Narrow Results

Brane transport provides a way to realize functors between the categories of B-branes of different phases in gauged linear sigma model (GLSM). When appropriately designed, these functors induce derived equivalence between Calabi–Yau manifolds. In some cases, brane transport can also be used to extract the homological projective dual (HPD) of certain projective embeddings. We describe the GLSMs realizing derived equivalence between different geometries via brane transport with emphasis on the nonabelian models, and how to construct the GLSMs where brane transport gives rise to the embedding of HPD category into the derived category of universal hyperplane section. The latter gives rise to an explanation of the relationship between GLSM and HPD, and also a noncommutative geometric interpretation to HPD.

A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type 𝔸 or Euclidean type . We classify all finite-dimensional algebras which are derived equivalent to gentle cluster tilted algebras.

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the category of left modules over the coproduct of R and S^op in an appropriate category. Letting K denote a field, we apply this to show that the category of K-representations of a quiver that is reflection equivalent to a path algebra KΓ is equivalent to a full subcategory of left and right K-representations of KΓ.

Given a noetherian abelian k-category of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category and the abelian category of modules over End(T)^op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

In this paper, we construct derived equivalences between subalgebras of some $\mathrm{\Phi }$-Auslander–Yoneda algebras from a class of triangles in idempotent complete triangulated categories. The derived equivalences are obtained by transferring subalgebras induced by triangles to endomorphism algebras induced by approximation sequences.

In this paper, we characterize all the finite-dimensional algebras that are derived equivalent to an $m$-cluster tilted algebras of type $\widetilde{𝔸}$. This generalizes a result of Bobiński and Buan [The algebras derived equivalent to gentle cluster tilted algebras, J. Algebra Appl.11(1) (2012), Article ID:1250012, 26 pp.].

In this note, we construct two-sided tilting complexes corresponding to one-sided tilting complexes for Brauer tree algebras.

In this paper, we compute the dimension of the Hochschild cohomology groups of any $m$-cluster tilted algebra of type $\widetilde{𝔸}$. Moreover, we give conditions on the bounded quiver of an $m$-cluster tilted algebra $\mathrm{\Lambda }$ of type $\widetilde{𝔸}$ such that the Gerstenhaber algebra ${HH}^{\ast }(\mathrm{\Lambda })$ has nontrivial multiplicative structures. We also show that the derived class of gentle $m$-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

Crawley-Boevey introduced in [Poisson structure on moduli spaces of representations, J. Algebra 325 (2011) 205–215.], the notion of $H_0$-Poisson structure for associative algebras, which is the weakest condition that induces a Poisson structure on the moduli spaces of their representations. In this paper, by using a result of Armenta and Keller in [Derived invariance of the Tamarkin-Tsygan calculus of an algebra, C. R. Math. Acad. Sci. Paris 357(3) (2019) 236–240.], we show that an $H_0$-Poisson structure is preserved under derived Morita equivalence.

Let $A=A(\alpha ,\beta )$ be a graded down-up algebra with $(degx,degy)=(1,n)$ and $\beta \ne 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n\ge 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether (10) $\left(\right.\begin{array}{cc}\alpha \hfill & 1\hfill \\ \beta \hfill & 0\hfill \end{array}{\left)\right.}^n$$\left(\right.\begin{array}{c}1\hfill \\ 0\hfill \end{array}\left)\right.$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\ge 3$ in the context of Grothendieck groups.