Gorenstein coalgebras
Iwanaga-Gorenstein rings are left and right Noetherian rings with finite self-injective dimension on both sides, introduced by \textit{Y. Iwanaga} [Osaka J. Math. 15, 33-46 (1978; Zbl 0402.16017), Tsukuba J. Math. 4, 107-113 (1980; Zbl 0459.16011)]. For them relative homological theory is established, cf. \textit{E. E. Enochs, O. M. G. Jenda, B. Torrecillas} [J. Nanjing Univ., Math. Biq. 10, 1-9 (1993; Zbl 0794.16001)], \textit{E. E. Enochs, O. M. G. Jenda} [Math. Z. 220, No. 4, 611-633 (1995; Zbl 0845.16005)]. The aim of the paper is to find a class of coalgebras in which relative homology can be studied as in the case of modules. An \(n\)-Gorenstein coalgebra is a left and right semiperfect coalgebra with finite self-projective dimension not exceeding \(n\) on both sides, hence generalizes quasi-co-Frobenius coalgebras on both sides, studied by \textit{J. Gómez Torrecillas, C. Năstăsescu} [J. Algebra 174, No. 3, 909-923 (1995; Zbl 0833.16038)]. The authors first study the dimensions of a left comodule over an \(n\)-Gorenstein coalgebra. Next they introduce and characterize Gorenstein injective comodules, obtaining analogous results as in the case of modules and graded modules, and derive their main result: A coalgebra \(C\) is \(n\)-Gorenstein if and only if every left (right) \(C\)-comodule \(M\) has a projective cover, and for any minimal resolution of \(M\), the \(n\)-th comodule is Gorenstein injective. Moreover, just as in injective and coflat comodules, they obtain that Gorenstein injective and Gorenstein coflat comodules coincide.
scientific article; zbMATH DE number 1446808