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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.)
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.)
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.)
Let $k$$\mathbf{k}$ be a commutative ring.
Let $C$ be a filtered $k$$\mathbf{k}$-coalgebra. This means a $k$$\mathbf{k}$-coalgebra equipped with an increasing $k$$\mathbf{k}$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$$C^0 \subseteq C^1 \subseteq C^2 \subseteq \cdots$ satisfying $\bigcup\limits_{n\geq 0}C^n = C$ and $\Delta\left(C^n\right) \subseteq \sum\limits_{k=0}^n C^k \otimes C^{n-k}$ for all nonnegative integers $n$, where $C^k \otimes C^{n-k}$ really means the image of $C^k \otimes C^{n-k}$ insidethe canonical map $C \otimes C$$C^k \otimes C^{n-k} \to C \otimes C$ by abuse of notation (even if thethis map $C^k \otimes C^{n-k} \to C \otimes C$ fails to be injective).
Assume further that $C$ is connected, i. ee., the counit $\varepsilon$ restricted to $C^0$ is an isomorphism to $k$$C^0 \to \mathbf{k}$. This allows us to define an element $1_C \in C^0$ (often just called $1$ when no confusion can arise) as the preimage of $1 \in k$$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 $k$$\mathbf{k}$-submodule of $C$ formed by all primitive elements is called $\mathrm{Prim} C$$\operatorname{Prim} C$. Every primitive element $x\in C$ satisfies $\varepsilon x = 0$ (this is easy to check).
Conjecture 1: If $D$ is a further $k$-coalgebra, and $f : C \to D$ is a surjective homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.
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 $I \cap \mathrm{Prim}C = 0$, then $I = 0$. (Here, a coideal of $C$ means a $k$-submodule $J$ of $C$ such that $\Delta J \subseteq J \otimes C + C \otimes J$, where again an abuse of notation is made.)
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 $k$$\mathbf{k}$ is a field, and the easy proof (by induction) generalizes to the case when "everything in sight is flat" (I don't want to go into the details of what this means, but I can give the proof if needed). Moreover, Conjecture 2 is also true when $C$ is a graded coalgebra and $I$ is homogeneous (more or less because direct sums are just as good as flatness when it comes to preserving injectivity under tensor products). I am unable to come up with a counterexample in any other case, but this is because I have no real idea how to construct non-flat counterexamples to anything (a good reference for non-flat perversion would be highly appreciated). It seems to me that the conjectures should be true, if only because they are highly useful.
Note that when $k$$\mathbf{k}$ is a field, Conjecture 1 holds even without assuming that $f$ be surjective. But this does not generalize to general commutative rings $k$$\mathbf{k}$. Here is a counterexample (a variation of the example on pp. 56-57 of Warren Nichols and Moss Sweedler, Hopf Algebras and Combinatorics, in Robert Morris (ed.), Umbral Calculus and Hopf Algebras): Let $k = \mathbb Z$$\mathbf{k} = \mathbb Z$. Let $C = \mathbb Z \oplus \mathbb Z/2 \oplus \mathbb Z/2$ as $k$$\mathbf{k}$-module, with basis vectors $e_0 = \left(1,0,0\right)$, $e_1 = \left(0,1,0\right)$ and $e_2 = \left(0,0,1\right)$. Make $C$ into a graded $k$$\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) $k$$\mathbf{k}$-coalgebra with $1_C = e_0$ and $\mathrm{Prim}C = \left\langle e_1 \right\rangle$$\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) $k$$\mathbf{k}$-coalgebra with $1_D = f_0$. Let now $f : C \to D$ be the $k$$\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 $\mathrm{Prim}C = \left\langle e_1 \right\rangle$$\operatorname{Prim}C = \left\langle e_1 \right\rangle$; but clearly, $f$ is not injective on the whole of $C$.
Let $k$ be a commutative ring.
Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ 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 $C^k \otimes C^{n-k}$ inside $C \otimes C$ by abuse of notation (even if the map $C^k \otimes C^{n-k} \to C \otimes C$ fails to be injective).
Assume further that $C$ is connected, i. e., the counit $\varepsilon$ restricted to $C^0$ is an isomorphism to $k$. 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 $1 \in k$ 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 $k$-submodule of $C$ formed by all primitive elements is called $\mathrm{Prim} C$. Every primitive element $x\in C$ satisfies $\varepsilon x = 0$ (this is easy to check).
Conjecture 1: If $D$ is a further $k$-coalgebra, and $f : C \to D$ is a surjective homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.
Conjecture 2: If $I$ is a coideal of $C$ such that $I \cap \mathrm{Prim}C = 0$, then $I = 0$. (Here, a coideal of $C$ means a $k$-submodule $J$ of $C$ such that $\Delta J \subseteq J \otimes C + C \otimes J$, where again an abuse of notation is made.)
Background: Conjectures 1 and 2 are known to be true if $k$ 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 $k$ is a field, Conjecture 1 holds even without assuming that $f$ be surjective. But this does not generalize to general commutative rings $k$. 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 $k = \mathbb Z$. Let $C = \mathbb Z \oplus \mathbb Z/2 \oplus \mathbb Z/2$ as $k$-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 $k$-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) $k$-coalgebra with $1_C = e_0$ and $\mathrm{Prim}C = \left\langle e_1 \right\rangle$. 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) $k$-coalgebra with $1_D = f_0$. Let now $f : C \to D$ be the $k$-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 $\mathrm{Prim}C = \left\langle e_1 \right\rangle$; but clearly, $f$ is not injective on the whole of $C$.
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).
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 $C$.
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 $k$-coalgebra, and $f : C \to D$ is a surjective homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.
Background: Conjecture 2 is equivalent to the special case of ConjectureConjectures 1 when $f$ is assumed surjective. Both conjecturesand 2 are known to be true if $k$ 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 $k$ is a field, Conjecture 1 holds even without assuming that $f$ be surjective. But this does not generalize to general commutative rings $k$. 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 $k = \mathbb Z$. Let $C = \mathbb Z \oplus \mathbb Z/2 \oplus \mathbb Z/2$ as $k$-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 $k$-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) $k$-coalgebra with $1_C = e_0$ and $\mathrm{Prim}C = \left\langle e_1 \right\rangle$. 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) $k$-coalgebra with $1_D = f_0$. Let now $f : C \to D$ be the $k$-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 $\mathrm{Prim}C = \left\langle e_1 \right\rangle$; but clearly, $f$ is not injective on the whole of $C$.
Conjecture 1: If $D$ is a further $k$-coalgebra, and $f : C \to D$ is a homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.
Background: Conjecture 2 is equivalent to the special case of Conjecture 1 when $f$ is assumed surjective. Both conjectures are known to be true if $k$ 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.
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 $k$-coalgebra, and $f : C \to D$ is a surjective homomorphism of $k$-coalgebras such that $f\mid_{\mathrm{Prim}C}$ is injective, then $f$ is injective.
Background: Conjectures 1 and 2 are known to be true if $k$ 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 $k$ is a field, Conjecture 1 holds even without assuming that $f$ be surjective. But this does not generalize to general commutative rings $k$. 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 $k = \mathbb Z$. Let $C = \mathbb Z \oplus \mathbb Z/2 \oplus \mathbb Z/2$ as $k$-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 $k$-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) $k$-coalgebra with $1_C = e_0$ and $\mathrm{Prim}C = \left\langle e_1 \right\rangle$. 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) $k$-coalgebra with $1_D = f_0$. Let now $f : C \to D$ be the $k$-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 $\mathrm{Prim}C = \left\langle e_1 \right\rangle$; but clearly, $f$ is not injective on the whole of $C$.
