Mapping cones of monomial ideals over exterior algebras

Let $K$ be a field, $V$ a finite dimensional $K$-vector space and $E$ the exterior algebra of $V$. We analyze iterated mapping cone over $E$. If $I$ is a monomial ideal of $E$ with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution $F$ of $I$ via the Cartan complex. Moreover, we provide an explicit description of the differentials in $F$ when the ideal $I$ has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.