In the Milnor and Moore paper, "On the structure of Hopf algebras" proposition 1.7 said the following:

1. $A$ a connected $K$-algebra.
2. $N$ a left $A$ module that is connected as a $K$-graded module i.e. there is an isomorphism $\eta_N:K\to N_0$ which results in an augmentation $\varepsilon_N:N\to K$.
3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \xrightarrow{\cong} A \xrightarrow{\varepsilon_A} K$.
4. $\Delta :N \to N\otimes C$ a morphism of left $A$ modules.
5. $\pi:N\to C$ the canonical epimorphism. Let $f:C\to N$ such that $\pi f = \operatorname{id}_C$. In this paper they used $C$ the object letter to indicate the identity morphism.
6. $(\varepsilon_N\otimes C)\circ \Delta =f$.
7. $(N\otimes \varepsilon_C)\circ \Delta = N$.
8. $A\otimes C \xrightarrow{i\otimes C} N\otimes C $ is a monomorphism.

If $\tilde{f}$ is the composition $$A\otimes C \xrightarrow{A\otimes f} A \otimes N \xrightarrow{\varphi_N} N$$ then $\tilde f$ is an isomorphism.

The map $i$ is defined as the composition $$A\xrightarrow{\cong} A\otimes K \xrightarrow{A\otimes \eta_N} A\otimes N \xrightarrow{\varphi_N} N$$ and $\varphi_N$ is the left action of $A$ on $N$.

My first question: in the process to show that $\tilde{f}$ is a monomorphism, they showed $\Delta \tilde{f}$ is a monomorphism.

They define a filtration on $$F_p(A\otimes C) = \sum_{q\geq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\geq p} N\otimes C_q$$ Then they said: "let $E^0 (A\otimes C)$ be the associated bigraded module." Is this the grading associated with the filtration i.e. $\operatorname{gr}_p(A\otimes C) = F_p/F_{p-1}$? Or it is just the grading on $A$ and $C$ as $K$-modules? I guess it is the second since they said $E^0_{p,q} (A\otimes C) = A_p \otimes C_q$.

After that they said we identify $E^0(A\otimes C)$ with $A\otimes C$. Then why did they introduce the $E^0$?

Second question: How come the two following morphisms are the same: $$A\otimes C \xrightarrow{A\otimes f} A\otimes N \xrightarrow{\varphi_N} N\to_{\Delta} N\otimes C$$ and $i\otimes C$ which is
$$A\otimes C \xrightarrow{\cong\otimes C } A\otimes K\otimes C \xrightarrow{A\otimes \eta_N\otimes C } A\otimes N \otimes C \xrightarrow{\varphi_N} N\otimes C. $$

In the Milnor and Moore paper, "On the structure of Hopf algebras" proposition 1.7 said the following:

1. $A$ a connected $K$-algebra.
2. $N$ a left $A$ module that is connected as a $K$-graded module i.e. there is an isomorphism $\eta_N:K\to N_0$ which results in an augmentation $\varepsilon_N:N\to K$.
3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \xrightarrow{\cong} A \xrightarrow{\varepsilon_A} K$.
4. $\Delta :N \to N\otimes C$ a morphism of left $A$ modules.
5. $\pi:N\to C$ the canonical epimorphism. Let $f:C\to N$ such that $\pi f = \operatorname{id}_C$. In this paper they used $C$ the object letter to indicate the identity morphism.
6. $(\varepsilon_N\otimes C)\circ \Delta =\pi$.
7. $(N\otimes \varepsilon_C)\circ \Delta = N$.
8. $A\otimes C \xrightarrow{i\otimes C} N\otimes C $ is a monomorphism.

If $\tilde{f}$ is the composition $$A\otimes C \xrightarrow{A\otimes f} A \otimes N \xrightarrow{\varphi_N} N$$ then $\tilde f$ is an isomorphism.

The map $i$ is defined as the composition $$A\xrightarrow{\cong} A\otimes K \xrightarrow{A\otimes \eta_N} A\otimes N \xrightarrow{\varphi_N} N$$ and $\varphi_N$ is the left action of $A$ on $N$.

My first question: in the process to show that $\tilde{f}$ is a monomorphism, they showed $\Delta \tilde{f}$ is a monomorphism.

They define a filtration on $$F_p(A\otimes C) = \sum_{q\leq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\leq p} N\otimes C_q$$ Then they said: "let $E^0 (A\otimes C)$ be the associated bigraded module." Is this the grading associated with the filtration i.e. $\operatorname{gr}_p(A\otimes C) = F_p/F_{p-1}$? Or it is just the grading on $A$ and $C$ as $K$-modules? I guess it is the second since they said $E^0_{p,q} (A\otimes C) = A_p \otimes C_q$.

After that they said we identify $E^0(A\otimes C)$ with $A\otimes C$. Then why did they introduce the $E^0$?

Second question: How come the two following morphisms are the same: $$A\otimes C \xrightarrow{A\otimes f} A\otimes N \xrightarrow{\varphi_N} N\to_{\Delta} N\otimes C$$ and $i\otimes C$ which is
$$A\otimes C \xrightarrow{\cong\otimes C } A\otimes K\otimes C \xrightarrow{A\otimes \eta_N\otimes C } A\otimes N \otimes C \xrightarrow{\varphi_N} N\otimes C. $$

In the Milnor and Moore paper, "On the structure of Hopf algebra" proposition 1.7. Said the following:

Blockquote

1. $A$ connected $K$-algebra
2. $N$ a left $A$ module and it is connected $K$-graded module i.e. there is an isomorphism $\eta_N:K\to N_0$ which results in an augmentation $\varepsilon_N:N\to K$
3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \xrightarrow{\cong} A \xrightarrow{\varepsilon_A} K$
4. $\Delta :N \to N\otimes C$ morphism of left $A$ modules.
5. $\pi:N\to C$ canonical epimorphism, let $f:C\to N$ such that $\pi f = id_C$ in this paper they used $C$ the object letter to indicate the identity morphism
6. $(\varepsilon_N\otimes C)\circ \Delta =f$
7. $(N\otimes \varepsilon_C)\circ \Delta = N$
8. $A\otimes C \xrightarrow{i\otimes C} N\otimes C $ is monomorphism.

If $\tilde{f}$ to be the composition $$A\otimes C \xrightarrow{A\otimes f} A \otimes N \xrightarrow{\varphi_N} N$$ then it is an isomorphism

The map $i$ is defined as the composition $$A\xrightarrow{\cong} A\otimes K \xrightarrow{A\otimes \eta_N} A\otimes N \xrightarrow{\varphi_N} N$$ and $\varphi_N$ the left action of $A$ on $N$

My first question: in the process to show that $\tilde{f}$ is a monomorphism, they showed $\Delta \tilde{f}$ is a monomorphism.

They define a filtration on $$F_p(A\otimes C) = \sum_{q\geq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\geq p} N\otimes C_q$$ Then they said: "let $E^0 (A\otimes C)$ be the associated bigraded module" is this the grading associated with the filtration i.e. $gr_p(A\otimes C) = F_p/F_{p-1}$ ? or it is just the grading on $A$ and $C$ as $K$-modules? I guess it is the second since they said $E^0_{p,q} (A\otimes C) = A_p \otimes C_q$ After that they said we identify $E^0(A\otimes C)$ with $A\otimes C$ ? then why did they introduce the $E^0$?

Second question: How come the two following morphisms are the same: $$A\otimes C \xrightarrow{A\otimes f} A\otimes N \xrightarrow{\varphi_N} N\to_{\Delta} N\otimes C$$ and $i\otimes C$ which is
$$A\otimes C \xrightarrow{\cong\otimes C } A\otimes K\otimes C \xrightarrow{A\otimes \eta_N\otimes C } A\otimes N \otimes C \xrightarrow{\varphi_N} N\otimes C $$

The paper: https://t.ly/G1dL

In the Milnor and Moore paper, "On the structure of Hopf algebras" proposition 1.7 said the following:

1. $A$ a connected $K$-algebra.
2. $N$ a left $A$ module that is connected as a $K$-graded module i.e. there is an isomorphism $\eta_N:K\to N_0$ which results in an augmentation $\varepsilon_N:N\to K$.
3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \xrightarrow{\cong} A \xrightarrow{\varepsilon_A} K$.
4. $\Delta :N \to N\otimes C$ a morphism of left $A$ modules.
5. $\pi:N\to C$ the canonical epimorphism. Let $f:C\to N$ such that $\pi f = \operatorname{id}_C$. In this paper they used $C$ the object letter to indicate the identity morphism.
6. $(\varepsilon_N\otimes C)\circ \Delta =f$.
7. $(N\otimes \varepsilon_C)\circ \Delta = N$.
8. $A\otimes C \xrightarrow{i\otimes C} N\otimes C $ is a monomorphism.

If $\tilde{f}$ is the composition $$A\otimes C \xrightarrow{A\otimes f} A \otimes N \xrightarrow{\varphi_N} N$$ then $\tilde f$ is an isomorphism.

The map $i$ is defined as the composition $$A\xrightarrow{\cong} A\otimes K \xrightarrow{A\otimes \eta_N} A\otimes N \xrightarrow{\varphi_N} N$$ and $\varphi_N$ is the left action of $A$ on $N$.

My first question: in the process to show that $\tilde{f}$ is a monomorphism, they showed $\Delta \tilde{f}$ is a monomorphism.

They define a filtration on $$F_p(A\otimes C) = \sum_{q\geq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\geq p} N\otimes C_q$$ Then they said: "let $E^0 (A\otimes C)$ be the associated bigraded module." Is this the grading associated with the filtration i.e. $\operatorname{gr}_p(A\otimes C) = F_p/F_{p-1}$? Or it is just the grading on $A$ and $C$ as $K$-modules? I guess it is the second since they said $E^0_{p,q} (A\otimes C) = A_p \otimes C_q$.

After that they said we identify $E^0(A\otimes C)$ with $A\otimes C$. Then why did they introduce the $E^0$?

Second question: How come the two following morphisms are the same: $$A\otimes C \xrightarrow{A\otimes f} A\otimes N \xrightarrow{\varphi_N} N\to_{\Delta} N\otimes C$$ and $i\otimes C$ which is
$$A\otimes C \xrightarrow{\cong\otimes C } A\otimes K\otimes C \xrightarrow{A\otimes \eta_N\otimes C } A\otimes N \otimes C \xrightarrow{\varphi_N} N\otimes C. $$