On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories

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Abstract

Abstract: In this note we prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated category (Formula Presented.) that is compactly generated by a single object G is weakly approximable if (Formula Presented.) for (Formula Presented.) (we say that G is weakly negative if this assumption is fulfilled; the case where the equality (Formula Presented.) is fulfilled as well was mentioned by Neeman himself). Moreover, if (Formula Presented.) whenever (Formula Presented.) is also approximable. The latter result can be useful since (under a few more additional assumptions) it allows to characterize a certain explicit subcategory of (Formula Presented.) as the category of finite cohomological functors from the subcategory (Formula Presented.) of compact objects of (Formula Presented.)-modules (for a noetherian commutative ring R such that (Formula Presented.)-linear). One may apply this statement to the construction of certain adjoint functors and t-structures. Our proof of (weak) approximability of (Formula Presented.) under the aforementioned assumptions is closely related to (weight decompositions for) certain (weak) weight structures, and we discuss this relationship in detail

Original languageEnglish
Pages (from-to)151-159
Number of pages9
JournalLobachevskii Journal of Mathematics
Volume41
Issue number2
DOIs
StatePublished - 13 Jul 2020

Scopus subject areas

  • Mathematics(all)

Keywords

  • approximable triangulated category
  • heart
  • negative subcategory
  • t-structure
  • Triangulated category
  • weak weight structure
  • weight decomposition
  • weight structure

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