2014/09/23 by Carlos E. Parra, Parra, Carlos E.
Mathematics · #13Dxx #16Exx #18E15 #18E30 #18Gxx #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.CT #msc:13Dxx #msc:16Exx #msc:18E15 #msc:18E30 #msc:18Gxx
paper · pdf · doi:10.48550/arxiv.1409.6639
Ph. D. thesis, 201 pages
arxiv created 2014/09/23 · openalex publication_date 2014/09/23 · arxiv updated 2014/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This thesis deals with the general problem of determining when the heart H of a t-structure in a triangulated category D is a Grothendieck or a module category. As preliminaries, we study Grothendieck conditions AB3-AB5 for H in a very general setting. We then concentrate on two familiar examples of smashing t-structures. First, we consider that D=D(G) is the (unbounded) derived category of a Grothendieck category G and that the t-structure is the one associated to a torsion pair t=(T,F) in G, usually known as Happel-Reiten-Smal∅ t-structure. In the second example studied, we assume that D=D(R) is the derived category of a commutative Noetherian ring R and that the t-structure is compactly generated. On what concern the Happel-Reiten-Smal∅ example, we show that if H=Ht is AB5, then F is closed under taking direct limits in G. Moreover, the converse is true, even implying that Ht is a Grothendieck category, for a wide class of torsion pairs in G which includes the hereditary, tilting and cotilting ones. When G=R-Mod is a module category, we are able to identify the hereditary torsion pairs t in R-Mod for which Ht is a module category. When R is a commutative noetherian ring, we show that all compactly generated t-structures in D(R) whose associated filtration by supports is left bounded have a heart H which is a Grothendieck category. This is used to identify all compactly generated t-structures in D(R) whose heart is a module category.