  Let $\Lambda$ be an artin algebra and $X$ a finitely generated
$\Lambda$-module. Iyama has shown that there exists a module $Y$ such that the
endomorphism ring $\Gamma$ of $X\oplus Y$ is quasi-hereditary, with a heredity
chain of length $n$, and that the global dimension of $\Gamma$ is bounded by
this $n$. In general, one only knows that a quasi-hereditary algebra with a
heredity chain of length $n$ must have global dimension at most $2n-2$. We want
to show that Iyama's better bound is related to the fact that the ring $\Gamma$
he constructs is not only quasi-hereditary, but even left strongly
quasi-hereditary: By definition, the left strongly quasi-hereditary algebras
are the quasi-hereditary algebras with all standard left modules of projective
dimension at most~1.
