TY - JOUR

T1 - On purely generated α-smashing weight structures and weight-exact localizations

AU - Bondarko, Mikhail V.

AU - Sosnilo, Vladimir A.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects P of a triangulated category C_ that satisfies a certain (countable) negativity condition (there are no C_-extensions of positive degrees between elements of P; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both “halves” of which are closed either with respect to C_-coproducts of less than α objects (where α is a fixed regular cardinal) or with respect to all coproducts (provided that C_ is closed with respect to coproducts of this sort). This construction gives all “reasonable” weight structures satisfying the latter conditions. In particular, one can obtain certain weight structures on spectra (in SH) consisting of less than α cells, and on certain localizations of SH; these results are new.

AB - This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects P of a triangulated category C_ that satisfies a certain (countable) negativity condition (there are no C_-extensions of positive degrees between elements of P; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both “halves” of which are closed either with respect to C_-coproducts of less than α objects (where α is a fixed regular cardinal) or with respect to all coproducts (provided that C_ is closed with respect to coproducts of this sort). This construction gives all “reasonable” weight structures satisfying the latter conditions. In particular, one can obtain certain weight structures on spectra (in SH) consisting of less than α cells, and on certain localizations of SH; these results are new.

KW - Compact objects

KW - Generalized universal localizations of rings

KW - Localizations

KW - Localizing subcategories

KW - Perfect classes

KW - Stable homotopy category

KW - Triangulated categories

KW - Weight structures

KW - Weight-exact functors

KW - α-smashing classes

KW - T-STRUCTURES

KW - alpha-smashing classes

KW - FILTRATIONS

UR - http://www.scopus.com/inward/record.url?scp=85068551035&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/purely-generated-%CE%B1smashing-weight-structures-weightexact-localizations

U2 - 10.1016/j.jalgebra.2019.07.003

DO - 10.1016/j.jalgebra.2019.07.003

M3 - Article

AN - SCOPUS:85068551035

VL - 535

SP - 407

EP - 455

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -