TY - JOUR

T1 - A Nullstellensatz for triangulated categories

AU - Bondarko, M. V.

AU - Sosnilo, V. A.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - The paper is aimed at proving the following: for a triangulated category C and E ⊂ ObjC, there exists a cohomological functor F (with values in some Abelian category) such that E is its set of zeros if (and only if) E is closed with respect to retracts and extensions (so, a certain Nullstellensatz is obtained for functors of this type). Moreover, if C is an R-linear category (where R is a commutative ring), this is also equivalent to the existence of an R-linear functor F: Coop → R- mod with this property. As a corollary, it is proved that an object Y belongs to the corresponding "envelope" of some D ⊂ ObjC whenever the same is true for the images of Y and D in all the categories Cp obtained from C via "localizing the coefficients" at maximal ideals p R. Moreover, certain new methods are developed for relating triangulated categories to their (nonfull) countable triangulated subcategories. The results of this paper can be applied to weight structures and triangulated categories of motives.

AB - The paper is aimed at proving the following: for a triangulated category C and E ⊂ ObjC, there exists a cohomological functor F (with values in some Abelian category) such that E is its set of zeros if (and only if) E is closed with respect to retracts and extensions (so, a certain Nullstellensatz is obtained for functors of this type). Moreover, if C is an R-linear category (where R is a commutative ring), this is also equivalent to the existence of an R-linear functor F: Coop → R- mod with this property. As a corollary, it is proved that an object Y belongs to the corresponding "envelope" of some D ⊂ ObjC whenever the same is true for the images of Y and D in all the categories Cp obtained from C via "localizing the coefficients" at maximal ideals p R. Moreover, certain new methods are developed for relating triangulated categories to their (nonfull) countable triangulated subcategories. The results of this paper can be applied to weight structures and triangulated categories of motives.

KW - Cohomological functors

KW - Envelopes

KW - Localization of the coefficients

KW - Separating functors

KW - Triangulated categories

UR - http://www.scopus.com/inward/record.url?scp=84999233013&partnerID=8YFLogxK

U2 - 10.1090/spmj/1425

DO - 10.1090/spmj/1425

M3 - Article

AN - SCOPUS:84999233013

VL - 27

SP - 889

EP - 898

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 6

ER -