Related papers: Proper resolutions and Gorensteinness in extriangu…

For Gorenstein quotient spaces $C^d/G$, a direct generalization of the classical McKay correspondence in dimensions $d\geq 4$ would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not…

In this paper, we study ideal quotients of triangulated categories by higher cluster tilting subcategories. Koenig and Zhu proved that the ideal quotient by a $2$-cluster tilting subcategory is an abelian category; moreover, by Morita's…

We proposed, in our previous paper, to characterize the Hirota-Miwa equation by means of the theory of triangulated category. We extend our argument in this paper to support the idea. In particular we show in detail how the singularity…

In the paper, we investigate the lifting of recollements with respect to Gorenstein-projective modules. Specifically, a homological ring epimorphism can induce a lifting of the recollement of the stable category of finitely generated…

We construct a generalization of the Dolbeault-Grothendieck resolution on a singular complex space. The same construction yields, for each morphism of analytic spaces, a pullback mapping between the respective Dolbeault-Grothendieck…

Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We the propose the definition of a categorical resolution of singularities. Our main example is the derived category $D(X)$ of quasi-coherent…

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms…

Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting…

Let $B \subseteq A$ be an extension of finite dimensional algebras. We provide a sufficient condition for the existence of triangle equivalences of singularity categories (resp. Gorenstein defect categories) between $A$ and $B$. This result…

We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et…

Let $\mathscr{C}$ be an extriangulated category with enough projectives and injectives. We give the definitions of Wakamatsu-tilting subcategories and Wakamatsu-cotilting subcategories of $\mathscr{C}$ and show that they coincide with each…

We classify exactly when the toric algebras $\C[S_{\tree}(\br)]$ are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of $n-1$ points on $\mathbb{P}^{n-3}$, or…

We investigate purity within the Frobenius category of Gorenstein flat cotorsion modules, which can be seen as an infinitely generated analogue of the Frobenius category of Gorenstein projective objects. As such, the associated stable…

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory…

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q},C^{\ast}(X;\mathbb{Q}))$ with a…

The aim of this paper is twofold. On one hand, we prove a slight generalization of the stability for Gorenstein categories in [SWSW] and [Huang]; and show that the relative Auslander algebra of a CM-finite algebra is CM-free. On the other…

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the…

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…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…