Corings and comodules

348 Pages·2003·1.619 MB·English

by  Tomasz Brzezinski, Robert Wisbauer

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210 HilbertC*-modules, E.C.LANCE
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226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds)
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265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART
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271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV
272 Charactertheoryfortheoddordertheorem, T.PETERFALVI
273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds)
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277 Computationalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal
278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO
279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds)
280 CharactersandautomorphismgroupsofcompactRiemannsurfaces, THOMASBREUER
281 Explicitbirationalgeometryof3-folds, ALESSIOCORTI&MILESREID(eds)
282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO
283 Nonlinearelasticity, Y.FU&R.W.OGDEN(eds)
284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SÜLI(eds)
285 Rationalpointsoncurvesoverfinitefields, H.NIEDERREITER&C.XING
286 Cliffordalgebrasandspinors2ed, P.LOUNESTO
287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MARTÌNEZ(eds)
288 Surveysincombinatorics,2001, J.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE
290 QuantumgroupsandLieTheory, A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed)
292 Aquantumgroupsprimer, S.MAJID
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294 Introductiontothetheoryofoperatorspaces, G.PISIER
London Mathematical Society Lecture Note Series. 309
Corings and Comodules
Tomasz Brzezinski
University of Wales Swansea
Robert Wisbauer
Heinrich Heine Universität Düsseldorf
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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© T. Brzezinski & R. Wisbauer 2003
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Contents
Preface vii
Notations xi
1 Coalgebras and comodules 1
1 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Coalgebra morphisms . . . . . . . . . . . . . . . . . . . . . . . 8
3 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 C-comodules and C∗-modules . . . . . . . . . . . . . . . . . . 41
5 The finite dual of an algebra . . . . . . . . . . . . . . . . . . . 55
6 Annihilators and bilinear forms . . . . . . . . . . . . . . . . . 62
7 The rational functor . . . . . . . . . . . . . . . . . . . . . . . 66
8 Structure of comodules . . . . . . . . . . . . . . . . . . . . . . 75
9 Coalgebras over QF rings . . . . . . . . . . . . . . . . . . . . . 80
10 Cotensor product of comodules . . . . . . . . . . . . . . . . . 93
11 Bicomodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
12 Functors between comodule categories . . . . . . . . . . . . . 110
2 Bialgebras and Hopf algebras 129
13 Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
14 Hopf modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
15 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
16 Trace ideal and integrals for Hopf algebras . . . . . . . . . . . 158
3 Corings and comodules 169
17 Corings and their morphisms . . . . . . . . . . . . . . . . . . . 170
18 Comodules over corings . . . . . . . . . . . . . . . . . . . . . . 180
19 C-comodules and C∗-modules . . . . . . . . . . . . . . . . . . . 195
20 The rational functor for corings . . . . . . . . . . . . . . . . . 212
21 Cotensor product over corings . . . . . . . . . . . . . . . . . . 217
22 Bicomodules over corings . . . . . . . . . . . . . . . . . . . . . 223
23 Functors between comodule categories . . . . . . . . . . . . . 230
24 The category of corings . . . . . . . . . . . . . . . . . . . . . . 240
4 Corings and extensions of rings 251
25 Canonical corings and ring extensions . . . . . . . . . . . . . . 251
26 Coseparable and cosplit corings . . . . . . . . . . . . . . . . . 256
27 Frobenius extensions and corings . . . . . . . . . . . . . . . . 264
28 Corings with a grouplike element . . . . . . . . . . . . . . . . 276
v
vi
29 Amitsur complex and connections . . . . . . . . . . . . . . . . 288
30 Cartier and Hochschild cohomology . . . . . . . . . . . . . . . 299
31 Bialgebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5 Corings and entwining structures 323
32 Entwining structures and corings . . . . . . . . . . . . . . . . 323
33 Entwinings and Hopf-type modules . . . . . . . . . . . . . . . 335
34 Entwinings and Galois-type extensions . . . . . . . . . . . . . 342
6 Weak corings and entwinings 357
35 Weak corings . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
36 Weak bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 366
37 Weak entwining structures . . . . . . . . . . . . . . . . . . . . 382
Appendix 395
38 Categories and functors . . . . . . . . . . . . . . . . . . . . . . 395
39 Modules and Abelian categories . . . . . . . . . . . . . . . . . 409
40 Algebras over commutative rings . . . . . . . . . . . . . . . . 416
41 The category σ[M] . . . . . . . . . . . . . . . . . . . . . . . . 425
42 Torsion-theoretic aspects . . . . . . . . . . . . . . . . . . . . . 435
43 Cogenerating and generating conditions . . . . . . . . . . . . . 445
44 Decompositions of σ[M] . . . . . . . . . . . . . . . . . . . . . 451
Bibliography 457
Index 471
Preface
Corings and comodules are fundamental algebraic structures that can be
thought of as both dualisations and generalisations of rings and modules.
Corings were introduced by Sweedler in 1975 as a generalisation of coalgebras
and as a means of presenting a semi-dual version of the Jacobson-Bourbaki
Theorem, but their origin can be traced back to 1968 in the work of Jonah on
cohomology of coalgebras in monoidal categories. In the late seventies they
resurfaced under the name of bimodules over a category with a coalgebra
structure, BOCSs for short, in the work of Rojter and Kleiner on algorithms
for matrix problems. For a long time, essentially only two types of examples
of corings truly generalising coalgebras were known – one associated to a
ring extension, the other to a matrix problem. The latter example was also
studiedinthecontextofdifferentialgradedalgebrasandcategories. Thislack
of examples hindered the full appreciation of the fundamental role of corings
in algebra and obviously hampered their progress in general coring theory.
On the other hand, from the late seventies and throughout the eighties
and nineties, various types of Hopf modules were studied. Initially these were
typically modules and comodules of a common bialgebra or a Hopf algebra
with some compatibility condition, but this evolved to modules of an algebra
and comodules of a coalgebra with a compatibility condition controlled by
a bialgebra. In fact, even the background bialgebra has been shown now to
be redundant provided some relations between a coalgebra and an algebra
are imposed in terms of an entwining. The progress and interest in such
categories of modules were fuelled by the emergence of quantum groups and
their application to physics, in particular gauge theory in terms of principal
bundles and knot theory.
By the end of the last century, M. Takeuchi realised that the compatibility
condition between an algebra and a coalgebra known as an entwining can
be recast in terms of a coring. From this, it suddenly became apparent
that various properties of Hopf modules, including entwined modules, can be
understoodandmoreneatlypresentedfromthepointofviewoftheassociated
coring. It also emerged, on the one hand, that coring theory is rich in many
interesting examples and, on the other, that – based on the knowledge of
Hopf-type modules – there is much more known about the general structure
of corings than has been previously realised. It also turned out that corings
might have a variety of unexpected and wide-ranging applications, to topics
in noncommutative ring theory, category theory, Hopf algebras, differential
graded algebras, and noncommutative geometry. In summary, corings appear
to offer a new, exciting possibility for recasting known results in a unified
general manner and for the development of ring and module theory from a
completely different point of view.
vii
viii
Asindicatedabove, coringscanbeviewedasgeneralisationsofcoalgebras,
the latter an established and well-studied theory, in particular over fields.
More precisely, a coalgebra over a commutative ring R can be defined as a
coalgebrainthemonoidalcategoryofR-modules–anotionthatiswellknown
in general category theory. On the other hand, an A-coring is a coalgebra
in the monoidal category of (A,A)-bimodules, where A is an arbitrary ring.
With the emergence of quantum groups in the works of Drinfeld [110], Jimbo
[135] and Woronowicz [214], new interest arose in the study of coalgebras,
mainly those with additional structures such as bialgebras and Hopf algebras,
because of their importance in various applications. In the majority of books
onHopfalgebrasandcoalgebras, suchasthenowclassictextsofSweedler[45]
and Abe [1] or in the more recent works of Montgomery [37] and Daˇscaˇlescu,
Naˇstaˇsescu and Raianu [14] together with texts motivated by quantum group
theory (e.g., Lusztig [30]; Majid [33, 34]; Chari and Pressley [11]; Shnider and
Sternberg [43]; Kassel [25]; Klimyk and Schmu¨dgen [26]; Brown and Goodearl
[7]), coalgebrasareconsideredoverfields. Thevastvarietyofapplicationsand
new developments, in particular in ring and module theory, manifestly show
thatthereisstillaneedforabetterunderstandingofcoalgebrasoverarbitrary
commutative rings, as a preliminary step towards the theory of corings. Let
us mention a few aspects of particular interest to the classical module and
ring theory.
There are parts of module theory over algebras A that provide a perfect
setting for the theory of comodules. Given any left A-module M, denote
by σ[M] the full subcategory of the category M of left A-modules that is
A
subgenerated by M. This is the smallest Grothendieck subcategory of M
A
containing M. Internal properties of σ[M] strongly depend on the module
properties of M, and there is a well-established theory that explores this
relationship. Although, in contrast to M, there need not be projectives
A
in σ[M], its Grothendieck property enables the use of techniques such as
localisation and various homological methods in σ[M]. Consequently, one
can gain a very good understanding of the inner properties of σ[M]. On the
other hand, by definition, σ[M] is closed under direct sums, submodules and
factor modules in M, and so it is a hereditary pretorsion class in M. If
A A
σ[M] is also closed under extensions in M, it is a (hereditary) torsion class.
A
Torsion theory then provides many characterisations of the outer properties
of σ[M], that is, the behaviour of σ[M] as a subclass of M.
A
Both the inner and outer properties of the categories of type σ[M] are
important in the study of coalgebras and comodules. If C is a coalgebra
over a commutative ring R, then the dual C∗ is an R-algebra and C is a
left and right module over C∗. The link to the module theory mentioned
above is provided by the basic observation that the category MC of right
C-comodules is subgenerated by C, and there is a faithful functor from MC