Related papers: A note on gluing cosilting objects
We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat$^1$-objects that are both obtained by pullback diagrams with extra…
In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring…
In this paper we develop the theory of Artin-Wraith glueings for topological spaces. As an application, we show that some categories of compactifications of coarse spaces that agree with the coarse structures are invariant under coarse…
Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for…
It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite…
In this paper, we define what it means for an object in an abstract module category to be dualizable and we give a homological description of the direct limit closure of the dualizable objects. Our description recovers existing results of…
In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules.…
Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in \'etale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more…
Given an elementary simple-minded collection in the derived category of a non-positive dg algebra with finite-dimensional total cohomology, we construct a silting object via Koszul duality.
This paper studies the associativity of gluing of trajectories in Morse theory. We show that the associativity of gluing follows from of the existence of compatible manifold with face structures on the compactified moduli spaces. Using our…
This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a…
We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case…
We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the Gysin morphism associated to a projective…
We construct relative $3$-Calabi--Yau categories related with higher Teichm\"uller theory. We further study their corresponding cosingularity categories and the additive categorification of the corresponding cluster algebras. The input for…
We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we…
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…
We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these…
We study certain toric Gorenstein varieties with isolated singularities which are the quotient spaces of generic unimodular representations by the one-dimensional torus, or by the product of the one-dimensional torus with a finite abelian…
We provide various ways to characterise $\Sigma$-pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two…
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.