Asymptotic prime divisors of torsion-free symmetric powers of modules

Let R be a Noetherian ring, F:=Rr and M F a submodule of rank r. Let denote the stable value of , for n large, where Fn is the nth symmetric power of Fn and Mn is the image of the nth symmetric power of M in Fn. We provide a number of characterizations for a prime ideal to belong to . We also show that , where A*(M) denotes the stable value of Ass(Fn/Mn).