Intuition behind basic facts about homogeneous ideals?

Question

What is the intuition (hopefully, geometric) behind these basic facts about homogeneous ideals? An ideal $I$ in $S$ is homogeneous if an element $f = \sum_{n \ge 0} f_n$ of $S$ lies in $I$ if and only if each homogeneous part $f_n$ lies in $I$. Here, we let $S = \bigoplus_{n \ge 0} S_n$ be an $\mathbf{N}$-graded ring.

• An ideal generated by homogeneous elements is a homogeneous ideal.

• A homogeneous ideal $I$ in $S$ is prime if and only if it is a proper ideal and $$fg \in I \implies f \in I \text{ or }g \in I$$for homogeneous $f, g \in S$.

• The kernel of $\mathbf{N}$-graded rings is a homogeneous ideal.

• For any homogeneous ideal $I$ of $S$ there is a natural $\mathbf{N}$-grading on $S/I$.

I am not an algebraic geometrer by trade, but I need to use these results.

Answer 1

If you have $S=K[x_1,\ldots, x_n]$ with the usual grading, then ideals will correspond to closed subsets of $\mathbb{A}^n$. An ideal $I$ is homogeneous if and only the zero set $V(I)$ satisfies the condition that if $(a_1,\ldots a_n)\in V(I)$ and $\lambda\in K\backslash 0$ then $(\lambda a_1\ldots, \lambda a_n)\in V(I)$. If you think of the natural projection $\mathbb{A}^n\backslash 0\rightarrow \mathbb{P}^{n-1}$, then these are exactly the inverse images of the Zariski closed sets in $\mathbb{P}^{n-1}$.

Answer 2

One thing to grok is that, in my opinion, graded rings are a hack. Occasionally graded rings as defined are of interest, but usually, at least in the fields I'm familiar with, the inhomogeneous elements don't play a role at all, and their presence is only a trick so that you can leverage the existing body of knowledge of commutative algebra.

The actual algebraic structure you care about is the essentially algebraic one made up out of the graded components, and the arithmetic with them (i.e. addition should only be defined for things of the same grading). This can be phrased in terms of preadditive categories, with one object per grading. A ring, for example, is a preadditive category with one object.

(also, I believe this foundation for graded rings wasn't known for a long time, so the version you cite is less of a trick and more of the only thing people had to work with)

The intuition behind the theorems you list is simply that when using this hack, the usual definitions of things have some relevance to what you want for graded rings (e.g. that a homogeneous prime ideal when represented as an ordinary ideal is actually prime). That is, the inclusion of the inhomogeneous things don't get in the way of the theory working in a simple way.

Answer 3

Let $X$ be an affine variety and $A$ its coordinate ring. Then a $ \mathbb{Z} $ grading on $A$ is the same thing as a $ \mathbb{G}_m = k^{\times} $ action on $X$. If $Y$ is a closed subvariety cut out by the ideal $I$, then $I$ is graded if and only if $Y$ is invariant under the torus action. This should give you some intuition for graded ideals: they are exactly those which cut out torus invariant subvarieties.