Cohomology theories in triangulated categories

Avramov-Martsinkovsky type exact sequences for extriangulated categories

“…Note that extriangulated categories are a simultaneous generalization of abelian categories and triangulated categories. It follows that Theorem 4.4 here unifies Theorem 7.1 proved by Avramov and Martsinkovsky [4] in the category of modules, and Theorem 4.10 proved by Ren, Zhao and Liu [21] in a triangulated category. It should be noted that our results here are new for exact categories and extension-closed subcategories of triangulated categories.…”

“…Ren and Liu established the global ξ-Gorenstein dimension for a triangulated category in [20] by introducing E-Gorenstein cohomology groups Ext i GP (−, −) and Ext i GI (−, −) for objects with finite E-Gorenstein dimension. Motivated by Avramov-Martsinkovsky type exact sequences constructed over a ring R in [4], Ren, Zhao and Liu [21] proved that Beligiannis's E-cohomology, Asadallahi and Salarian's E-Tate cohomology and Ren and Liu's Gorenstein cohomology can be connected by a long exact sequence.…”

Complete cohomology for extriangulated categories

“…They showed that this theory not only shares basic properties with ordinary cohomology, but also enjoys some distinctive features. We refer to [31,32] for a more discussion on this matter.…”

Balance of complete cohomology in extriangulated categories

“…However, the complete cohomology of modules is not balanced in the way Ext is balanced by [20,Theorem 5.2]. Recently, the authors [14] developed a complete cohomology theory in an extriangulated category and demonstrated that this theory shared some basic properties of complete cohomology in the category of modules [6,13,18,20,24] and Tate cohomology in the triangulated category [1,21,22]. It seems natural to characterize when complete cohomology in extriangulated categories is balanced.…”