For a finitedimensional algebra
$\Lambda $
and a nonnegative integer
$n$,
we characterize when the set
${\text{tilt}}_{n}\Lambda $
of additive equivalence classes of tilting modules with projective dimension at most
$n$ has a minimal
(or equivalently, minimum) element. This generalizes results of Happel and Unger. Moreover, for
an
$n$Gorenstein
algebra
$\Lambda $ with
$n\ge 1$, we construct a
minimal element in
${\text{tilt}}_{n}\Lambda $.
As a result, we give equivalent conditions for a
$k$Gorenstein
algebra to be Iwanaga–Gorenstein. Moreover, for a
$1$Gorenstein algebra
$\Lambda $ and its factor algebra
$\Gamma =\Lambda \u2215\left(e\right)$, we show that there is
a bijection between
${\text{tilt}}_{1}\Lambda $
and the set
$\text{s}\tau \text{tilt}\Gamma $
of additive equivalence classes of basic support
$\tau $tilting
$\Gamma $modules, where
$e$ is an idempotent
such that
$e\Lambda $
is the additive generator of the category of projectiveinjective
$\Lambda $modules.
