  A new definition of analytic adjoint ideal sheaves for quasi-plurisubharmonic
(quasi-psh) functions with only neat analytic singularities is studied and
shown to admit some residue short exact sequences which are obtained by
restricting sections of the newly defined adjoint ideal sheaves to some unions
of $\sigma$-log-canonical ($\sigma$-lc) centres. The newly defined adjoint
ideal sheaves induce naturally some residue $L^2$ norms on the unions of
$\sigma$-lc centres which are invariant under log-resolutions. They can also
describe unions of $\sigma$-lc centres without the need of log-resolutions even
if the quasi-psh functions in question are not in a simple-normal-crossing
configuration. This is hinting their potential use in discussing the
$\sigma$-lc centres even when the quasi-psh functions in question have more
general singularities. Furthermore, their relations between the algebraic
adjoint ideal sheaves of Ein--Lazarsfeld as well as those of Hacon--McKernan
are described in order to illustrate their role as a (potentially finer)
measurement of singularities in the minimal model program. In the course of the
study, a local $L^2$ extension theorem is proven, which shows that holomorphic
sections on any unions of $\sigma$-lc centres can be extended holomorphically
to some neighbourhood of the unions of $\sigma$-lc centres with some $L^2$
estimates. The proof does not rely on the techniques in the
Ohsawa--Takegoshi-type $L^2$ extension theorems.
