Balance of complete cohomology in extriangulated categories

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $\end{document}</tex-math></inline-formula> be an extriangulated category with a proper class <inline-formula><tex-math id="M2">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{E} $\end{document}</tex-math></inline-formula>-triangles. In this paper, we study the balance of complete cohomology in <inline-formula><tex-math id="M4">\begin{document}$ (\mathcal{C}, \mathbb{E}, \mathfrak{s}) $\end{document}</tex-math></inline-formula>, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an Artin algebra to be <inline-formula><tex-math id="M5">\begin{document}$ F $\end{document}</tex-math></inline-formula>-Gorenstein.</p>