Derived Equivalence induced by $n$-tilting modules

Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T’R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E{\perp}$ of $\D(\End(T’))$ equivalent to the quotient category of $\D(\End(T’))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S’}T’$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel’s results.

💡 Research Summary

The paper investigates the relationship between n‑tilting modules and derived equivalences over an arbitrary associative ring R. Starting from a right n‑tilting module T_R, the authors construct a new right n‑tilting module T′_R that is equivalent to T_R in the sense that it generates the same tilting class and satisfies the same Ext‑vanishing conditions. The endomorphism ring of T′_R is denoted S′ = End_R(T′). The central result is that the total left derived functor \