Gorenstein flat comodules
Let \(C\) be a coalgebra over a field \(K\) and Comod-\(C\) (resp. comod-\(C\)) be the category of (resp. finite dimensional) right C-comodules. Assume that \(C\) is left and right semiperfect, using the notion of flat comodule as introduced by \textit{J. Cuadra} and \textit{D. Simson} [Commun. Algebra 35, No. 10, 3164--3194 (2007; Zbl 1131.16018)], the authors call a right comodule \(M\) Gorenstein flat if there exist an exact sequence \(\cdots F_1\to F_0\to F^0\to F^1\to\cdots\) of flat right \(C\)-comodules with \(M\cong \mathrm{Ker}(F^0\to F^1)\) and such that Com\(_C(-,I)\) leaves the sequence exact whenever \(I\in \text{comod-}C\) and \(id_C(I)<\infty\), where \(id_C(-)\) denotes the injective dimension. Several characterizations of Gorenstein flat right \(C\)-comodules are given (Theorem 3.7). Next, the behaviour of Gorenstein flat right comodules under colocalization is studied. It is shown that if \(M\) is Gorenstein flat right \(C\)-comodule, then \(eM\) is a Gorenstein flat right \(eCe\)-comodule where \(e\) is an idempotent in the algebra \(C^*=\mathrm{Hom}_K(C,k)\).
scientific article; zbMATH DE number 7595373
scientific article