$(m,n)$-Quasitilted and $(m,n)$-almost hereditary algebras
Motivated by the study of $(m,n)$-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension 2 by a sequence of (co)tiltings involving $n-1$ tilting modules and $m-1$ cotilting modules, we introduce $(m,n)$-almost hereditary algebras. These are the algebras with global dimension $m+n$ and such that any indecomposable module has projective dimension at most $m$ or injective dimension at most $n$. We relate these two classes of algebras, among which $(m,1)$-almost hereditary ones play a special role. For the latter, we prove that any indecomposable module lies in the right part of the module category or in an $m$-analogue of the left part. This is based on the more general study of algebras the module categories of which admit a torsion-free subcategory such that any indecomposable module lies in that subcategory or has injective dimension at most $1$.