Homology, Homotopy and Applications

Volume 20 (2018)

Number 1

On constructing weight structures and extending them to idempotent completions

Pages: 37 – 57

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n1.a3

Authors

Mikhail V. Bondarko (Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia)

Vladimir A. Sosnilo (Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia)

Abstract

In this paper we describe a new method for constructing a weight structure $w$ on a triangulated category $\underline{C}$.

For a given $\underline{C}$ and $w$ it allows us to give a fairly comprehensive (and new) description of triangulated categories containing $\underline{C}$ as a dense subcategory (i.e., of subcategories of the idempotent completion of $\underline{C}$ that contain $\underline{C}$; we call them idempotent extensions of $\underline{C}$) to which $w$ extends. In particular, any bounded above or below $w$ extends to any idempotent extension of $\underline{C}$; however, we illustrate by an example that $w$ does not extend to the idempotent completion of $\underline{C}$ in general.

We also describe an application of our results to certain triangulated categories of (relative) motives.

Keywords

weight structure, triangulated category, idempotent completion, Voevodsky motive, Chow motive, Beilinson motive

2010 Mathematics Subject Classification

14C15, 18B15, 18E30, 18E40

Received 30 May 2016

Received revised 23 January 2017

Published 20 December 2017