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Since
, the

are
-lattices and
-lattices.
Since
is regular, the * saturation*
of
is a
-lattice and, hence,
is a finite number.
For any the inclusions
imply inclusions
and
and
are
-invariant. Hence,
and
induce endomorphisms
such that the V-filtration
defined by
on
induces the V-filtration on the subquotient
.

*Christoph Lossen*

*2001-03-21*