Let $ V $ be a vector space with a decreasing filtration $$ V = F_0 V \supseteq F_1 V \supseteq F_2 V \supseteq\cdots $$ We define the associated graded of $ V $ to be $$ gr V := \oplus_{k=0}^\infty F_k V / F_{k+1} V $$ Of course $ gr V $ can also be regarded as a filtered vector space and we have a canonical isomorphism $gr (gr V) = gr V $.

We say that $ V $ ``admits an expansion'' if there is an isomorphism of filtered vector spaces between $ gr V $ and $ V $, which becomes the identity map after applying $ gr $ to both $ gr V $ and $ V $.

This condition is equivalent to the existence of subspaces $ W_k \subset F_k V $ such that $ F_k V = W_k \oplus F_{k+1} V $ and $ V = \oplus_k W_k $.

Note that not every filtered vector space admits an expansion. For example, the vector space $ V = \mathbb C[[x]] $ with the filtration $ F_k V = x^k \mathbb C[[x]]$ does not admit an expansion. On the other hand, $ V = \mathbb C[x] $ with the same filtration does admit an expansion.

Here are my questions:

- Does this property have a different name in the literature?
- Let $V, W $ be two filtered vector spaces which admit expansions. Suppose that I have a filtration-preserving map $ \phi : V \rightarrow W $ such that $ gr \phi : gr V \rightarrow gr W $ is an isomorphism. Can I conclude that $ \phi $ is an isomorphism?