When a ring is not enough
Some higher-term tilting complexes have a leftover structure that is both nonzero and non-abelian in their derived tensor kernels. In the constructed examples, no homological ring epimorphism can realize that kernel. The result points to exact categories as a more flexible way to describe higher-term cases than the ordinary-ring picture used for simpler two-term constructions.
The paper asks how to turn such a kernel into a concrete mathematical object. Its setting begins with a ring A and a big tilting complex T with a bounded projective representative. From T, the authors form an endomorphism ring B and a derived tensor functor. They define E as the extension-closed part of B-Mod that lies inside the functor's kernel. An exact category is a setting with a controlled collection of exact sequences, without requiring the full structure of an abelian category.
The proof combines a differential-graded double-centralizer argument, which keeps the algebra's grading in view, with a functorial realization of the kernel on unbounded complexes whose degrees are not confined to a fixed finite interval.
The main payoff is a precise equivalence: the derived category built from E is triangle-equivalent to the tensor kernel. At the same time, the derived category of B sits in a recollement with the derived categories of E and A. A recollement is a formal arrangement that links one category to two related pieces, giving the kernel a concrete exact-category home.
The dividing line
The paper identifies a clear boundary between exact-category and ordinary-ring descriptions. The kernel comes from a homological ring epimorphism exactly when E is abelian. The same condition can be expressed by saying that the standard t-structure, which separates complexes by cohomological degree, restricts to the kernel. When that condition fails, the exact-category description remains available, but the ordinary-ring realization does not.
The familiar two-term picture appears as a special case. When the relevant projective model has right projective amplitude at most one, the abelian, t-structure and epimorphism conditions all hold. For two-term big tilting complexes, the result recovers homological universal localization. Higher-term complexes extend that picture while allowing the kernel to lose its abelian form.
Examples that force the issue
One family comes from an idempotent-chain construction. It turns an infinite amplification into an n-term big tilting complex exactly when the stated Hom and Ext vanishing conditions hold. If every L_i in the construction is nonzero, the complex is genuinely n-term. In this family, a nonzero quotient Cn is a homological ring epimorphism, the tensor kernel is described by annihilation by the final idempotent en, and the exact heart En is equivalent to Cn-Mod.
A different construction uses a connected singular local ring. It produces a connected, non-compact big tilting complex whose projective representative runs from degree minus d through degree zero. No representative can be supported in fewer than d+1 consecutive degrees, and its cohomology is nonzero in degree zero and in every negative degree from minus 1 through minus d. The lower bound makes these examples genuinely higher-term.
In that singular family, E is nonzero but not an abelian subcategory of B-Mod. Consequently, the tensor kernel is nonzero and cannot be realized by a homological ring epimorphism. The same example also rules out a different route back to ordinary rings: no compact tilting complex satisfies the required orthogonality condition against T, so T cannot be obtained from a module through a derived equivalence between ordinary rings.
A framework with clear boundaries
The general framework also gives structural control short of an ordinary-ring model. E is definable in B-Mod and is closed under filtered colimits, pure submodules, direct summands, products, coproducts and extensions. If a bounded projective right-B model is supported between degrees a and b, E is d-symmetric whenever d is at least the width b minus a. For semiperfect endomorphism rings, reduction modulo the Jacobson radical determines the optimal right projective amplitude and supplies a finite sufficient test for ordinary-ring realization.
These conclusions are conditional rather than universal. The recollement and abelian-kernel statements require the paper's big-tilting hypotheses, the finite amplitude test is restricted to semiperfect endomorphism rings, and the non-abelian kernel and transport obstruction are established for constructed examples. The examples therefore show what can happen at higher amplitude, not what happens for every finite-term big tilting complex.
The front matter identifies the document as arXiv version 2 in math.RT, dated 9 October 2026.
Paper data and sources
Original title: Recollements of derived categories from $n$-term big tilting complexes
Authors: Shengyong Pan, Huabo Xu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text