Partial Serre duality and cocompact objects

• Steffen Oppermann ^[2] ; Chrysostomos Psaroudakis ^[1] ; Torkil Stai ^[2]
  1. [1] Aristotle University of Thessaloniki

     Aristotle University of Thessaloniki

     Dimos Thessaloniki, Grecia

     
  2. [2] Institutt for matematiske fag, NTNU, Noruega
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 4, 2023
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in Oppermann et al. (Adv Math 350:190–241, 2019), motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities. We introduce “partial Serre duality”, which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an Auslander–Reiten theory, which in turn implies a weaker notion of duality which we call “non-degenerate composition”, and throughout this entire hierarchy of dualities the objects involved are 0-(co)compact. Furthermore, we produce explicit partial Serre functors for multiple flavors of homotopy categories, thus illustrating that this type of duality, as well as the resulting 0-cocompact objects, are abundant in prevalent triangulated categories.