Comodules over Coalgebras

The topic of the first part of this semester's seminar will be the structure and representation theory of coalgebras. Juan Cuadra has made up a program consisting of the following six talks:

1. Algebras and coalgebras

• Definition of coalgebra. Sigma notation. Substructures of a coalgebra. Coalgebra morphisms.
• Examples of coalgebras [S, pages 3-26], [C1, pages 1-6]:
  • Group-like coalgebra.
  • Incidence coalgebra over a poset.
  • Comatrix coalgebra.
  • Monoid coalgebra (power divided coalgebra).
    Path coalgebra C(Q) of a quiver Q.
    The comultiplication is defined by: Δ(γ) = Σ_{αβ = γ} α \otimes β
    The category of C(Q)-comodules is the full subcategory of all "successor-finite" representations of Q.
    A representation (V_x,V_α) is successor-finite, provided for any v in V_x, there are only finitely many paths w starting at x such that w(x) is non-zero. A quiver has representations which are not successor-finite, in case there is an infinite path starting at some vertex, and also in case there are infinitely many arrows starting at some vertex.
    Warning. Not every finite dimensional representation of a quiver has to be successor-finite. Example 1: If Q has oriented cycles, only locally nilpotent representations can be successor-finite. Example 2: If there are infinitely many arrows from a vertex x to a vertex y, then there are many 2-dim representations which are not successor-finite.
    Note that the category of locally finite representations of the quiver Q is also the category of comodules for a coalgebra C', but in general C' is different from C(Q).
    See Note on representations of a quiver and right comodules over the path coalgebra of the quiver (by Asashiba)
• Duality between algebras and coalgebras.
  • The dual algebra of a coalgebra [S, pages 7-22], [C1, pages 18-20]
  • Examples [C1,6-7]
  • The finite dual coalgebra of an algebra [S, 109-118], [C1, 13-18].
  • The representative coalgebra of a monoid. [A, 70-74], [C1, 23-24].
  • The finite dual coalgebra of the polynomial ring: Linearly recursive sequences (Optative). [T1,T2]

2. Modules and comodules

• Definition of comodule, comodule morphisms, subcomodules.
• Rational modules
• The fundamental theorem of comodules and coalgebras [S, 29-48]
  Any finitely generated comodule is finite dimensional.
  Consequence: Any comodule is a directed union of finite dimensional subcomodules.
  Any finitely generated coalgebra is finite-dimensional.
• Examples of categories of comodules:
  • Comodules over the power divided coalgebra [S, 43-44], [L, 363]
  • Locally finite modules [M1, 11]
  • Comodules over the path coalgebra. Locally finite representations of quivers [CH, 15], [Ns, 458-465], [W, 2788-2797]

3. Structure theory of coalgebras and comodules

• The coefficient space of a comodule
• Simple comodules.
• Loewy series
• Injectives in the category of comodules.
  Theorem:
  • The category of C-comodules has injective envelopes.
  • Any injective C-comodule is a direct sum of indecomposable injective C-comodules.
  • Any indecomposable injective right C-comodule is the injective hull I(S) of a simple right C-comodule S.
  • Any direct sum of injective comodules is injective.
• Decomposition of coalgebras: Brauer decomposition; Ext-quiver decomposition. [G, 138-156], [M2, 2344-2349]
  Theorem: Any coalgebra C is isomorphic (as a right comodule) to a direct sum of indecomposable injective right C-modules, each appearing with finite multiplicity (the multiplicity of I(S) is equal to the dimension of S over its endomorphism ring).
• Locally finite representations of algebras [G, 169-171]
• Representations of linear algebraic groups (Optative) [G, 164-166]
• The coradical filtration of a coalgebra [M1, 56-63]

4. Equivalences between categories of comodules

• Cotensor functor [T, 632], [D, 31-34]
• Quasi-finite comodules. Co-hom functor.
• Morita-Takeuchi contexts [T, 633-638], [C6, 2408-2413]
• Morita-Takeuchi Theorem [T, 638-640]
  Theorem. Let C, D be k-coalgebras. The categories of C-comodules and of D-comodules are equivalent iff there exists a quasi-finite injective C-cogenerator P such that the co-endomorphism coalgebra of P is isomorphic to D. ("quasi-finite" means: any simple submodule occurs with finite multiplicity; one needs to define the "coendomorphism coalgebra" ...)
• Basic coalgebras [CM, 1-8], [C6, 2421-2424], [W, 2788-2793]
  Theorem: For any coalgebra C, there exists a basic coalgebra B such that their comodule categories are equivalent. In case k is algebraically closed, B is a subcoalgebra of the path coalgebra of the Ext-quiver of the category of all C-comodules.
  Theorem: An abelian k-category is k-linear equivalent to the category of finite-dimensional comodules over a k-coalgebra iff it is a length category and all Hom-sets are finite dimensional k-spaces. ("length category" means that every object has finite length). [Si, Theorem 2.13]
• Locally finite abelian categories [T, 640-644]
  Theorem. An abelian k-category is k-linear equivalent to the category of all comodules over a k-coalgebra if and only if it is locally finite and the endomorphism ring of every simple object is finite-dimensional over k. ("locally finite" means: any object is a union of finite length objects)

5. Special classes of coalgebras

• Cosemisimple coalgebras [A, 130-131]
• Semiperfect coalgebras [L, 363-370], [GN, 915-919]
• Co-Frobenius and quasi-co-Frobenius coalgebras [L, 359-363], [D, 37-39], [GN, 910-917]
• Hereditary coalgebras [NTZ], [CH2]
• Serial coalgebras [CG2]
• Pure semisimple coalgebras [NS]

6. Kleiner-Reiten's characterization of categories having almost split sequences

• [KR]

References