Recollements of derived categories from $n$-term big tilting complexes

Abstract

Let $A$ be a ring and let $\bf T$ be an $n$-term big tilting complex over $A$, represented by a bounded complex $\cpx{P}$ of projective $A$-modules. Set $B=\End_{\D{A}}(\cpx{P}), \Lambda:=\dotEnd_A(\cpx{P})$ and $\Delta:=\tau_{\leq 0}\Lambda$. The associated complex of $A$-$B$-bimodule $\cpx{T}=\cpx{P}\otimesL_{\Delta}B$ is given. We construct an extension-closed exact subcategory $\mathscr E$ of $B\Modcat$ and prove that the derived category $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof passes through a dg double-centralizer description of a projective model of $\cpx{T}$ and an exact realisation theorem identifying $\D{\mathscr E}$ with the kernel of the derived tensor functor. We further show that $\mathscr E$ is $d$-symmetric for every $d$ not smaller than the amplitude of a perfect right $B$-model of $\cpx{T}$. Moreover, $\mathscr E$ is abelian if and only if the kernel is stable under the standard $t$-structure, equivalently, the recollement is induced by a homological ring epimorphism. In right $B$-amplitude at most one, the kernel is therefore abelian, and our construction specializes to the recollement arising from universal localisation in the two-term case. Thus the classical two-term picture extends to arbitrary finite-term big tilting complexes, with exact categories replacing abelian kernels in higher amplitude. Finally, we construct genuinely $n$-term non-compact big tilting complexes for every $n\geq2$, including an explicit three-term example whose left-hand term is determined by an algebraic Calkin-type quotient.