  The main conjectures in Iwasawa theory predict the relationship between the
Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved
formulation of the main conjecture, Greither and Kurihara described explicitly
the (initial) Fitting ideals of the Iwasawa modules for the cyclotomic
$\mathbb{Z}_p$-extensions of finite abelian extensions of totally real fields.
In this paper, we generalize the algebraic theory behind their work by
developing the theory of ``shifts of Fitting invariants.'' As applications to
Iwasawa theory, we obtain a noncommutative version and a two-variable version
of the work of Greither and Kurihara.
