Abstract
We consider the relative Bruce–Roberts number
$\mu _{\textrm {BR}}^{-}(f,\,X)$
of a function on an isolated hypersurface singularity
$(X,\,0)$
. We show that
$\mu _{\textrm {BR}}^{-}(f,\,X)$
is equal to the sum of the Milnor number of the fibre
$\mu (f^{-1}(0)\cap X,\,0)$
plus the difference
$\mu (X,\,0)-\tau (X,\,0)$
between the Milnor and the Tjurina numbers of
$(X,\,0)$
. As an application, we show that the usual Bruce–Roberts number
$\mu _{\textrm {BR}}(f,\,X)$
is equal to
$\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$
. We also deduce that the relative logarithmic characteristic variety
$LC(X)^{-}$
, obtained from the logarithmic characteristic variety
$LC(X)$
by eliminating the component corresponding to the complement of
$X$
in the ambient space, is Cohen–Macaulay.