  We study the topology of the prime spectrum of an algebra supporting a
rational torus action. More precisely, we study inclusions between prime ideals
that are torus-invariant using the $H$-stratification theory of Goodearl and
Letzter on one hand and the theory of deleting derivations of Cauchon on the
other. We apply the results obtained to the algebra of $m \times n$ generic
quantum matrices to show that the dimensions of the $H$-strata described by
Goodearl and Letzter are bounded above by the minimum of $m$ and $n$, and that
moreover all the values between 0 and this bound are achieved.
