Separability and triangulated categories
type: Item
instance of: scholarly article
title: Separability and triangulated categories
zbMATH Open document ID: 1236.18013
author: Paul Balmer
published in: Advances in Mathematics
publication date: 2011-03-14T00:00:00Z
review text: 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 \(\Delta_1\)-dia…
reviewed by: Matthias Künzer
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 5865624
zbMATH Keywords: triangulated category
zbMATH Keywords: separable monad
zbMATH Keywords: separable ring object
zbMATH Keywords: étale algebra, Kleisli categories
zbMATH Keywords: Eilenberg-Moore categories
zbMATH Keywords: higher octahedra
zbMATH Keywords: Verdier triangulated category
zbMATH Keywords: Puppe triangulated category
zbMATH Keywords: octahedral axiom
MaRDI profile type: Publication
full work available at URL: j.aim.2010.12.003
OpenAlex ID: W2057285450
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DOI: 10.1016/J.AIM.2010.12.003
description: scientific article; zbMATH DE number 5865624