We prove that if
B
⊆
A
B\subseteq A
is an extension of finite dimensional algebras such that the projective dimension of
A
/
B
A/B
as a
B
B
-bimodule is finite, then the finitistic dimension of
B
B
is finite whenever the finitistic dimension of
A
A
is finite. We exhibit examples demonstrating that the algebra
B
B
appearing in such an extension can be more complicated than
A
A
.
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