Vol. 313, No. 2, 2021

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Prime thick subcategories and spectra of derived and singularity categories of noetherian schemes

Hiroki Matsui

Vol. 313 (2021), No. 2, 433–457
DOI: 10.2140/pjm.2021.313.433
Abstract

For an essentially small triangulated category 𝒯, we introduce the notion of prime thick subcategories and define the spectrum of 𝒯, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme X. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces of X are embedded into their spectra as topological spaces.

Keywords
complete intersection, derived category, hypersurface, noetherian scheme, prime thick subcategory, singularity category, spectrum, triangulated category
Mathematical Subject Classification
Primary: 14F08
Secondary: 13D09, 13H10, 18G80
Milestones
Received: 27 March 2021
Accepted: 13 June 2021
Published: 12 October 2021
Authors
Hiroki Matsui
Graduate School of Mathematical Sciences
University of Tokyo
Tokyo
Japan