In this paper, we prove that every invertible 2-local or local automorphism of a simple generalized Witt algebra over any field of characteristic 0 is an automorphism. Furthermore, every 2-local or local automorphism of Witt algebras W_n is an automorphism for all n∈N. But some simple generalized Witt algebras indeed have 2-local (and local) automorphisms that are not automorphisms.

In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

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.