Trace invariants associated with quotient modules of the Hardy module

We consider the quotient module Q of the Hardy module H²(S) defined by an analytic set M˜ satisfying certain conditions. A representation for the orthogonal projection Q:L²(S,dσ)→Q was derived in [26], which allowed us to prove the geometric Arveson-Douglas conjecture for Q. In this paper we derive a new representation for Q, which makes it possible for us to take the next step: We show that for f,g∈Lip(S), the double commutator [M_f,[M_g,Q]] is in the Schatten class C_p for p>dim_CM˜. This Schatten-class membership leads to a number of results for trace invariants on Q and H²(S). In addition, we report an unexpected discovery: if dim_CM˜=1, then Q is 1-essentially normal. This is a stronger result than the prediction of the Arveson-Douglas conjecture.