On the Jacobian ring of a complete intersection

Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient J:=K[x,y]/I. We describe J by computing the cohomology of a certain complex whose top cohomology group is J.