Related papers: Detecting derived equivalences with the CHZ criter…

For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in Z}, is equivalent to the category of "quasi-coherent modules" over an associated Z-algebra A, we show that under suitable cohomological conditions "taking…

An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are…

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support…

Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support {\tau}-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given…

This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an…

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our…

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian} matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian {matrices is invariant under addition, switching,…

We study equivalences induced by a silting module $T$ or, equivalently, by a complex of projectives $\mathbb{P}$, concentrated in $-1$ and $0$ which is silting in the derived category $\mathbf{D}(R)$ of a ring $R$.

Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the…

In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin-Zhang equivalence conjecture. Our tautological relations have…

The main theme of this paper is to study $\tau$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $\tau$-cotorsion torsion triples and show a bijection between the…

We study Homological Mirror Symmetry (HMS) for $A_n$-resolutions from the SYZ viewpoint. Let $X\to\bC^2/\bZ_{n+1}$ be the crepant resolution of the $A_n$-singularity. The mirror of $X$ is given by a smoothing $\check{X}$ of…

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and…

For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$…

For a balanced pair $(\mathcal{X},\mathcal{Y})$ in an abelian category, we investigate when the chain homotopy categories ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ are triangulated equivalent. To this end, we realize these chain…

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting…

For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are…

We give a proof, based on the rigidity of tilting complexes, that the class of self-injective finite-dimensional algebras over an algebraically closed field is closed under derived equivalence.

We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by topological T-duality, Bouwknegt an Mathai study the complex of forms on an…

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a…