One can certainly verify Equation 6 on several instances, but this holds true in general as **a consequence of Theorem 3.1 (Correspondence Theorem)** on page 7.

Indeed, $\mathcal{M}_k \Doteq \{C_k^{l},\, f^l\}$ and $\mathcal{M}_{k - 1} \Doteq \{C_{k - 1}^{l},\, f^l\}$ are persistent modules in the sense of Definition 3.2 on page 7 and $\partial_k: \mathcal{M}_k \rightarrow \mathcal{M}_{k - 1}$ is a homomorphism in the category of persistence modules. By the Correspondence Theorem, $\alpha(\partial_k): \alpha(\mathcal{M}_k) \rightarrow \alpha(\mathcal{M}_{k - 1})$ is homomorphism of $\mathbb{N}$-graded $R[t]$-modules, where $R$ is the ground ring of homology coefficients and $\alpha$ is the correspondence of Section 3.1 (this is an equivalence of categories). The matrice $M_k$ is the matrix of $\alpha(\partial_k)$ with respect to some *homogeneous bases* $\{e_j\}$ and $\{\hat{e}_i\}$ (see definition in Lemma 1) of $\alpha(\mathcal{M}_k)$ and $\alpha(\mathcal{M}_{k - 1})$ respectively.

The fact that the coefficients of $M_k$ are unique and homogeneous is explained in Lemma 1 below, as well as the fact that $\alpha(\mathcal{M}_{k - 1})$ is free over $R[t]$ with basis $\{\hat{e}_i\}$ in the category of (non-graded) $R[t]$-modules.

By definition, $\alpha(\partial_k)(e_j) = \sum_i M_k(i, j)\hat{e}_i$. Since $\alpha(\partial_k)$ is a graded $R[t]$-module homomorphism, it maps a homogeneous element to a homogeneous element with the same degree. Therefore, every non-zero term in the previous sum is homogeneous of degree $\deg e_j$, that is $$\deg e_j = \deg M_k(i, j)\hat{e}_i = \deg M_k(i, j) + \deg\hat{e}_i$$ whenever $M_k(i,j) \neq 0$.

Let us explain now why $M_k$ is well-defined by unique homogeneous coefficients. Let $\Lambda = \Lambda_0 \oplus \Lambda_1 \oplus \cdots$ be an $\mathbb{N}$-graded ring and let $f:M \rightarrow N$
an $\mathbb{N}$-graded homomorphism of $\mathbb{N}$-graded $\Lambda$-modules with $M$ and $N$ finitely generated. The coefficients $\lambda_{ij}$ of a matrix that represents $f$ with respect to sets of generators $\{e_j\}$ and $\{\hat{e}_i\}$ of minimal cardinalities are not necessarily unique but we can always assume that they are homogeneous (consider the homogeneous term of degree $\deg e_j$ in $\lambda_{ij}\hat{e}_i$ when it exists). In the context of persistent homology, uniqueness of coefficients is guaranteed by

**Lemma 1** Let $\Lambda = R[t]$ where $R$ is commutative ring with identity. Let $N = \alpha(\mathcal{N})$ where $\mathcal{N} = \{N^j, \varphi^j\}$ is a persistent module over $R$. Assume that $N$ is finitely generated over $\Lambda$, that $\varphi^i$ is injective, $\varphi^j(N^j)$ and $N^{j + 1}/\varphi^j(N^j)$ are free over $R$ for every $j$. Then there is a generating set $\{\hat{e}_i\}$ of minimal cardinality whose elements are homogeneous. In addition, $N$ is free over $\Lambda$ with basis $\{\hat{e}_i\}$ in the category of (non-graded) $\Lambda$-modules for any such generating set. We call such a set a *homogeneous basis of $N$*.

Beware: the module $N$ of Lemma 1 is not necessarily a free $\mathbb{N}$-graded module over $\Lambda$.

*Proof of Lemma 1.* Since $N$ is finitely generated over $\Lambda$, the existence of $\{\hat{e}_i\}$ is immediate. Because of our assumptions on $N$, the set of elements of the form $t^l\hat{e}_k$ of degree $j$ is a generating set of $N^j$ over $R$ with minimal cardinality. Since $N^j$ is free over $R$, these elements form an $R$-basis of $N^j$ [1, Theorem 2.4]. Now take coefficient $\lambda_i \in \Lambda$ such that $\sum_i \lambda_i \hat{e}_i = 0$. By splitting $\lambda_i \hat{e}_i$ in a sum of homogeneous terms and by using the previous fact and the injectivity of $\varphi^j$, it readily follows that $\lambda_i = 0$ for every $i$.

[1] H. Matsumura, "Commutative Ring Theory", 1989.