Homogeneous ideal

Question

I am reading “Algebraic geometry” by Hartshorne. By Hartshorne’s definition, in a graded ring $S$, an ideal $a$ is homogeneous if $a=\oplus _{d\geq 0}(a\cap S_d)$. And then, it is written that an ideal $a$ is homogeneous if and only if it can be generated by homogeneous elements. In order to prove this, if $a$ is homogenous, then since $a=\oplus_{d\geq 0}(a\cap S_d)$, we conclude that $a$ is generated by the elements of $a\cap S_d$’s $d\geq 0$ and since these are homogeneous elememts, we conclude that $a$ is generated by homogeneous elements. For the other direction, trivially for each $d\geq 0$, we have $a\cap S_d\subset a$ and thus $\oplus_{d\geq 0}(a\cap S_d)\subset a$. On the other hand, since by our assumption, $a$ is generated by homogeneous elements, we get that $a\subset \oplus_{d\geq} (a\cap S_d)$ and thus we conclude $a=\oplus_{d\geq}(a\cap S_d)$ and thus we are done. Now, my question is that am I true or am I missing something?

Answer 1

At least for me, $a\subset\oplus_{d\ge 0}(a\cap S_d)$ isn't so clear. For example, this requires if $x+x^2+y^3+x^2y^4\in F[x,y]$ is in the ideal, then all of $x$, $x^2, y^3, x^2y^4$ must be in the ideal too.

This is not hard to show though: if $a$ is generated by homogeneous elements $\{a_i\}_{i\in I}$, then for any $x\in a$, $x=\sum_{i=1}^n a_ib_i = \sum_{i=1}^n a_i (\sum_{j=1}^m b_{ij}) = \sum_{i,j} a_ib_j$, where $a_ib_j\in a$ are all homogeneous, and we can regroup them by grading, i.e. $$x=\sum_{d\ge 0}\sum_{\deg(a_ib_j)=d}a_ib_j\in \sum_{d\ge 0} a\cap S_d$$