Locally finitely presented and coherent hearts

Abstract

Given a torsion pair t = (T , F ) in a Grothendieck category G, we study when the heart H t of the associated Happel-Reiten-Smalø t-structure in the derived category D(G) is a locally finitely presented or a locally coherent Grothendieck category. Since H t is Grothendieck precisely when t is of finite type (i.e., F is closed under direct limits), we first study the latter torsion pairs showing that, as in modules, they are precisely the quasi-cotilting ones, that in turn coincide with the cosilting ones.We then prove that, for G chosen in a wide class of locally finitely presented Grothendieck categories that includes the locally coherent ones, the module categories and several categories of quasi-coherent sheaves over schemes, the heart H t is locally finitely presented if, and only if, t is generated by finitely presented objects. For the same class of Grothendieck categories, it is then proved that if F is a generating class in G, in which case it is known that t is given by a (1-)cotilting object Q, the heart H t is locally coherent if, and only if, it is generated by finitely presented objects and there is a set X ⊂ F ∩ fp(G) that is a set of generators of G and satisfies the following two conditions:(1) Ext 1 G (X, −) vanishes on direct limits of objects in Prod(Q), for all X ∈ X ; (2) each epimorphism p :has a kernel which is a direct summand of (1 : t)(N ), for some N ∈ fp(G). A consequence of this is that, when G = Mod-A is the module category over small pre-additive category A (e.g., over an associative unital ring) and F is generating in Mod-A, the heart H t is locally coherent if, and only if, t is generated by finitely presented modules and, for each M ∈ mod-A := fp(Mod-A), the module (1 : t)(M ) admits a projective resolution with finitely generated terms.A further consequence is that if R is a right coherent ring and t = (T , F ) is a torsion pair such that the torsion ideal t(R) is finitely generated on the right, then H t is a locally coherent Grothendieck category if, and only if, t is generated by finitely presented modules and it is given by a cosilting module which is an elementary cogenerator in Mod-R.

Reflective and coreflective subcategories

“…The canonical finitely accessible subcategory of a Grothendieck category. The following is an extended version of [26,Lemma 1.11]. Proposition 9.16 (Generalized Lazard's Trick).…”

Tilting preenvelopes and cotilting precovers in general Abelian categories

“…We apply an argument due to Lazard (see [PSV21,Lem. 1.11]), that we just outline here, to show that T ∈ lim − → pres(V ).…”

“…Remark 6.10. In [PSV21] (see the discussion right after Proposition 8.19 in [Op.Cit. ]), we have shown that the answer to the above question is negative when Mod-R is replaced by a locally coherent Grothendieck category.…”

Local Coherence of Hearts Associated with Thomason Filtrations

“…i.e., by definition of Yoneda ext-group, there is an index γ ∈ Λ such that ξ i factors as the pushout diagram (see again [PSV19] for details)…”

“…On the other hand, crucial results concerning the finiteness conditions have been achieved e.g. in [Sao17,PSV19]. An exhaustive survey devoted to the study of HRS hearts and related topics is [PS20].…”