We investigate the problem of representing moment sequences by measures in the context of Polynomial Optimization Problems. This consists in finding the infimum of a real polynomial on a real semialgebraic set defined by polynomial inequalities. We analyze the exactness of Moment Matrix (MoM) relaxations, dual to the Sum of Squares (SoS) relaxations, which are hierarchies of convex cones introduced by Lasserre to approximate measures and positive polynomials. We investigate in particular flat truncation properties, which allow testing effectively when MoM exactness holds. We consider the quadratic module Q generated by the inequalities. We show that the dual of the MoM relaxation coincides with the SoS relaxation extended with the real radical of the support of Q, and focus on the zero-dimensional case, generalizing results for equations defining a finite real variety. We deduce sufficient and necessary conditions for flat truncation, under the finite convergence assumption: flat truncation happens if and only if the support of the quadratic module associated with the minimizers is of dimension zero. We also bound the order of the relaxation at which flat truncation holds. As corollaries, we conclude that flat truncation holds: • when regularity conditions, known as Boundary Hessian Conditions, hold: this result implies that flat truncation and MoM exactness holds generically; • when the support of the quadratic module Q is zero-dimensional; • in singular cases, flat truncation holds for the MoM relaxation extended with the polar constraints when the real variety of polar points is finite. Effective numerical computations illustrate these flat truncation properties.