Homological transfer between additive categories and higher differential additive categories (Q6541372)

Let \(\mathcal C\) be an additive category and \(n\geq 2\) be an integer. The higher differential additive category consists of objects \(X\) in \(\mathcal C\) equipped with an endomorphism \(\varepsilon_X\) satisfying \(\varepsilon_X^n=0.\) Let \(R\) be a finite-dimensional basic algebra over an algebraically closed field, mod\(\,R\) the category of finitely generated left \(R\)-modules, and \(T\) be the augmenting functor from mod\(\,R\) to mod\(\,R[t]/(t^n)\).\N\NWith the help of the theory of higher differential objects in additive categories, the article is concerned with investigating the transfer of some homological properties between \(R\) and \(R[t]/(t^n)\). The following theorem is the main result of the paper.\N\NTheorem. Let \(M\in\) mod\(\,R\). Then the following statements hold.\N\N(1) \(M\) is a \(\tau\)-rigid \(R\)-module if and only if \(T(M)\) is a \(\tau\)-rigid \(R[t]/(t^n)\)-module.\N\N(2) \(M\) is a \(\tau\)-tilting \(R\)-module if and only if \(T(M)\) is a \(\tau\)-tilting \(R[t]/(t^n)\)-module.\N\N(3) \(M\) is an almost complete \(\tau\)-tilting \(R\)-module if and only if \(T(M)\) is an almost complete \(\tau\)-tilting \(R[t]/(t^n)\)-module.\N\N(4) \(M\) is a support \(\tau\)-tilting \(R\)-module if and only if \(T(M)\) is a support \(\tau\)-tilting \(R[t]/(t^n)\)-module.