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

Abstract

Let $A$ be a ring, let ${\bf T}$ be a finite-term big tilting complex over $A$, and choose a bounded projective representative $\cpx{P}$ of ${\bf T}$. Put $B=\End_{\D{A}}(\cpx{P})$. We prove that the kernel of the associated derived tensor functor ${\bf R}:\D{B}\to\D{A}$ is equivalent, through the derived inclusion, to $\D{\mathscr E}$, where $\mathscr E=\Ker({\bf R})\cap B\Modcat$ carries the inherited exact structure. Consequently, $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category $\mathscr E$ is definable and $d$-symmetric whenever $d$ is at least the projective amplitude of the underlying perfect right $B$-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when $\mathscr E$ is an abelian subcategory, or equivalently when the standard $t$-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.