Let $R$ be a Noetherian ring. Let $I=(x_1,x_2,...,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n,x_2^n,...,x_t^n)$. Are there any results about finiteness of $\cup_n Ass_R(I^n/I_n)$?
More generally, if $M$ is finitely generated $R$ module, do we know anything about $\cup_n Ass_R(I^nM/I_nM)$?
$ \bigcup_n Ass(I^n/I^n)$ need not be finite. Note that we have an exact sequence
$$ 0 \rightarrow \bigoplus_n \frac{I^n}{I_n} \rightarrow \bigoplus_n \frac{R}{I_n} \rightarrow \bigoplus_n \frac{R}{I^n} \rightarrow 0. $$
By Brodmann's result we have $ \bigcup_n Ass(R/I^n)$ is a finite set. So if $ \bigcup_n Ass(R/I_n)$ is an infinite set then $ \bigcup_n Ass(I^n/I^n)$ is also an infinite set.
Let $H^t_I(R)$ is the $t^{th}$ local co-homology module of $R$ with respect to $I$. It can be shown that $$ Ass \ H^t_I(R) \subseteq \bigcup_n Ass(R/I_n)$$ ,see Proposition 2.1 in the paper "Associated primes of local cohomology modules and of Frobenius powers", Anurag Singh and Irena Swanson, International Mathematics Research Notices 33 (2004) 1703–1733.
So if $Ass \ H^t_I(R)$ is an infinite set then $ \bigcup_n Ass(I^n/I^n)$ is also an infinite set.
For an example when this occurs see the paper referred above.
