Basic question on graded quotients

Question

Let $A$ be a graded algebra, and let $I$ be a graded (two-sided) ideal.

($A=\bigoplus A_k$, $I=\bigoplus I_k$).

Is it true that $A/I=\bigoplus A_k/I_k$?

I can't seem to see it.

Thanks for any help.

Answer 1

Yes, it is true. For each $k$, the image of $A_k$ in $A/I$ is $A_k/(I\cap A_k)=A_k/I_k$, so each $A_k/I_k$ is a subgroup of $A/I$. Since $A=\sum A_k$, $A/I$ is the sum of the images in the quotient. It remains to be shown that if $a_k\in A_k/I_k$ for each $k$ and $\sum a_k=0$, then $a_k=0$ for each $k$ (so that the representation of an element of $A/I$ as a sum of elements of the $A_k/I_k$ is unique). To show this, lift each $a_k$ to an element $b_k\in a_k$. Since $\sum a_k=0$, $\sum b_k\in I$. Since $I=\bigoplus I_k$, this means $b_k\in I_k$ for each $k$. Since $a_k$ is the image of $b_k$, this means $a_k=0$.

You can similarly see that this grading on $A/I$ respects the ring structure, so $A/I$ is a graded ring. That is, if $a\in A_k/I_k$ and $b\in A_\ell/I_\ell$, then $ab\in A_{k+\ell}/I_{k+\ell}$. To prove this, lift $a$ and $b$ to elements $c\in A_k$ and $d\in A_\ell$. Then $cd\in A_{k+\ell}$, and the image of $cd$ in $A/I$ is $ab$. Thus $ab$ is in the image of $A_{k+\ell}$, which is $A_{k+\ell}/I_{k+\ell}$.