Support $τ$-tilting subcategories in exact categories

Abstract

Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $τ$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support $τ$-tilting modules (subcategories) in various context. It is also a generalization of tilting subcategories of exact categories. We show that there is a bijection between support $τ$-tilting subcategories and certain $τ$-cotorsion pairs. Given a support $τ$-tilting subcategory $\mathcal{T}$, we find a subcategory $\mathcal{E}_{\mathcal{T}}$ of $\mathcal{E}$ which is an exact category and $\mathcal{T}$ is a tilting subcategory of $\mathcal{E}_{\mathcal{T}}$. If $\mathcal{E}$ is Krull-Schmidt, we prove the cardinal $|\mathcal{T}|$ is equal to the number of isomorphism classes of indecomposable projectives $Q$ such that ${\rm Hom}_{\mathcal{E}}(Q,\mathcal{T})\neq 0$. We also show a functorial version of Brenner-Butler's theorem.