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

Let $k$ be a field and let $\mathrm{\Lambda }={\left(\begin{array}{cc}A& N\\ M& B\end{array}\right)}_{\phi ,\psi }$ be a finite-dimensional Morita context algebra. We introduce a bilateral approximation construction which glues support $\tau$-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 $\tau$-tilting $\mathrm{\Lambda }$-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.