Injections between successive kernel quotients ($\ker T^{i+1}/\ker T^i\hookrightarrow \ker T^i/\ker T^{i-1}$)

For a linear map $T:V\to V$, set $K_i=\ker T^i$ and $K_0=0$. Since $T(K_{i+1})\subseteq K_i$ and $T(K_i)\subseteq K_{i-1}$, the rule $v+K_i\mapsto Tv+K_{i-1}$ defines a linear map $K_{i+1}/K_i\to K_i/K_{i-1}$. It is injective: if $Tv\in K_{i-1}$, then $T^{i}v=0$, so $v\in K_i$. In finite dimension, this provesThe linear map need not be surjective: for $T=0$ on a nonzero vector space, its $i=1$ instance is $0\to V$. For a nilpotent linear map, these dimensions are the increments in the kernel dimensions of powers of a nilpotent map.