2016/11/02 by Mikhail V. Bondarko, Bondarko, Mikhail V.
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.1611.00754
The paper contains a collection of results related to weight structures, t-structures, and (more generally) to torsion pairs. For any weight structure w we study (co)homological pure functors; these "ignore all weights except weight zero" and have already found several applications. We also study virtual t-truncations of cohomological functors coming from w. These are closely related to t-structures; so we prove in several cases (including certain categories of coherent sheaves) that w "gives" a t-structure (that is adjacent or Φ-orthogonal to it). We also study in detail "well generated" weight structures (and prove that any perfect set of objects generates a weight structure). The existence of weight structures right adjacent to compactly generated t-structures (and constructed using Brown-Comenetz duality) implies that the hearts of the latter have injective cogenerators and satisfy the AB3* axiom; actually, "most of them" are Grothendieck abelian (due to the existence of "regularly orthogonal" weight structures). It is convenient for us to use the notion of torsion pairs; these essentially generalize both weight structures and t-structures. We prove several properties of torsion pairs (that are rather parallel to that of weight structures); we also generalize a theorem of D. Pospisil and J. Stovicek to obtain a classification of compactly generated torsion pairs.