  Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise
hereditary algebras obtained from quasitilted algebras of global dimension two
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 else injective dimension at most n. We
relate these two classes of algebras, among which (m,1)-almost hereditary ones
play a special role. For these, we prove that any indecomposable module lies in
the right part of the module category, or else in an m-analog 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 else has injective dimension at most n.
