Separability and triangulated categories
Given a separable monad in a triangulated category satisfying some compatibilities, it is shown that the category of modules over that monad is again triangulated. This confirms a private conjecture of Giordano Favi. Consider a category \(\mathcal{C}\). Let \(M = (M:\mathcal{C}\to\mathcal{C}, \mu : M^2 \to M, \eta : \text{id}\to M)\) be a monad in \(\mathcal{C}\), i.e., suppose that \(\mu\circ M\mu = \mu\circ \mu M\) and \(\mu\circ M\eta = \mu\circ\eta M = \text{id}_M\,\). Call \(M\) separable if there exists \(\sigma : M\to M^2\) such that \(\mu\circ\sigma = \text{id}_M\) and \(M\mu\circ\sigma M = \sigma\circ\mu = \mu M\circ M\sigma\). A left \(M\)-module in \(\mathcal{C}\) is a pair \((X,\lambda : MX\to X)\) with \(X\in\text{Ob}\mathcal{C}\) such that \(\lambda\circ M\lambda = \lambda\circ\mu X\) and \(\lambda\circ \eta X = \text{id}_X\,\). For instance, given a field \(k\) and a \(k\)-algebra \(A\), the functor \(A\otimes_{k} - : k\text{-Mod}\to k\text{-Mod}\), together with multiplication and identity, is a monad in \(k\text{-Mod}\). It is separable if and only if the multiplication map \(A\otimes_{k} A\to A\) is a split epimorphism of \(A\)-\(A\)-bimodules, i.e., if and only if \(A\) is separable as a \(k\)-algebra. A left \((A\otimes_{k} -)\)-module in \(A\text{-Mod}\) is just a left \(A\)-module. Suppose given an endofunctor \(\Sigma : \mathcal{C}\to\mathcal{C}\). Call \(M\) stable if \(M\), \(\mu\) and \(\eta\) commute with \(\Sigma\). Call \(M\) stably separable if there exists a \(\sigma\) as above that moreover commutes with \(\Sigma\). Now suppose that \(\mathcal{C}\) is an idempotent complete Puppe triangulated category. Suppose that \(M\) is a stable and stably separable monad with \(M : \mathcal{C}\to\mathcal{C}\) exact. Then the category of left \(M\)-modules carries a unique Puppe triangulation such that the forgetful functor to \(\mathcal{C}\) is exact (4.3). Recall that in a Puppe triangulated category, every morphism, i.e., every \(\Delta_1\)-diagram, can be extended to a distinguished triangle, and every morphism of \(\Delta_1\)-diagrams can be extended to a morphism of distinguished triangles. A Verdier triangulated category is a Puppe triangulated category that satisfies the octahedral axiom. There is a slightly stronger version of a Verdier triangulated category, in which certain octahedra become distinguished in such a way that every commutative triangle, i.e., every \(\Delta_2\)-diagram, can still be extended to a distinguished octahedron, and that every morphism of \(\Delta_2\)-diagrams can be extended to a morphism of distinguished octahedra. In order to fit into a general framework, its distinguished triangles are renamed \(2\)-triangles, its distinguished octahedra are renamed \(3\)-triangles, and it is called a \(3\)-triangulated category. In said general framework then, given \(N\in [2,\infty]\), an \(N\)-triangulated category contains \(n\)-triangles for \(n\leqslant N\) with the expected properties. So \[ \dots\;\;\Rightarrow\;\;\text{\(4\)-triangulated}\;\Rightarrow\; \text{\(3\)-triangulated}\;\Rightarrow\;\text{Verdier}\;\Rightarrow\;\;\text{\(2\)-triangulated}\;\;\Leftrightarrow\;\;\text{Puppe}\;. \] (The notion of an \(N\)-triangulated category is not to be confused with the notion of an \(N\)-angulated category of Geiss, Keller and Oppermann, and also not, if \(N = \infty\), with the notion of a stable \((\infty,1)\)-category of Lurie.) Suppose that \(\mathcal{C}\) is an idempotent complete \(N\)-triangulated category and that \(M\) is a stable and stably separable monad with \(M : \mathcal{C}\to\mathcal{C}\) exact in the \(N\)-triangulated sense. Then the category of left \(M\)-modules carries a unique \(N\)-triangulation such that the forgetful functor to \(\mathcal{C}\) is exact in the \(N\)-triangulated sense (5.17). Applied, e.g., to a separable \(k\)-algebra \(A\), an equivalence, exact in the \(\infty\)-triangulated sense, between the category of left \((A\otimes_k -)\)-modules in \(\text{D}(k)\) and the category \(\text{D}(A)\) results (6.5).
scientific article; zbMATH DE number 5865624
scientific article