Related papers: Higher exact dg-categories
Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting…
We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We…
We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These…
The paper constructs new Hecke endomorphism algebras with a stratified structure. A novel feature of the proof is to approach difficult Ext^1 vanishing conditions by building entire exact category structures in which the analogous vanishing…
The paper focuses on investigating how certain relations between strict $n$-categories are preserved in a particular implementation of $(\infty,n)$-categories, given by saturated $n$-complicial sets. In this model, we show that the…
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple…
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors,…
For any CDG-ring $B^\bullet=(B^*,d,h)$, we show that the homotopy category of graded-projective (left) CDG-modules over $B^\bullet$ is equivalent to the quotient category of the homotopy category of graded-flat CDG-modules by its full…
Let k be a commutative noetherian ring. We construct a strictly-functorial presheaf of small dg-categories over k on the category of k-schemes of finite type, which gives dg-enhancements of the derived categories of perfect complexes.
We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure…
This paper establishes the homological and geometric foundations of non-commutative n-ary Gamma-semirings, unifying two previously distinct directions in Gamma-algebra: the derived Gamma-geometry developed for the commutative ternary case…
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact…
Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in…
We provide an equivalence between the dg category of coherent matrix factorizations and a certain dg category of absolute singularities. As an application, we compute the l-adic cohomology of the dg category of coherent matrix…
A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If $\Phi$ is a finite…
We generalise $\tau$-cluster morphism categories to the setting of non-positive dg algebras with finite dimensional cohomology in all degrees. The compatibility of silting reduction with support $\tau$-tilting reduction will be an essential…
A characteristic property of cohomology with compact support is the long exact sequence that connects the compactly supported cohomology groups of a space, an open subspace and its complement. Given an arbitrary cohomology theory of…
In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include…
We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…