Comodule

In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

Formal definition

Let K be a field, and C be a coalgebra over K. A (right) comodule over C is a K-vector space M together with a linear map

such that

  1. ,

where Δ is the comultiplication for C, and ε is the counit.

Note that in the second rule we have identified with .

Examples

  • A coalgebra is a comodule over itself.
  • If M is a finite-dimensional module over a finite-dimensional K-algebra A, then the set of linear functions from A to K forms a coalgebra, and the set of linear functions from M to K forms a comodule over that coalgebra.
  • A graded vector space V can be made into a comodule. Let I be the index set for the graded vector space, and let be the vector space with basis for . We turn into a coalgebra and V into a -comodule, as follows:
  1. Let the comultiplication on be given by .
  2. Let the counit on be given by .
  3. Let the map on V be given by , where is the i-th homogeneous piece of .

In algebraic topology

One important result in algebraic topology is the fact that homology over the dual Steenrod algebra forms a comodule.[1] This comes from the fact the Steenrod algebra has a canonical action on the cohomology

When we dualize to the dual Steenrod algebra, this gives a comodule structure

This result extends to other cohomology theories as well, such as complex cobordism and is instrumental in computing its cohomology ring .[2] The main reason for considering the comodule structure on homology instead of the module structure on cohomology lies in the fact the dual Steenrod algebra is a commutative ring, and the setting of commutative algebra provides more tools for studying its structure.

Rational comodule

If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C, but the converse is not true in general: a module over C is not necessarily a comodule over C. A rational comodule is a module over C which becomes a comodule over C in the natural way.

Comodule morphisms

Let R be a ring, M, N, and C be R-modules, and

be right C-comodules. Then an R-linear map is called a (right) comodule morphism, or (right) C-colinear, if

This notion is dual to the notion of a linear map between vector spaces, or, more generally, of a homomorphism between R-modules.[3]

See also

References

  1. Liulevicius, Arunas (1968). "Homology Comodules" (PDF). Transactions of the American Mathematical Society. 134 (2): 375–382. doi:10.2307/1994750. ISSN 0002-9947.
  2. Mueller, Michael. "Calculating Cobordism Rings" (PDF). Archived (PDF) from the original on 2 Jan 2021.
  3. Khaled AL-Takhman, Equivalences of Comodule Categories for Coalgebras over Rings, J. Pure Appl. Algebra,.V. 173, Issue: 3, September 7, 2002, pp. 245–271
  • Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", Revue Roumaine de Mathématiques Pures et Appliquées, 43: 591–603
  • Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
  • Sweedler, Moss (1969), Hopf Algebras, New York: W.A.Benjamin
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