Some topological aspects of a general spectra construction of Matsui and Takahashi

Matsui and Takahashi introduce a general spectra construction for triangulated categories in [J. Math. Soc. Japan, 4:2121-2150,2020], which is later used to establish Matsui’s theory of triangular geometry. In this paper, we study several topological aspects of this general construction and give criteria for soberness and spectralness of the spectra. Furthermore, we discuss and generalize the immersion phenomenon for Noetherian schemes as appeared in [Pacific. J. Math., 313(2):433-457, 2021]. The last section illustrates that similar immersions also appear if the underlying category has a well-defined finite group action. We work in the extriangulated context to incorporate similar ideas from the triangulated and abelian contexts.