This is related to my earlier question regarding the category of graded modules.
Let $A$ be a commutative unital $\mathbb{Z}$-graded ring, and $f:M\to N$ be an $A$-linear map where $f=\sum_if_i$ and each $f_i$ is homogeneous of degree $i$. To show that $\text{ker}(f)$ is graded, we need to show that $m\in\text{ker}(f)$ implies $m_i\in\text{ker}(f)$ for all $i$.
We have that $$f(m)=\sum_{i,j}f_i(m_j)=\sum_k\sum_{i+j=k}f_i(m_j),$$ where $\sum_{i+j=k}f_i(m_j)\in N_k$ for each $k$. So $\sum_{i+j=k}f_i(m_j)=0$ for all $k$. But here I am stuck and am doubting whether or not the kernel is graded.. what can I do from here?
Best Answer
Such an $f$ is not what people use when talking about the category of graded modules, and its kernel need not be graded. For example if $A=k$ is a field and $M = k \oplus \Sigma k$, $N = k$, and $f$ the map sending $(a, \Sigma b)$ to $a - b$, then the kernel is not graded.