  We consider a class of extensions of both abstract and pseudocompact
algebras, which we refer to as "strongly proj-bounded extensions". We prove
that the finiteness of the left global dimension and the support of the
Hochschild homology is preserved by strongly proj-bounded extensions,
generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we
show that the finiteness of the big left finitistic dimension is preserved by
strongly proj-bounded extensions. In order to construct examples, we describe a
new class of extensions of algebras of finite relative global dimension, which
may be of independent interest.
