it follows that the above functor provides a bijective correspondence between irreducible left C-comodules and irreducible left C-contramodules. 232 A. Contramodules over Coalgebras over Fields Comparing the cobar complex for comodules with the bar complex for con- tramodules, one discovers that for any finite-dimensional left C-comodules L and M there is a natural isomorphism C,i i ∗∗ Ext (L, M) ExtC(L, M) . In other words, the Ext spaces between finite-dimensional C-comodules in the cate- gory of arbitrary C-contramodules are the completions of the Ext spaces in the cat- egory of finite-dimensional C-comodules with respect to the profinite-dimensional topology. A.2 Nakayama’s Lemma The exposition below is based on the structure theory of coalgebras, see [82, Chap- ters VIII–IX] or [68, Chapter 5]. A coalgebra is called cosimple if it has no nontrivial proper subcoalgebras. A coalgebra C is called cosemisimple if it is a union of finite-dimensional coalgebras dual to semisimple k-algebras, or equivalently, if the abelian category of (left or right) C-comodules is semisimple. Any cosemisimple coalgebra can be decomposed into an (infinite) direct sum of cosimple coalgebras in a unique way. For any coalgebra C,letCss denote its maximal cosemisimple subcoalgebra; it contains all other cosemisimple subcoalgebras of C.
Load more

Recommended publications
  • Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology SUBMITTED in PA
    J- Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology (1963) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY August, 1965 Signature of Author . .-. .. Department of Mathematics, August 31, 1965 Certified by ....-.. Thesis Supervisqr Accepted by .......... 0................................... Chairman, Departmental Committee on Graduate Students v/I 2. Cocommutative Hopf Algebras 19 with Antipode by Moss Eisenberg Sweedler Submitted to the Department of Mathematics on August 31, 1965, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract In the first chapter the preliminaries of the theory of Hopf algebras are presented. The notion and properties of the antipode are developed. An important filtration is induced in the Hopf algebra by its dual when the Hopf alge- bra is split. It is shown conilpotence and an algebraically closed field insure a Hopf algebra is split. The monoid of grouplike elements is studied. In the second chapter conditions for an algebra A -- which is a comodule for a Hopf algebra H --to be of the form A 'E B ® H (linear isomorphism) are given. The dual situation is studied. The graded Hopf algebra associated with a split Hopf algebra decomposes in the above manner. Chapter III contains the cohomology theory of a commutative algebra which is a module for a cocommutative Hopf algebra. There is extension theory and specialization to the situation the Hopf algebra is a group algebra. Chapter IV is dual to chapter III. Chapter V is devoted to coconnected cocommutative Hopf algebras, mostly in characteristic p > 0 .
    [Show full text]
  • Module and Comodule Categories - a Survey
    Module and Comodule Categories - a Survey Robert Wisbauer University of D¨usseldorf,Germany e-mail [email protected] Abstract The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly indepen- dently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule structure of algebras and comodule categories are sketched. 1. Module theory: Homological classification, the category σ[M], Morita equivalence, the functor ring, Morita dualities, decompositions, torsion theo- ries, trace functor. 2. Bimodule structure of an algebra: Multiplication algebra, Azumaya rings, biregular algebras, central closure of semiprime algebras. 3. Coalgebras and comodules: C-comodules and C∗-modules, σ-decom- position, rational functor, right semiperfect coalgebras, duality for comodules. B 4. Bialgebras and bimodules: The category B, coinvariants, B as B M projective generator in B, fundamental theorem for Hopf algebras, semiper- fect Hopf algebras. M 5. Comodule algebras: (A-H)-bimodules, smash product A#H∗, coin- H variants, A as progenerator in A . 6. Group actions and moduleM algebras: Group actions on algebras, A GA as a progenerator in σ[A GA], module algebras, smash product A#H, ∗ ∗ A#H A as a progenerator in σ[A#H A]. 1 1 Module theory In this section we recall mainly those results from module categories which are of interest for the applications to bimodules and comodules given in the subsequent sections.
    [Show full text]
  • Arxiv:Q-Alg/9509023V1 22 Sep 1995
    July 1993 Published in Advances in Hopf Algebras, Marcel Dekker Lec. Notes Pure and Applied Maths 158 (1994) 55-105. ALGEBRAS AND HOPF ALGEBRAS IN BRAIDED CATEGORIES1 SHAHN MAJID2 Department of Applied Mathematics & Theoretical Physics University of Cambridge, Cambridge CB3 9EW, U.K. ABSTRACT This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, as well as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braided-commutative’ or ‘braided-cocommutative’ Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane. One of the main motivations of the theory of Hopf algebras is that they provide a gener- arXiv:q-alg/9509023v1 22 Sep 1995 alization of groups. Hopf algebras of functions on groups provide examples of commutative Hopf algebras, but it turns out that many group-theoretical constructions work just as well when the Hopf algebra is allowed to be non-commutative. This is the philosophy associated to some kind of non-commutative (or so-called quantum) algebraic geometry. In a Hopf algebra context one can say the same thing in a dual way: group algebras and enveloping algebras are cocommutative but many constructions are not tied to this.
    [Show full text]
  • Algebras and Cyclic Homology
    Homology of L∞-algebras and Cyclic Homology Masoud Khalkhali∗ A celebrated theorem of Loday and Quillen [LQ] and (independently) Tsygan [T] states that the Lie algebra homology of the Lie algebra of stable matrices over an associative algebra is canonically isomorphic, as a Hopf algebra, to the exterior power of the cyclic homology of the associative algebra. The main point of this paper is to lay the ground such that an extension of this theorem to the category of A∞-algebras becomes possible (theorem 3.1). The category of L∞- (respectively, A∞-) algebras extend the category of differential graded (DG) Lie (respectively, DG associative) algebras. These concepts are both due to J. Stasheff. See [S], [LS], and references therein, and also [HS] where an alternative approach to L∞-algebras is given. In [Kh], we proposed an approach to homological invariants of A∞- algebras (Hochschild, cyclic, periodic cyclic, etc.) based on the notion of X-complex due to Cuntz and Quillen [CQ]. It seems that it is now possible to extend most of the tools of noncommutative geometry of Connes [C] to the homotopical setting of A∞ and L∞-algebras. There is, however, a notable exception in that so far we don’t know how the K-theory of an A∞-algebra should be defined. 1 L∞ and A∞ algebras Let V be a vector space (not graded). Let ScV denote the cofree cocommutative counital coassociative coalgebra generated by V . Over fields of characteristic zero there are two c c ⊗n different constructions for S V that we recall now. First, let T V = M V be the cofree n≥0 counital coassociative coalgebra generated by V .
    [Show full text]
  • The Freedom of Yetter-Drinfeld Hopf Algebras
    Advances in Pure Mathematics, 2014, 4, 522-528 Published Online September 2014 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2014.49060 The Freedom of Yetter-Drinfeld Hopf Algebras Yanhua Wang School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China Email: [email protected] Received 1 August 2014; revised 2 September 2014; accepted 13 September 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, the fundamental theorem of Yetter-Drinfeld Hopf module is proved. As applications, the freedom of tensor and twisted tensor of two Yetter-Drinfeld Hopf algebras is given. Let A be a Yetter-Drinfeld Hopf algebra. It is proved that the category of A-bimodule is equivalent to the cat- egory of AA⊗ -twisted module. Keywords Hopf Algebra, Hopf Module, Yetter-Drinfeld Module, Yetter-Drinfeld Hopf Algebra 1. Introduction Let k be a field and A an algebra. A left A -module is a k -vector space V together with a k -linear map ⊗→ such that →=→ → and →=. The category of left -module is denoted by AV V abvabv( ) 1 vv A ∆ ε A M . Dually, let (C,,) be a coalgebra. A left C -comodule is a k -vector space V together with a k -linear map ρ :V→⊗ VC such that −10 a−−1⊗⊗ a 1 a 0= a − 1⊗ a 0⊗ a 0, ε aa−10= a. ∑∑( )12( ) ( ) ( ) ∑( ) C The category of left C -comodule is denoted by M .
    [Show full text]
  • On the Structure of Irreducible Yetter-Drinfeld Modules Over Quasi
    On the Structure of Irreducible Yetter-Drinfeld Modules over Quasi-Triangular Hopf Algebras Zhimin Liu∗ Shenglin Zhu †‡ Fudan University, Shanghai 200433, China Abstract Let (H, R) be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field k. In this paper, we give the structure of irreducible objects of H the Yetter-Drinfeld module category H YD. Let HR be the Majid’s transmuted braided group of (H, R) , we show that HR is cosemisimple. As a coalgebra, let HR = D1 ⊕ ···⊕Dr be the sum of minimal H-adjoint-stable subcoalgebras. For each i (1 ≤ i ≤ r), we choose a minimal left coideal Wi of Di, and we can define the R-adjoint-stable algebra NWi of Wi. Using Ostrik’s theorem on characterizing module categories over H monoidal categories, we prove that V ∈ H YD is irreducible if and only if there exists an i (1 ≤ i ≤ r) and an irreducible right N -module U , such that V ∼ U ⊗ (H ⊗ W ). Wi i = i NWi i Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If k is an algebraically closed field of characteristic, we stress that the R-adjoint-stable algebra NWi is an al- gebra over which the dimension of each irreducible right module divides its dimension. KEYWORDS: Quasi-triangular Hopf algebra, Yetter-Drinfeld module, Transmuted braided group 2000 MATHEMATICS SUBJECT CLASSIFICATION: 16W30 1 Introduction arXiv:1811.05593v2 [math.RA] 17 Jun 2019 Yetter-Drinfeld modules over a bialgebra were introduced by Yetter [25] in 1990.
    [Show full text]
  • Indecomposable Coalgebras, Simple Comodules, And
    proceedings of the american mathematical society Volume 123, Number 8, August 1995 INDECOMPOSABLE COALGEBRAS,SIMPLE COMODULES, AND POINTED HOPF ALGEBRAS SUSAN MONTGOMERY (Communicated by Lance W. Small) Abstract. We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C . This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element. Introduction A basic structure theorem for cocommutative coalgebras asserts that any such coalgebra C is a direct sum of its irreducible components; as a consequence, it can be shown that any pointed cocommutative Hopf algebra is a skew group ring of the group G of group-like elements of H over the irreducible component of the identity element. These results were proved independently by Cartier and Gabriel and by Kostant in the early 1960's; see [Di, SI]. This paper is concerned with versions of these results for arbitrary coalgebras and for arbitrary (pointed) Hopf algebras. In fact much is already known about the coalgebra problem. In 1975 Ka- plansky [K] showed that any coalgebra C is (uniquely) a direct sum of inde- composable ones; moreover when C is cocommutative, the indecomposable components are irreducible. In 1978 Shudo and Miyamoto [ShM] defined an equivalence relation on the set of simple subcoalgebras of C and showed that the equivalence classes correspond to the indecomposable summands. A weaker version of this equivalence relation was studied recently in [XF].
    [Show full text]
  • Arxiv:1911.12847V1 [Math.QA] 28 Nov 2019 Omto Ruod Ekbagba Ekhp Algebra
    ALGEBRAIC STRUCTURES IN COMODULE CATEGORIES OVER WEAK BIALGEBRAS CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON Abstract. For a bialgebra L coacting on a k-algebra A, a classical result states that A is a right L-comodule algebra if and only if A is an algebra in the monoidal category ML of right L-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras H. The category MH admits a monoidal structure by work of Nill and B¨ohm-Caenepeel-Janssen, but the algebras in MH are not canonically k-algebras. Nevertheless, we prove that there is an isomorphism between the category of right H-comodule algebras and the category of algebras in MH . We also recall and introduce the formulaic notion of H coacting on a k-coalgebra and on a Frobenius k-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor cate- gories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra L to the corepresenta- tion category of a weak bialgebra built from L (a “quantum transformation groupoid”), thereby creating weak quantum symmetries from ordinary quantum symmetries. Contents 1. Introduction 2 2. Preliminaries on weak bialgebras 5 2.1. Algebraic structures over a field 5 2.2. Terminology and properties 6 2.3. Modules and comodules 7 2.4.
    [Show full text]
  • HOPF ALGEBRAS 1. Algebras and Coalgebras 1.1. Algebras. We Start by Formulating a Familiar Definition in a Possibly Unfa- Miliar
    HOPF ALGEBRAS 1. Algebras and coalgebras 1.1. Algebras. We start by formulating a familiar de¯nition in a possibly unfa- miliar way: De¯nition. A k-algebra is a k-vector space with two linear maps m : A ­k A ! A and u : k ! A satisfying the following conditions hold. These respectively encode associativity and the unit element: (a) the diagram m­id A ­ A ­ A / A ­ A id­m m ² m ² A ­ A / A is commutative; and (b) the diagram A ­ A u: dII u­id uu IIid­u uu II uu II uu I m k ­ AJ A ­ k JJ tt JJ tt JJ tt s JJ tt s J$ ² ztt A (where s denotes scalar multiplication) is commutative. We get 1A as u(1k) via the second diagram. 1.2. Coalgebras. To de¯ne these, we reverse the arrows in (7.1): De¯nition. A k-coalgebra is a k-vector space, C, with two k-linear maps, ¢ (co- multiplication or coproduct) and " (counit), with ¢ : C ! C ­ C and " : C ! k; such that the following conditions, respectively called the coassociativity and the counit axioms, hold: 1 2 HOPF ALGEBRAS a) the diagram ¢o ­id C ­ CO ­ CC ­O C id­¢ ¢ C ­ C o C ¢ commutes; b) the diagram C ­ C t O JJ "­id tt JJid­" tt JJ tt JJ ztt J$ k ­ CCJd ¢ : ­ k JJ tt JJ1­ ­1 tt JJ tt JJ tt J tt C commutes, where 1 ­ is the map x 7! 1 ­ x. 1.3. Morphisms. De¯nition. Let A and B be k-algebras.
    [Show full text]
  • Graded Twisting of Comodule Algebras and Module Categories
    GRADED TWISTING OF COMODULE ALGEBRAS AND MODULE CATEGORIES JULIEN BICHON, SERGEY NESHVEYEV, AND MAKOTO YAMASHITA Abstract. Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of G ⊂ SL−1(2) G on K−1[x; y] and show that in most cases the corresponding invariant rings K−1[x; y] are 0 invariant rings K[x; y]G for the action of a classical subgroup G0 ⊂ SL(2). As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are graded twistings of the Poisson boundaries. Introduction Let A, B be Hopf algebras, and assume that B is a 2-cocycle twisting of A. Then we have a monoidal equivalence between the categories of left comodules F : Comod(A) '⊗ Comod(B) It is a simple observation that F induces an equivalence between the respective categories of comodule algebras, and that if R is an A-comodule algebra, we have an isomorphism between the fixed point algebras coBF(R) ' coAR In other words, the invariant theory of A determines that of B. One of the main goals of this paper is to give a version of this principle when B is a graded twisting of A [BNY16]. In this case there is also an equivalence Comod(A) ' Comod(B), but it is not monoidal in general, so we cannot expect a correspondence between the categories of comodule algebras.
    [Show full text]
  • COTENSOR PRODUCTS of MODULES 1. Introduction Cotensor Product and Its Derived Functors Cotori Were Originally Introduced by Eile
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 6, Pages 2173{2185 S 0002-9947(02)02883-0 Article electronically published on February 1, 2002 COTENSOR PRODUCTS OF MODULES L. ABRAMS AND C. WEIBEL Abstract. Let C beacoalgebraoverafieldk and A its dual algebra. The category of C-comodules is equivalent to a category of A-modules. We use this to interpret the cotensor product MN of two comodules in terms of the appropriate Hochschild cohomology of the A-bimodule M ⊗ N,whenA is finite-dimensional, profinite, graded or differential-graded. The main applica- tions are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations. 1. Introduction Cotensor product and its derived functors Cotori were originally introduced by Eilenberg-Moore [EM] to calculate the homology of an induced fibration. These functors have been discussed in the literature, primarily in homotopy theory, but they have developed a reputation of being inaccessible. To counter this, we describe cotensor product and Cotor in terms of Hochschild cohomology, and provide a new class of examples to which they are relevant. Let C be a coalgebra over a field k and A its dual algebra. It is classical [Car] that one can identify comodules over C with a subcategory of modules over A. Thus comodule constructions may be translated into module constructions. In this paper, we interpret the cotensor product MN of two comodules as the vector space HomAe (A; M ⊗ N) of bimodule maps from A ⊗ ∗ to M N. We also interpret the right derived functors CotorC (M;N)intermsof the Hochschild cohomology of the A-bimodule M ⊗ N.
    [Show full text]
  • Koszulity of Cohomology= $ K (\Pi, 1) $-Ness+ Quasi-Formality
    KOSZULITY OF COHOMOLOGY = K(π, 1)-NESS + QUASI-FORMALITY LEONID POSITSELSKI Abstract. This paper is a greatly expanded version of [37, Section 9.11]. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the “Koszulity implies K(π, 1)” claim, discuss the differences between two versions of Massey op- erations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion. Contents Introduction 1 1. Koszulity Implies K(π, 1)-ness 5 2. Koszulity Implies Quasi-Formality 10 3. Noncommutative (Rational) Homotopy Theory 18 4. K(π, 1)-ness + Quasi-Formality Imply Koszulity 23 5. Self-Consistency of Nonhomogeneous Quadratic Relations 26 6. Koszulity Does Not Imply Formality 33 References 37 arXiv:1507.04691v8 [math.KT] 21 Apr 2017 Introduction A quadratic algebra is an associative algebra defined by homogeneous quadratic relations. In other words, a positively graded algebra A = k ⊕ A1 ⊕ A2 ⊕··· over a field k is called quadratic if it is generated by its first-degree component A1 with relations in degree 2. A positively graded associative algebra A is called Koszul [41, A 4, 32] if one has Torij(k,k) = 0 for all i =6 j, where the first grading i on the Tor spaces is the usual homological grading and the second grading j, called the internal grading, is induced by the grading of A.
    [Show full text]