On locally coherent hearts (Q514629)

The notion of t-structure in a triangulated category was introduced by \textit{A. A. Beilinson} et al. [Astérisque 100, 172 p. (1982; Zbl 0536.14011)] as a a pair of full subcategories satisfying some axioms which guarantee that their intersection is an abelian category, called the heart of the t-structure. In the literature about t-structures a relevant question is the following: given a t-structure, what are the conditions that permit to assert that its heart is a Grothendieck category with good finiteness conditions? In the paper under review, the author tackle the question for the locally coherent condition, assuming that the t-structure lives in the (unbounded) derived category \({\mathcal D}({\mathcal G})\) of a Grothendieck category \({\mathcal G}\) which is itself locally coherent. More concretely, if the t-structure restricts to \({\mathcal D}^{b}(fp({\mathcal G}))\), the bounded derived category of the category of finitely presented (i.e., coherent) objects, in Proposition 4.5, the author gives a precise list of sufficient conditions on a t-structure in \({\mathcal D}({\mathcal G})\) so that its heart \({\mathcal H}\) is a locally coherent Grothendieck category on which \({\mathcal H}\cap {\mathcal D}^{b}(fp({\mathcal G}))\) is the class of its finitely presented objects. As a consequence the following results are proved: \textbf{1)} (see Theorem 5.2) Let \({\mathcal G}\) be a locally coherent Grothendieck category and \(\mathbf t =({\mathcal T}, {\mathcal F})\) be a torsion pair in \({\mathcal G}\). The associated Happel-Reiten-Smalø-t-structure in \({\mathcal D}({\mathcal G})\) restricts to \({\mathcal D}^{b}(fp({\mathcal G}))\) and has a heart which is a locally coherent Grothendieck category if, and only if, \({\mathcal F}\) is closed under taking direct limits in \({\mathcal G}\) and \(\mathbf t\) restricts to \(fp({\mathcal G})\); \textbf{2)} (see Theorem 6.2) If \(R\) is a commutative noetherian ring, then any compactly generated t-structure in \({\mathcal D}(R)\) which restricts to \({\mathcal D}^{b}_{fg}(R)\backsimeq {\mathcal D}^{b}(R-mod)\) has a heart \({\mathcal H}\) which is a locally coherent Grothendieck category on which \({\mathcal H}\cap {\mathcal D}^{b}_{fg}(R)\) is the class of its finitely presented objects. Finally, as a corollary (see Corollary 6.4) of this last theorem, the author proved that, if \(R\) is a commutative noetherian ring, then the heart of each t-structure in\({\mathcal D}^{b}_{fg}(R)\) is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.