Derived equivalence induced by infinitely generated 𝑛-tilting modules

Abstract

Let T R T_R be a right n n -tilting module over an arbitrary associative ring R R . In this paper we prove that there exists an n n -tilting module T R ′ T’_R equivalent to T R T_R which induces a derived equivalence between the unbounded derived category D ( R ) \mathcal {D}(R) and a triangulated subcategory E ⊥ \mathcal E_{\perp } of D ( End ⁡ ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) equivalent to the quotient category of D ( End ⁡ ( T ′ ) ) \mathcal {D}(\operatorname {End}(T’)) modulo the kernel of the total left derived functor − ⊗ S ′ L T ′ -\otimes ^{\mathbb L}_{S’}T’ . If T R T_R is a classical n n -tilting module, we have that T = T ′ T=T’ and the subcategory E ⊥ \mathcal E_{\perp } coincides with D ( End ⁡ | ( T ) ) \mathcal {D}(\operatorname {End}|(T)) , recovering the classical case.