We show that many invariant subspaces M for
d-shifts (S_1,...,S_d) of finite rank
have the property that the orthogonal projection
P_M onto M satisfies
$$
P_MS_k - S_kP_M\in\mathcal L^p,\qquad 1\leq k\leq d
$$
for every p>2d, $\mathcal L^p$ denoting the
Schatten-von Neumann class of all compact operators
having p-summable singular value lists.
In such cases, the d tuple of operators
\bar T=(T_1,...,T_d) obtained by compressing
(S_1,...,S_d) to M^\perp generates a
*-algebra whose
commutator ideal is contained in
\mathcal L^p for every p>d.
It follows that
the C^*-algebra generated by {T_1,...,T_d}
and the identity is commutative modulo compact operators,
the Dirac operator associated with
\bar T is Fredholm,
and the index formula for the curvature invariant
is stable under compact perturbations and homotopy
for this restricted class of finite rank d-contractions.
Though this class is limited,
we conjecture that the same conclusions persist under
much more general circumstances.