Related papers: Stable categories and reconstruction
If two cluster-tilting objects of an acyclic cluster category are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent [Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a simple module".…
We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of the class of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain…
Recently, we obtained in [7] a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an…
By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e. the orbit category of the derived…
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 develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the…
We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$…
We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic $d$-Calabi-Yau triangulated category with a $d$-cluster tilting object $T$ such that its shifted sum…
We give a general construction of realization functors for $t$-structures on the base of a strong stable derivator. In particular, given such a derivator $\mathbb D$, a $t$-structure $\mathbf t=(\mathcal D^{\leq0},\mathcal D^{\geq0})$ on…
Several authors have studied the filtered colimit closure lim(B) of a class B of finitely presented modules. Lenzing called lim(B) the category of modules with support in B, and proved that it is equivalent to the category of flat objects…
A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ is fully faithful on the full subcategory of projective $A$-modules, for a given projective $A$-module $P$. By definition,…
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an…
In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on…
We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…
Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra ({\em full centre}) in the monoidal centre of the monoidal…
The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods…
We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash…
We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…