AbstractLet
$\mathcal S$
be a Serre subcategory of the category of
$R$
modules, where
$R$
is a commutative Noetherian ring. Let
$\mathfrak a$
and
$\mathfrak b$
be ideals of
$R$
and let
$M$
and
$N$
be finite
$R$
modules. We prove that if
$N$
and
$H^i_{\mathfrak a}(M,N)$
belong to
$\mathcal S$
for all
$i\lt n$
and if
$n\leq \mathrm {f}$

$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$
, then
$\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$
. We deduce that if either
$H^i_{\mathfrak a}(M,N)$
is finite or
$\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$
is finite for all
$i\lt n$
, then
$\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$
is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal
${\mathfrak {p}}$
of
$R$
, there exists an integer
$n_{\mathfrak {p}}$
such that
$\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$
for every
$i$
less than a fixed integer
$t$
, then does there exist an integer
$n$
such that
$\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$
for all
$i\lt t$
? A formulation of this question is referred to as the localglobal principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are localglobal principles for the finiteness and Artinianness of generalised local cohomology modules.