On tilting and cotilting-type modules

Gabriella D'Este

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Let F: R → R/G be a Galois covering and $m o d_{1} (R / G)$ (resp. $m o d_{2} (R / G)$ ) be a full subcategory of the module category mod (R/G), consisting of all R/G-modules of first (resp. second) kind with respect to F. The structure of the categories $(m o d (R / G)) / [m o d_{1} (R / G)]$ and $m o d_{2} (R / G)$ is given in terms of representation categories of stabilizers of weakly-G-periodic modules for some class of coverings.

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For every module M we have a natural monomorphism $Φ : ∐_{i \in I} H o m_{R} (A_{i}, M) \to H o m_{R} (\prod_{i \in I} A_{i}, M)$ and we focus attention on the case when Φ is also an epimorphism. The corresponding modules M depend on thickness of the cardinal number card(I). Some other limits are also considered.

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For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_{A}$ is a tilting module and $B = E n d T_{A}$ , then T(A) is tame if and only if T(B) is tame.

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Let Fl(R) denote the category of flat right modules over an associative ring R. We find necessary and sufficient conditions for Fl(R) to be a Grothendieck category, in terms of properties of the ring R.