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Из книжки по полубесконечной гомологической алгебре, выписано для письма коллеге:
Let C be a coring over a ring A. It makes sense to assume that C is a projective left and a flat right A-module. There are the operations of:
- cotensor product of right and left C-comodules;
- Cohom from a left C-comodule to a left C-contramodule;
- contratensor product of a right C-comodule and a left C-contramodule.
The terminology in the book is:
- a comodule is coflat if the functor of cotensor product with it is exact on the category of comodules;
- a comodule is coprojective if the functor Cohom from it is exact on the category of contramodules;
- a comodule is quasicoflat if the functor of cotensor product with it is right exact;
- a comodule is quasicoprojective if the functor Cohom from it is left exact;
- a comodule is relatively coflat if the functor of cotensor product with it preserves exactness of short exact sequences of A-flat C-comodules;
- a comodule is relatively coprojective if the functor Cohom from it preserves exactness of short exact sequences of A-injective C-contramodules;
- a comodule is relatively injective if the functor Hom into it preserves exactness of short exact sequences of A-projective C-comodules;
- a comodule is quite relatively injective if the functor Hom into it preserves exactness of A-split short exact sequences of C-comodules;
- a contramodule is coinjective if the functor Cohom into it is exact on the category of left comodules;
- a contramodule is quasicoinjective if the functor Cohom into it is left exact on the category of comodules;
- a contramodule is relatively coinjective if the functor Cohom into it preserves exactness of short exact sequences of A-projective C-comodules;
- a contramodule is relatively projective if the functor Hom from it preserves exactness of short exact sequences of A-injective C-contramodules;
- a contramodule is quite relatively projective if the functor Hom from it preserves exactness of A-split short exact sequences of C-contramodules;
- a contramodule is contraflat if the functor of contratensor product with it is exact on the category of right comodules;
- a contramodule is relatively contraflat if the functor of contratensor product with it preserves exactness of short exact sequences of A-flat C-comodules;
- a contramodule is quite relatively contraflat if the functor of contratensor product with it preserves exactness of A-pure short exact sequences of C-comodules.
[Lemma 5.2] tells that a comodule is quasicoprojective if and only if it is quite relatively injective. Dually, a contramodule is quasicoinjective if and only if it is quite relatively projective. The lemma also describes coprojective comodules and coinjective contramodules.
[Lemma 5.3.2] tells that, under the usual additional assumption that the left global dimension of the ring A is finite, a comodule is relatively coprojective if and only if it is relatively injective, and a contramodule is relatively coinjective if and only if it is relatively projective.
Let C be a coring over a ring A. It makes sense to assume that C is a projective left and a flat right A-module. There are the operations of:
- cotensor product of right and left C-comodules;
- Cohom from a left C-comodule to a left C-contramodule;
- contratensor product of a right C-comodule and a left C-contramodule.
The terminology in the book is:
- a comodule is coflat if the functor of cotensor product with it is exact on the category of comodules;
- a comodule is coprojective if the functor Cohom from it is exact on the category of contramodules;
- a comodule is quasicoflat if the functor of cotensor product with it is right exact;
- a comodule is quasicoprojective if the functor Cohom from it is left exact;
- a comodule is relatively coflat if the functor of cotensor product with it preserves exactness of short exact sequences of A-flat C-comodules;
- a comodule is relatively coprojective if the functor Cohom from it preserves exactness of short exact sequences of A-injective C-contramodules;
- a comodule is relatively injective if the functor Hom into it preserves exactness of short exact sequences of A-projective C-comodules;
- a comodule is quite relatively injective if the functor Hom into it preserves exactness of A-split short exact sequences of C-comodules;
- a contramodule is coinjective if the functor Cohom into it is exact on the category of left comodules;
- a contramodule is quasicoinjective if the functor Cohom into it is left exact on the category of comodules;
- a contramodule is relatively coinjective if the functor Cohom into it preserves exactness of short exact sequences of A-projective C-comodules;
- a contramodule is relatively projective if the functor Hom from it preserves exactness of short exact sequences of A-injective C-contramodules;
- a contramodule is quite relatively projective if the functor Hom from it preserves exactness of A-split short exact sequences of C-contramodules;
- a contramodule is contraflat if the functor of contratensor product with it is exact on the category of right comodules;
- a contramodule is relatively contraflat if the functor of contratensor product with it preserves exactness of short exact sequences of A-flat C-comodules;
- a contramodule is quite relatively contraflat if the functor of contratensor product with it preserves exactness of A-pure short exact sequences of C-comodules.
[Lemma 5.2] tells that a comodule is quasicoprojective if and only if it is quite relatively injective. Dually, a contramodule is quasicoinjective if and only if it is quite relatively projective. The lemma also describes coprojective comodules and coinjective contramodules.
[Lemma 5.3.2] tells that, under the usual additional assumption that the left global dimension of the ring A is finite, a comodule is relatively coprojective if and only if it is relatively injective, and a contramodule is relatively coinjective if and only if it is relatively projective.
