Let $\mathbf{k}$ be a commutative ring.

Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq \cdots$ satisfying $\bigcup\limits_{n\geq 0}C^n = C$ and $\Delta\left(C^n\right) \subseteq \sum\limits_{k=0}^n C^k \otimes C^{n-k}$ for all nonnegative integers $n$, where $C^k \otimes C^{n-k}$ really means the image of the canonical map $C^k \otimes C^{n-k} \to C \otimes C$ by abuse of notation (even if this map fails to be injective).

Assume further that $C$ is connected, i.e., the counit $\varepsilon$ restricted to $C^0$ is an isomorphism $C^0 \to \mathbf{k}$. This allows us to define an element $1_C \in C^0$ (often just called $1$ when no confusion can arise) as the preimage of $1 \in \mathbf{k}$ under this isomorphism (even if $C$ has no algebra structure given).

An element $x\in C$ is said to be *primitive* if $\Delta x = x \otimes 1_C + 1_C \otimes x$. The $\mathbf{k}$-submodule of $C$ formed by all primitive elements is called $\operatorname{Prim} C$. Every primitive element $x\in C$ satisfies $\varepsilon x = 0$ (this is easy to check).

The following two conjectures are equivalent versions of each other (I have given them different numbers for easier reference):

> **Conjecture 1:** If $D$ is a further $\mathbf{k}$-coalgebra, and $f : C \to D$ is a surjective homomorphism of $\mathbf{k}$-coalgebras such that $f\mid_{\operatorname{Prim}C}$ is injective, then $f$ is injective.
>
> **Conjecture 2:** If $I$ is a coideal of $C$ such that $I \cap \operatorname{Prim}C = 0$, then $I = 0$. (Here, a *coideal* of $C$ means a $\mathbf{k}$-submodule $J$ of $C$ such that $\Delta J \subseteq J \otimes C + C \otimes J$, where we again abuse notation as above.)

**Background:** Conjectures 1 and 2 are known to be true if $\mathbf{k}$ is a field, and the easy proof (by induction) generalizes to the case when "everything in sight is flat" (I don't want to go into the details of what this means, but I can give the proof if needed). Moreover, Conjecture 2 is also true when $C$ is a graded coalgebra and $I$ is homogeneous (more or less because direct sums are just as good as flatness when it comes to preserving injectivity under tensor products). I am unable to come up with a counterexample in any other case, but this is because I have no real idea how to construct non-flat counterexamples to *anything* (a good reference for non-flat perversion would be highly appreciated). It seems to me that the conjectures should be true, if only because they are highly useful.

Note that when $\mathbf{k}$ is a field, Conjecture 1 holds even without assuming that $f$ be surjective. But this does not generalize to general commutative rings $\mathbf{k}$. Here is a counterexample (a variation of the example on pp. 56-57 of Warren Nichols and Moss Sweedler, *Hopf Algebras and Combinatorics*, in Robert Morris (ed.), *Umbral Calculus and Hopf Algebras*): Let $\mathbf{k} = \mathbb Z$. Let $C = \mathbb Z \oplus \mathbb Z/2 \oplus \mathbb Z/2$ as $\mathbf{k}$-module, with basis vectors $e_0 = \left(1,0,0\right)$, $e_1 = \left(0,1,0\right)$ and $e_2 = \left(0,0,1\right)$. Make $C$ into a graded $\mathbf{k}$-module by setting $\deg e_i = i$. Define a coproduct on $C$ by $\Delta e_0 = e_0 \otimes e_0$, $\Delta e_1 = e_0 \otimes e_1 + e_1 \otimes e_0$ and $\Delta e_2 = e_0 \otimes e_2 + e_1 \otimes e_1 + e_2 \otimes e_0$ (like in a divided-powers coalgebra). Define a counity on $C$ by $\varepsilon\left(e_i\right) = \delta_{i, 0}$. Thus, $C$ becomes a connected filtered (and even graded) $\mathbf{k}$-coalgebra with $1_C = e_0$ and $\operatorname{Prim}C = \left\langle e_1 \right\rangle$. Now, let $D = \mathbb Z \oplus \mathbb Z/4$, with basis vectors $f_0 = \left(1,0\right)$ and $f_1 = \left(0,1\right)$. Grade $D$ by $\deg f_i = i$. Define a coproduct on $D$ by $\Delta f_0 = f_0 \otimes f_0$ and $\Delta f_1 = f_0 \otimes f_1 + f_1 \otimes f_0$. Define a counity on $D$ by $\varepsilon\left(f_i\right) = \delta_{i, 0}$. This makes $D$ into a connected filtered (and, again, graded) $\mathbf{k}$-coalgebra with $1_D = f_0$. Let now $f : C \to D$ be the $\mathbf{k}$-module morphism defined by $f\left(e_0\right) = f_0$, $f\left(e_1\right) = 2f_1$ and $f\left(e_2\right) = 0$. It is straightforward to check that $f$ is a coalgebra homomorphism, and $f$ is injective on $\operatorname{Prim}C = \left\langle e_1 \right\rangle$; but clearly, $f$ is not injective on the whole of $C$.