This question concerns the asymptotic behaviour of associated primes of graded components of local cohomology modules. A survey of this can be found in M. Brodmann, Asymptotic behaviour of cohomology: tameness, supports and associarted primes, Contemp. Math. 390 (2005), 31-61.
Let $R=\bigoplus_{n\in\mathbb{N}}R_n$ be a noetherian homogeneous ring, let $R_+=\bigoplus_{n>0}R_n$ denote its irrelevant ideal, let $M$ be a $\mathbb{Z}$-graded $R$-module of finite type, and let $i\in\mathbb{N}$.
The $i$-th local cohomology module $H_{R_+}^i(M)$ of $M$ with respect to $R_+$ is a $\mathbb{Z}$-graded $R$-module. We consider the sets ${\rm Ass}_{R_0}(H_{R_+}^i(M)_n)$ of associated primes of the graded components of $H_{R_+}^i(M)$ and are interested in their behaviour for $n\to-\infty$. More precisely, we look at the following properties:
(1) We say that the associated primes of $H_{R_+}^i(M)_n$ are asymptotically stable for $n\to-\infty$ if there exists $n_0\in\mathbb{Z}$ such that ${\rm Ass}_{R_0}(H_{R_+}^i(M)_n)={\rm Ass}_{R_0}(H_{R_+}^i(M)_{n-1})$ for every $n\leq n_0$.
(2) We say that the associated primes of $H_{R_+}^i(M)_n$ are asymptotically increasing for $n\to-\infty$ if there exists $n_0\in\mathbb{Z}$ such that ${\rm Ass}_{R_0}(H_{R_+}^i(M)_n)\subseteq{\rm Ass}_{R_0}(H_{R_+}^i(M)_{n-1})$ for every $n\leq n_0$.
There are examples where the associated primes of $H_{R_+}^i(M)$ are asymptotically stable (e.g., if $\dim(R_0)=0$). There are also examples where the associated primes of $H_{R_+}^i(M)$ are not asymptotically increasing. But I do not know an example that distinguishes between the two properties. Thus:
Are there examples where the associated primes of $H_{R_+}^i(M)$ are asymptotically increasing, but not asymptotically stable?
