  We study the exterior depth of an $E$-module and its exterior generic
annihilator numbers. For the exterior depth of a squarefree $E$-module we show
how it relates to the symmetric depth of the corresponding $S$-module and
classify those simplicial complexes having a particular exterior depth in terms
of their exterior shifting. We define exterior annihilator numbers analogously
to the annihilator numbers over the polynomial ring introduced by Trung and
Conca, Herzog and Hibi. In addition to a combinatorial interpretation of the
annihilator numbers we show how they are related to the symmetric Betti numbers
and the Cartan-Betti numbers, respectively. We finally conclude with an example
which shows that neither the symmetric nor the exterior generic annihilator
numbers are minimal among the annihilator numbers with respect to a sequence.
