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We give a systematic construction of semiorthogonal decompositions of derived categories of coherent sheaves on quasi-smooth derived algebraic stacks over $\mathbb{C}$, where the summands are subcategories defined by weight conditions, and…
Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…
Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category $\mathcal{C}$ and $\mathcal{C}$-module category $\mathcal{M}$, we introduce a version…
We study the homological algebra of an R = Q/I module M using A-infinity structures on Q-projective resolutions of R and M. We use these higher homotopies to construct an R-projective bar resolution of M, Q-projective resolutions for all…
Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category…
The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a…
We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative…
Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this…
As an alternative perspective on the injectivity of a pure-injective module, a pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A…
For a finite quiver $Q$ without sinks, we consider the corresponding finite dimensional algebra $A$ with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective…
In this paper, we introduced a generalization of the derived category, which is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is…
For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived…
The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring $R$ and…
Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper,…
We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary…
Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…
In this paper we use A-infinity modules to study the derived category of a finite dimensional algebra over an algebraically closed field. We study varieties parameterising A-infinity modules. These varieties carry an action of an algebraic…
Forman's Discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoeldberg we show that this theory can be extended to chain complexes of free modules over a ring. We provide three applications…
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of…
Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the…