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

TL;DR

This paper explores topological properties of a general spectra construction for triangulated categories, providing criteria for soberness and spectralness, and extending immersion phenomena to categories with finite group actions within an extriangulated framework.

Contribution

It introduces criteria for soberness and spectralness of spectra, and generalizes immersion phenomena to categories with finite group actions in an extriangulated setting.

Findings

Criteria for soberness and spectralness established.

Immersion phenomena extended to categories with finite group actions.

Connections made between triangulated, abelian, and extriangulated contexts.

Abstract

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.