Support $τ$-tilting modules over trivial extensions of hereditary algebras

Let $A$ be a finite-dimensional basic hereditary algebra and let $T(A)=A\ltimes D(A)$ be its trivial extension. Building on the classification of indecomposable $τ$-rigid $T(A)$-modules, we give explicit Hom-vanishing conditions characterizing arbitrary basic $τ$-rigid $T(A)$-modules. For such a module $M$, we also determine its maximal projective complement in terms of the support of the underlying $A$-module $U(M)$, and hence obtain an explicit criterion for $M$ to be support $τ$-tilting. As an application, for the linearly oriented quiver of type $A_n$, we classify all basic rank-two $τ$-rigid modules over $T(\Bbbk A_n)$ and prove that their number is \[ \binom{n}{2}\binom{n+1}{2}. \] We also show that every basic $τ$-tilting $T(\Bbbk A_n)$-module contains an indecomposable projective direct summand. Finally, for two orientations of a quiver of type $D_4$, we determine the corresponding support $τ$-tilting compatibility graphs and their face distributions, from which we obtain and compare the associated F-triangles.