TY - JOUR

T1 - On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories

AU - Bondarko, M. V.

AU - Vostokov, S. V.

N1 - Bondarko, M.V., Vostokov, S.V. On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories. Lobachevskii J Math 41, 151–159 (2020). https://doi.org/10.1134/S1995080220020031

PY - 2020/2

Y1 - 2020/2

N2 - 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

AB - 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

KW - approximable triangulated category

KW - heart

KW - negative subcategory

KW - t-structure

KW - Triangulated category

KW - weak weight structure

KW - weight decomposition

KW - weight structure

KW - T-STRUCTURES

KW - SPECTRAL SEQUENCES

KW - TORSION THEORIES

KW - FILTRATIONS

KW - MOTIVES

UR - http://www.scopus.com/inward/record.url?scp=85087881282&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/179c25ec-3f27-3efd-a245-38ed22f8aa66/

U2 - 10.1134/S1995080220020031

DO - 10.1134/S1995080220020031

M3 - Article

AN - SCOPUS:85087881282

VL - 41

SP - 151

EP - 159

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 2

ER -