2019/07/08 by Bondarko, Mikhail V.
#16E65 #18E10 #18F20 #18G05 #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary 18E30 #Representation Theory (math.RT) #Secondary 18E40
paper · doi:10.48550/arxiv.1907.03686
A t-structure t=(Ct≤ 0,Ct≥ 0) on a triangulated category C is right adjacent to a weight structure w=(Cw≤ 0, Cw≥ 0) if Ct≥ 0=Cw≥ 0; then t can be uniquely recovered from w and vice versa. We prove that if C satisfies the Brown representability property then t that is adjacent to w exists if and only if w is smashing (i.e., coproducts respect weight decompositions); then the heart Ht is the category of those functors Hwop→ Ab that respect products. The dual to this statement is related to results of B. Keller and P. Nicolas. We also prove that an adjacent t exists whenever w is a bounded weight structure on a saturated R-linear category C (for a noetherian ring R); for C=Dperf(X), where the scheme X is regular and proper over R, this gives 1-to-1 correspondences between bounded weights structures on C and the classes of those bounded t-structures on it such that Ht has either enough projectives or injectives. We generalize this existence statement to construct (under certain assumptions) a t-structure t on a triangulated category C' such that C and C' are subcategories of a common triangulated category D and t is right orthogonal to w. In particular, if X is proper over R but not necessarily regular then one can take C=Dperf(X), C'=Dbcoh(X) or C'=D-coh(X), and D=Dqc(X). We also study hearts of orthogonal t-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal t-structures. The main tool of this paper are virtual t-truncations of (cohomological) functors; these are defined in terms of weight structures and "behave as if they come from t-truncations" whether t exists or not.