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Algebra

Monads and comonads on module categories

Profile image of Tomasz BrzezińskiTomasz Brzeziński

2009

https://doi.org/10.1016/J.JALGEBRA.2009.06.003
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30 pages

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Abstract

Let A be a ring and M A the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the

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References (40)

  1. 11. Theorem. Let N ∈ C M be a [C, -]-Galois comodule over an A-coring C, put T = End C (N ) and assume T to be a B-ring for some ring B. Assume that N is a projective generator of right B-modules. Then the following hold.
  2. -⊗ [C,-] N : M [C,-] → M B is an equivalence. (2) C is a projective right A-module.
  3. N is a finitely generated and projective left A-module.
  4. C is isomorphic to the comatrix A-coring N ⊗ B
  5. * N .
  6. If, in addition, C is a generator of right A-modules, then N is a faithfully flat left A-module.
  7. Proof. Assertions (1) and (2) are immediate by 5.9.
  8. Since -⊗ [C,-] N is an equivalence, it has a left adjoint Hom B (N, -) : M B → M [C,-] . The free functor Hom A (C, -) has a left adjoint -⊗ [C,-] C : M [C,-] → M A by 4.3. References
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