AI 中文总结
本文研究遗传代数平凡扩张上的支持τ-倾斜模，给出τ-刚性T(A)-模的Hom零化条件与支持τ-倾斜判据，分类A型n线性定向箭图对应平凡扩张的秩2τ-刚性模，确定D₄型两定向的支持τ-倾斜相容图及F-三角形。
AI 中文摘要
设A是有限维基础遗传代数，T(A)=A⋉D(A)是其平凡扩张。基于不可分解τ-刚性T(A)-模的分类，我们给出刻画任意基础τ-刚性T(A)-模的显式Hom零化条件。对这类模M，我们还基于其对应的A-模U(M)的支撑确定其极大投射补，从而得到M为支持τ-倾斜的显式判据。作为应用，对A型n的线性定向箭图，我们分类T(𝕜Aₙ)上所有基础秩2的τ-刚性模，证明其数量为组合数binom(n,2)binom(n+1,2)；还证明每个基础τ-倾斜T(𝕜Aₙ)-模都包含一个不可分解直和项。最后，对D₄型箭图的两个定向，我们确定对应的支持τ-倾斜相容图及其面分布，由此得到并比较相关的F-三角形。
英文摘要
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.
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