'Unusual' examples of graded rings?

Question

Almost all examples I have seen of graded rings are some variants of polynomial rings; either localisations or quotients, etc. I am looking for more 'unusual' examples, such as the cohomology example given here: https://en.wikipedia.org/wiki/Graded_ring#Basic_examples . What kind of graded rings do people encounter in surprising places? Are there 'organic' examples where the index monoid is something other than $(\mathbb{N},+)$? I am interested in any examples people might have.

Answer 1

Let's make a little list of interesting graded rings.

1. The ring of modular forms (https://en.wikipedia.org/wiki/Ring_of_modular_forms). In some cases this is (isomorphic to) a polynomial ring, but in general we only know details about generators of modular forms for congruence groups. And modular forms are nice objects anyways.

2. Let $k$ be a field and $R = k[x_1, ..., x_n]$ a polynomial ring. Write $x^a = x_1^{a_1}\cdot ... \cdot x_n^{a_n}$, and let $\deg(x^a) = a = (a_1,...,a_n)$. This is the standard multigrading on $R$, and is among other things the setting for an algebraic combinatorics (or combinatorial algebra). Multigraded Hilbert series and Betti numbers are used to assess properties of combinatorial objects such as abstract simplicial complexes.

3. The associated graded ring (https://en.wikipedia.org/wiki/Associated_graded_ring). Let's look at an example which has applications to solving partial differential equations, and which is mentioned as "multiplicative filtrations" in the wikipedia article (details are taken from Jan-Erik Björk's book Rings of Differential Operators). Let $k$ be a field of characteristic zero, and $R = A_n(k)$ the Weyl algebra of dimension $n$ over $k$. Let $F_i$ be the $k$-subspace of $R$ spanned by elements $x^a\partial^b$ (in the multivariate notation as above) such that $|a| + |b| \leq i$. The filtration $F = (F_i)$ is called the Bernstein filtration on $R$, and one checks that $\cup_i F_i = R$ and that $F_iF_j \subset F_{i+j}$. The vector space $\operatorname{gr}_F(R) = \oplus_i F_i/F_{i+1}$ is in fact a ring, the graded ring associated to $F$. Disappointingly, this graded ring is isomorphic to $k[y_1, ..., y_{2n}]$, but in return one can use its algebraic properties to study solutions to partial differential equations, which is a beautiful marriage of algebra and analysis. Note that $A_n(k)$, as well as many other important rings in this theory, are non-commutative.

4. For something with negative degree, let $R$ be a graded commutative unitary ring, $S$ a multiplicative subset of homogeneous elements, and $S^{-1}R$ the localisation. Then one can get negative degrees by declaring $\deg(f/g) = \deg(f) - \deg(g)$ on the localisation.