  In this paper we define module biprojctivity and module biflatness for a
Banach algebra which is a Banach module over another Banach algebra with
compatible actions, and find their relation to classical biprojectivity and
biflatness. As a typical example, We show that for an inverse semigroup $S$
with an upward directed set of idempotents $E$, the semigroup algebra $ \ell
^{1}(S)$, as an $\ell ^{1}(E)$-module, is module biprojective if and only if an
appropriate group homomorphic image of $S$ is finite. Also we show that $\ell
^{1}(S)$ is module biflat if and only if $S$ is amenable.
