  We improve upon the recent keystone result of Dasgupta-Kakde on the $\Bbb
Z[G(H/F)]^-$-Fitting ideals of certain Selmer modules $Sel_S^T(H)^-$ associated
to an abelian, CM extension $H/F$ of a totally real number field $F$ and use
this to compute the $\Bbb Z_p[[G(H_\infty/F)]]^-$-Fitting ideal of the Iwasawa
module analogues $Sel_S^T(H_\infty)_p^-$ of these Selmer modules, where
$H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$, for an odd prime $p$.
Our main Iwasawa theoretic result states that the $\Bbb
Z_p[[G(H_\infty/F]]^-$-module $Sel_S^T(H_\infty)_p^-$ is of projective
dimension $1$, is quadratically presented, and that its Fitting ideal is
principal, generated by an equivariant $p$-adic $L$-function
$\Theta_S^T(H_\infty/F)$. Further, we establish a perfect duality pairing
between $Sel_S^T(H_\infty)_p^-$ and a certain $\Bbb
Z_p[[G(H_\infty/F)]]^-$-module $\mathcal M_S^T(H_\infty)^-$, essentially
introduced earlier by Greither-Popescu. As a consequence, we recover the
Equivariant Main Conjecture for the Tate module $T_p(\mathcal
M_S^T(H_\infty))^-$, proved by Greither-Popescu under the hypothesis that the
classical Iwasawa $\mu$-invariant associated to $H$ and $p$ vanishes. As a
further consequence, we give an unconditional proof of the refined
Coates-Sinnott Conjecture, proved by Greither-Popescu under the same $\mu=0$
hypothesis, and also recently proved unconditionally but with different methods
by Johnston-Nickel, regarding the $\Bbb Z[G(H/F)]$-Fitting ideals of the higher
Quillen $K$-groups $K_{2n-2}(\mathcal O_{H,S})$, for all $n\geq 2$. Finally, we
combine the techniques developed in the process with the method of
''Taylor-Wiles primes'' to strengthen further the keystone result of
Dasgupta-Kakde and prove, as a consequence, a conjecture of Burns-Kurihara-Sano
on Fitting ideals of Selmer groups of CM number fields.

Number Theory;K-Theory and Homology