Relative left Bongartz completions and their compatibility with mutations

Abstract

In this paper, we introduce relative left Bongartz completions for a given basic $\tau$-rigid pair (U, Q) in $\textsf {mod} \,A$. They give a family of basic $\tau$-tilting pairs containing (U, Q) as a direct summand. More precisely, to each basic $\tau$-tilting pair (M, P) with $\textsf {Fac} \,M\subseteq ^{\bot }_{}{(\tau U)}\cap Q^\bot$, we prove that $\textsf {Fac} \,U*({\mathcal {W}}\cap \textsf {Fac} \,M)$ is a functorially finite torsion class in $\textsf {mod} \,A$ such that $\textsf {Fac} \,U\subseteq \textsf {Fac} \,U*({\mathcal {W}}\cap \textsf {Fac} \,M) \subseteq ^{\bot }_{}{(\tau U)}\cap Q^\bot$, where $\mathcal W:=U^\bot \cap ^{\bot }_{}({\tau U})\cap Q^\bot$ is the wide subcategory associated to (U, Q). Thus $\textsf {Fac} \,U*({\mathcal {W}}\cap \textsf {Fac} \,M)$ corresponds to a basic $\tau$-tilting pair $(M^-,P^-)$ containing (U, Q) as a direct summand. We call $(M^-,P^-)$ the left Bongartz completion (or Bongartz co-completion) of (U, Q) with respect to (M, P). Notice that $(M^-,P^-)$ coincides with the usual Bongartz co-completion of (U, Q) in $\tau$-tilting theory if $(M,P)=(0,A)$. We prove that relative left Bongartz completions have nice compatibility with mutations. Using this compatibility we are able to study the existence of maximal green sequences under $\tau$-tilting reduction. We also explain our construction and some of the results in the setting of silting theory.