Assume we have an R-Module A, with R a commutative ring and three descending filtrations F,G,H on this. We can take the associated graded module with respect to any of these, say F, by setting
$$\operatorname{Gr}_F(A)=\bigoplus_p F^p(A)/F^{p+1}$$
The other filtrations also induce filtrations on this object and one can look again at the associated graded modules. It can be shown that
$$\operatorname{Gr}_G^n\operatorname{Gr}_F^m(A)\cong \operatorname{Gr}_F^m\operatorname{Gr}_G^n(A).$$
But if we look at
$$\operatorname{Gr}_H\operatorname{Gr}_G\operatorname{Gr}_F(A)$$
the roles of $G$ and $F$ are in general not interchangeable, i.e. the filtrations induced by $H$ on the two bigraduated objects do not correspond each other via the above isomorphism.
Are there standard examples for this?
The question comes from reading about Hodge-theory, where $F$ in the last expression would be an ascending filtration (the weight filtration of a mixed Hodge structure) and $G$ and $H$ conjugate. An example arising naturally in this context would also be very nice.
here is an example that I found in "structure de Hodge mixte et fibrés sur le plan projectif complexe" by Olivier Penacchio:
Let $k$ be a field and $$V=k^2=<e,f>$$ equiped with three descending filtrations
$$F^{-2}=V, F^{-1}=F^0=<e>, F^1=\{0\},$$
$$G^0=V, G^1=<f+\lambda e>, F^2=\{0\},$$
$$H^0=V, H^1=<f+\mu e>, H^2=\{0\},$$
where $\lambda,\mu$ are two distinct elements of $k$.
Then we have:
$$\operatorname{Gr}^1_G\operatorname{Gr}^1_H\operatorname{Gr}^{-2}_F=<\bar{f}>,$$ so it has dimension $1$, but
$$\operatorname{Gr}^{-2}_F\operatorname{Gr}^1_G\operatorname{Gr}^{1}_H=\{0\}.$$
I do not, however, know if this is an example coming from geometry.