We study cluster categories arising from marked surfaces (with punctures and nonempty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of Ext1 as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

We give two homological proofs of the Noetherianity of the category FI, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.

We present a version of a proof by Andy Chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems. This proof differs from the one in [Ch2] in that it is based on the computation of derived functors of certain inverse limits. This leads to a much shorter proof, but one which is aimed mostly at researchers familiar with homological algebra.

In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.

Let $(R, \frak{m}, k_{R})$ be a regular local$k$-algebra satisfying the weak Jacobian criterion, and such that$k$_{$R$}/$k$is an algebraic field extension. Let $\mathcal{D}_{R}$ be the ring of$k$-linear differential operators of$R$. We give an explicit decomposition of the $\mathcal{D}_{R}$ -module $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}$ as a direct sum of simple modules, all isomorphic to $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}$ , where certain “Pochhammer” differential operators are used to describe generators of the simple components.