I'm reading these notes by Hutchings on spectral sequences. In the first section, he motivates spectral sequences with the long exact sequence in relative homology. Given a chain complex $C_*$ and a subcomplex $F_0C_*$, we form a long exact sequence in homology given by
$$\dotsb \to H_{k}(F_0C_*) \to H_k (C_*) \to H_k(C_*/F_0C_*) \to H_{k-1}(F_0C_*) \to \dotsb$$
Isolating at $H_k(C_*)$, we have a short exact sequence
$$0 \to \operatorname{cok} \delta_{k+1} \to H_k(C) \to \operatorname{ker}\delta_k \to 0$$
He then refers to $\operatorname{cok} \delta_{k+1}$ and $\operatorname{ker} \delta_k$ as $G_0$ and $G_1$ respectively--the associated graded modules of the homology of $C_*$.
It's clear to me why $\operatorname{cok} \delta_{k+1}$ is $G_0$. It's less clear to me why $\operatorname{ker} \delta_k$ is $G_1$. I would describe $\operatorname{ker} \delta_k$ as the subgroup of the relative $k$th homology classes whose boundaries in $F_0C_*$ are also boundaries of elements in $F_0C_*$. I'm not seeing why that is the same thing as $G_1H_k$, which I've learned is those $k$ homology classes modded out by those which have a representative in $F_0C_*$.
Can someone help me understand?
By exactness of the long exact sequence, $\ker(\delta_k)$ is equal to the image of the map $H_k(C_*)\to H_k(C_*/F_0C_*)$, which is in turn the quotient of $H_k(C_*)$ by the image of $H_k(F_0C_*)\to H_k(C_*)$. This is exactly your description of $G_1$.