$(m,n)$-Quasitilted and $(m,n)$-almost hereditary algebras
Résumé
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 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 $m$-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 has injective dimension at
most $1$.
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