Related papers: Morita theory and singularity categories

We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central…

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These…

This paper provides a new class of examples for the Koszul dualities established in~\cite{5}. We study quadratic monomial algebras from the perspective of Koszul duality, with particular emphasis on finitely presented and finitely…

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study…

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…

We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally…

Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories M\otimes - :D(S)--> D(R). In particular, this characterisation…

In this paper we homologically construct a (functorial) BGG resolution of the finite-dimensional simple module of the nilBrauer algebra by using infinity-categorical methods following the reconstruction-from-stratification philosophy, e.g.…

We describe applications of Koszul cohomology to the Brill-Noether theory of rank 2 vector bundles. Among other things, we show that in every genus g>10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other…

To an artin algebra with radical square zero, a regular algebra in the sense of von Neumann and a family of invertible bimodules over the regular algebra are associated. These data describe completely, as a triangulated category, the…

Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules…

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an…

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…

This paper studies several singularity categories of a locally bounded $k-$linear category $\mathscr{C}$ with radical square zero. Following the work of Bautista and Liu [6], we give a complete description of $D^{b}_{sg}(\mathscr{C})$,…

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial…

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new…

Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded…

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement…

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jorgensen, we define the Castelnuovo-Mumford…

Let $\Lambda = \left[\begin{array}{cc} A & 0 \\ M & B \end{array}\right] $ be an Artin algebra and $_BM_A$ a $B$-$A$-bimodule. We prove that there is a triangle equivalence $D_{sg}(\Lambda) \cong D_{sg}(A)\coprod D_{sg}(B)$ between the…