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.