Hereditary cotorsion pairs and silting subcategories in extriangulated categories
The paper under review studies hereditary cotorsion pairs in extriangulated categories, and their relation with silting subcategories and co-t-structures. Let \((\mathcal{C},\mathbb{E},\mathfrak{s})\) be an extriangulated category, where \(\mathcal{C}\) is an additive \(R\)-linear category (\(R\) an associative ring with identity), \(\mathbb{E} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \longrightarrow \mathsf{Mod}(R)\) is an \(R\)-bilinear functor, and \(\mathfrak{s}\) is a realization of \(\mathbb{E}\). Two subcategories \(\mathcal{X}\) and \(\mathcal{Y}\) of \(\mathcal{C}\) form a (complete) cotorsion pair \((\mathcal{X,Y})\) if \(\mathcal{X}\) and \(\mathcal{Y}\) are closed under direct summands, \(\mathbb{E}(\mathcal{X,Y}) = 0\), and for every object \(C \in \mathcal{C}\) there exist two \(\mathfrak{s}\)-conflations \(Y \to X \to C \dashrightarrow\) and \(C \to Y' \to X' \dashrightarrow\) with \(X, X' \in \mathcal{X}\) and \(Y, Y' \in \mathcal{Y}\). If in addition \(\mathbb{E}^k(\mathcal{X,Y}) = 0\) for every \(k \geq 2\), then \((\mathcal{X,Y})\) is said to be hereditary. The collection of cotorsion pairs form a partially ordered set with the following relation: given \(x_1 = (\mathcal{X}_1,\mathcal{Y}_1)\) and \(x_2 = (\mathcal{X}_2,\mathcal{Y}_2)\), then \(x_1 \leq x_2\) if \(\mathcal{Y}_1 \subseteq \mathcal{Y}_2\). In this case, \([x_1,x_2]\) denotes the interval of cotorsion pairs \(x\) for which \(x_1 \leq x \leq x_2\). In their first main result (Theorem 3.6), the authors show the existence of an isomorphism between the interval \([x_1,x_2]\) and the poset \(\mathsf{cotors}(\mathcal{H}_{[x_1,x_2]})\) of cotorsion pairs in the coheart \(\mathcal{H}_{[x_1,x_2]} := \mathcal{X}_1 \cap \mathcal{Y}_2\). Namely, \(\Phi \colon [x_1,x_2] \to \mathsf{cotors}(\mathcal{H}_{[x_1,x_2]})\) is given by \(\Phi(\mathcal{X,Y}) := (\mathcal{X} \cap \mathcal{Y}_2,\mathcal{X}_1 \cap \mathcal{Y})\) with inverse \(\Psi \colon \mathsf{cotors}(\mathcal{H}_{[x_1,x_2]}) \to [x_1,x_2]\) defined as \(\Psi(\mathcal{A,B}) := (\mathsf{add}(\mathcal{X}_2 \ast \mathcal{A}),\mathsf{add}(\mathcal{B}\ast \mathcal{Y}_1))\), where \(\mathsf{add}(\mathcal{X}_2 \ast \mathcal{A})\) denotes the smallest subcategory of \(\mathcal{C}\) containing \(\mathcal{X}_2 \ast \mathcal{A}\) (the subcategory of objects \(C\) in \(\mathcal{C}\) admitting an \(\mathfrak{s}\)-conflation \(X \to C \to A \dashrightarrow\) with \(X \in \mathcal{X}_2\) and \(A \in \mathcal{A}\)) which is closed under direct summands and finite direct sums. In the case where \(\mathcal{C}\) is a triangulated category, the previous isomorphism recovers a result by \textit{D. Pauksztello} and \textit{A. Zvonareva} [``Co-t-structures, cotilting and cotorsion pairs'', Preprint, \url{arXiv:2007.06536}], namely, an isomorphism between the interval of co-t-structures and the poset of cotorsion pairs in its coheart (see Corollary 3.9). The second main result in this article establishes a bijective correspondence between the set of bounded hereditary cotorsion pairs in an extriangulated category \(\mathcal{C}\) and the set of silting subcategories of \(\mathcal{C}\). The latter is a generalization, proposed by the authors, of the notion of silting subcategories in a triangulated category. Specifically, \(\mathcal{M} \subseteq \mathcal{C}\) is silting if \(\mathcal{M}\) is closed under direct summands and finite direct sums, \(\mathbb{E}^k(\mathcal{M,M}) = 0\) for every \(k \geq 1\), and \(\mathcal{C}\) is the smallest thick subcategory containing \(\mathcal{M}\). In this setting, it is proved in Theorem 5.7 that the map \((\mathcal{X,Y}) \mapsto \mathcal{X} \cap \mathcal{Y}\) gives rise to the bijection mentioned before, with inverse given by \(\mathcal{M} \mapsto (\mathcal{M}^\vee,\mathcal{M}^\wedge)\). Here, \(\mathcal{M}^\wedge\) and \(\mathcal{M}^\vee\) are the classes of objects in \(\mathcal{C}\) that can be finitely resolved or coresolved by objects in \(\mathcal{X}\), where (co)resolutions are thought in terms of \(\mathfrak{s}\)-conflations. For particular choices of \(\mathcal{C}\), the authors recover several well known results. For instance, if \(\mathcal{C}\) is triangulated, then there is a bijection between bounded co-t-structures in \(\mathcal{C}\) and silting subcategories in \(\mathcal{C}\), as proved by \textit{O. Mendoza Hernández} et al. [Appl. Categ. Struct. 21, No. 5, 417--440 (2013; Zbl 1291.18017)]. Moreover, the bijection between the set of isomorphism classes of basic tilting modules and the set of contravariantly finite resolving subcategories [\textit{M. Auslander} and \textit{I. Reiten}, Adv. Math. 86, No. 1, 111--152 (1991; Zbl 0774.16006)] is another application that follows from Theorem 5.7 and other bijections proved by the authors that involve silting subcategories of extriangulated categories.
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