Coconnected commutative Hopf algebras

Let $k$ be a field and let $(A,\mathrm{\Delta },\epsilon ,S)$ be a commutative Hopf algebra over $k$ (Commutative Hopf algebras over a field), with tensor products over $k$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums).

The Hopf algebra $A$ is coconnected if there is an increasing filtration $C_0\subseteq C_1\subseteq C_2\subseteq \cdots$ of $A$ by $k$-linear subspaces (Linear subspace of a vector space) with $C_0=k\cdot 1_A,{\bigcup }_{r\ge 0}{C}_r=A,\mathrm{\Delta }(C_r)\subseteq {\sum }_{i=0}^r{C}_i{\otimes }_k{C}_{r-i}\text{for all }r\ge 0.$ In words: the filtration starts at the constants, exhausts $A$, and the comultiplication of an element of filtration degree $r$ lands in the sum of tensor products whose degrees add up to $r$.

No reducedness, finite generation or smoothness of $A$ is imposed, and no choice principle is used. The filtration is the condition dual to the existence of a group-like element generating the simple comodules; the first step is the constants because a unipotent group has no nontrivial characters.