Support $\tau$-tilting modules over trivial extensions of hereditary algebras

Let $A$ be a finite-dimensional basic hereditary algebra and let $T(A)=A⋉D(A)$ be its trivial extension. Building on the classification of indecomposable $\tau$-rigid $T(A)$-modules, we give explicit Hom-vanishing conditions characterizing arbitrary basic $\tau$-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 $\tau$-tilting. As an application, for the linearly oriented quiver of type $A_n$, we classify all basic rank-two $\tau$-rigid modules over $T(\mathbb{k}{A}_n)$ and prove that their number is $$\left(\genfrac{}{}{0px}{}{n}{2}\right)\left(\genfrac{}{}{0px}{}{n+1}{2}\right)\mathrm{.}$$ We also show that every basic $\tau$-tilting $T(\mathbb{k}{A}_n)$-module contains an indecomposable projective direct summand. Finally, for two orientations of a quiver of type $D_4$, we determine the corresponding support $\tau$-tilting compatibility graphs and their face distributions, from which we obtain and compare the associated F-triangles.