AI 中文总结
本文通过分裂倾斜三元组，将τ-倾斜模构造为τ_2-倾斜模，并推广了Peng的构造，确定了自同态代数及半正交分解。
AI 中文摘要
设 $(A,T,B)$ 为一个分裂倾斜三元组。我们提供了一种从 $\tau$-倾斜 $A$-模构造 $\tau_2$-倾斜 $B$-模的方法。另一方面，若 $A$ 是遗传的，通过考虑 $A$ 的某个两项倾斜复形的突变及其在导出等价 $\mathbf{R}{\rm Hom}_A(T,-):D^b(A)\to D^b(B)$ 下的像，我们得到 $B$ 的一个三项倾斜复形 ${\bf W}$。在两个直和项之间态射锥的显式非消失条件下，我们证明 ${\rm H}^0\boldsymbol{W}$ 是一个基本的 $2$-倾斜 $B$-模，并且它与从第一种构造得到的模一致。作为应用，我们将 Peng 的构造从倾斜模推广到 $\tau$-倾斜模。我们还确定了所构造模的自同态代数，它作为一个显式的三角矩阵代数，当所构造模是 $2$-倾斜时，它与 $A$ 和 $B$ 导出等价，并给出了 $K^b(\operatorname{proj}A)$ 的完全半正交分解。
英文摘要
Let $(A,T,B)$ be a splitting tilting triple. We provide a construction of $τ_2$-tilting $B$-modules from $τ$-tilting $A$-modules. On the other hand, if $A$ is hereditary, by considering the mutation of a certain two-term silting complex of $A$ and its image under the derived equivalence $\mathbf{R}{\rm Hom}_A(T,-):D^b(A)\to D^b(B)$, we get a three-term silting complex ${\bf W}$ of $B$. Under an explicit non-vanishing condition on the cones of morphisms between the two summands, we prove that ${\rm H}^0\boldsymbol{W}$ is a basic $2$-tilting $B$-module and that it coincides with the module obtained from the first construction. As an application, we generalize the construction of Peng from tilting modules to $τ$-tilting modules. We also determine the endomorphism algebra of the constructed module as an explicit triangular matrix algebra, which is derived equivalent to $A$ and $B$ whenever the constructed module is $2$-tilting, and give a full semi-orthogonal decomposition of $K^b(\operatorname{proj}A)$.
Comments30 pages
