Support $τ$-tilting modules over Morita context algebras: A bilateral approximation approach

Abstract

Let $k$ be a field and let $Λ=\left(\begin{smallmatrix}A&N\\M&B\end{smallmatrix}\right)_{ϕ,ψ}$ be a finite-dimensional Morita context algebra. We introduce a bilateral approximation construction which glues support $τ$-tilting modules over $A$ and $B$ by alternately correcting the two corner components through minimal approximations and pushouts. When this process terminates, it yields a support $τ$-tilting $Λ$-module with the prescribed componentwise torsion class. The one-sided case recovers Zhang's triangular-matrix construction, while the two-sided compatibility conditions give direct corner induction and, for radical-valued connecting maps, are also necessary, extending the Gao--Huang criterion. Examples show that the bilateral correction process can terminate even when neither one-sided compatibility condition is satisfied, while in other examples the process never terminates.