We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau–N category D(Gamma_N Q) associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group Br(Q) acts freely upon it by spherical twists, in particular that the spherical twist group Br(Gamma_N Q) is isomorphic to Br(Q). This generalises the result of Brav–Thomas for the N=2 case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.

Universal extensions arise naturally in the Auslander bijections. For an abelian category having Auslander-Reiten duality, we exploit a bijection triangle, which involves the Auslander bijections, universal extensions and the Auslander-Reiten duality. Some consequences are given, in particular, a conjecture by Ringel is verified.

We show that the standard graded cover of the well-known category $\mathcal{O}$ of Bernstein–Gelfand–Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.

We propose the notion of partial resolution of a ring, which is by definition the endomorphism ring of a certain generator of the given ring. We prove that the singularity category of the partial resolution is a quotient of the singularity category of the given ring. Consequences and examples are given.

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.