Related papers: An Auslander-Reiten principle in derived categorie…
Let $\Lambda$ be an artin algebra and $S(\Lambda)$ the category of all embeddings $(A\subseteq B)$ where $B$ is a finitely generated $\Lambda$-module and $A$ is a submodule of $B$. Then $S(\Lambda)$ is an exact Krull-Schmidt category which…
Of the many interesting insights in the Auslander-Bridger Memoir of 1969, the theory of Gorenstein dimension has most often been taken up by commutative algebraists. Over a local ring, it deals with resolutions by modules which are totally…
Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even…
We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing…
We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field C as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some…
For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical…
Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point…
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely…
We extend the notion of type sequence to rings that are not necessarily residually rational. Using this invariant we characterize different types of rings as almost Gorenstein rings and rings of maximal length.
This paper is devoted to study the relationship between two important notions in ring theory, category theory, and representation theory of Artin algebras; namely, Gabriel topologies and higher Auslander(-Gorenstein) algebras. It is…
We introduce Auslander-Reiten sequences for group algebras and give several recent applications. The first part of the paper is devoted to some fundamental problems in Tate cohomology which are motivated by homotopy theory. In the second…
We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with…
For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A^{{\rm op}})$ and ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A)$. If $A=\Lambda$ is an…
Let $\Lambda$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $\Lambda$-mod with enough projective and injective objects and $\mathcal S (\mathcal C)$ be its monomorphism category. It turns…
We study the relationship between Serre duality and the Mukai pairing for smooth and proper dg-algebras. We introduce an alternative definition of the Mukai pairing and prove that it coincides with the Mukai pairings defined by…
We introduce a new combinatorial condition on a subinterval of a poset P (a clamped subinterval) that allows us to relate the Auslander-Reiten quiver of the bounded derived category of P to that of the subinterval. Applications include the…
We give a counterexample to a conjecture posed by S. Ding regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and…
In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example…
The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, \emph{J. Algebra} \textbf{323} (2010) 2041-2057] to the…
According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to…