Let M,N be real smooth manifolds and p:M-->N a smooth map. Then smooth functions on M form a module over the ring of smooth functions on N (via pullback). Is it know whether this module is flat when p is a submersion or fiber bundle?

Recall that flatness is equivalent to the following: whenever h₁ ... h_k are smooth functions on N and g₁ ... g_k are smooth functions on M such that:

h₁g₁ + ... + h_kg_k = 0 (as function on M)

then there are functions G₁ ... G_r on M and a_i,j on N such that:

g_i= Σ_j a_i,jG_j for all i

and Σ_i h_i a_i,j= 0 for all j.