Related papers: Completing perfect complexes
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the…
We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate…
For an abelian category and a distinguished object with a graded endomorphism ring a necessary and sufficient criterion is given so that the category is equivalent to the abelian quotient of the category of finitely presented graded modules…
These notes for a master class at Aarhus University (March 22--24, 2023) provide an introduction to the theory of completion for triangulated categories.
It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects,…
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in…
The reflexive completion of a category consists of the Set-valued functors on it that are canonically isomorphic to their double conjugate. After reviewing both this construction and Isbell conjugacy itself, we give new examples and revisit…
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…
We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with…
This work is concerned with approximability (\`{a} la Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes…
We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond…
In contrast with the Hovey correspondence of abelian model structures from two compatible complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from one hereditary complete cotorsion…
We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with…
A semiorthogonal decomposition for the bounded derived category (the category of perfect complexes in a non smooth case) of coherent sheaves on a Brauer Severi scheme is given. It relies on bounded derived categories (categories of perfect…
We revisit an old assertion due to Rouquier, characterizing the perfect complexes as bounded homological functors on the bounded complexes of coherent sheaves. The new results vastly generalize the old statement---first of all the ground…
The aim of this paper is to discus the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of…
We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…
Let $\mathcal C$ be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the category $\mathbf{C}(\mathcal C)$ of…
Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness which differ from proposals in previous works. Key to our approach is to treat them consequentially as…
The category of coherent sheaves over a noetherian scheme is very important for studying the properties of a given scheme. For noetherian schemes it is a well-known fact that the topology can be fully recovered from the corresponding…