Is there any reason why people consider different grading of a polynomial ring other than the canonical one? For example, one can define grading on $$k[s^4,s^3t, st^3,t^4]$$ by considering $$\operatorname{deg}(s)=\operatorname{deg}(t)=1$$, or considering $$\operatorname{deg}(s^4)=\operatorname{deg}(t^4)=1$$. What advantages do we get from different grading? I would also like to see more examples that put different grading into use to solve problems.
• There are many instances of graded rings arising naturally which are isomorphic to polynomial rings, but where the "natural" grading places the variables in degrees other than $1$ (often degree $2$). For example, the cohomology ring of a space $H^*(X) := \bigoplus_{i = 0}^{\infty} H^i(X)$ has a natural grading on it, as does the homotopy ring of a ring spectrum $\pi_* E := \bigoplus \pi_i(E).$ A particular important example is the topological Hochschild homology of a finite field $k$. It is known that $\pi_*THH(k)\cong k[u],$ but $u$ is in degree $2$ -- it generates $\pi_2 THH(k).$ Jan 15, 2020 at 5:34
• Another less fancy example is that $H^*(\Bbb{C}\Bbb{P}^n,\Bbb{Z})\cong\Bbb{Z}[x]/(x^{n+1}),$ where $\deg x = 2.$ Here $H^1(\Bbb{C}\Bbb{P}^n,\Bbb{Z}) = 0,$ and $x$ represents a generator of $H^2(\Bbb{C}\Bbb{P}^n,\Bbb{Z})\cong\Bbb{Z}.$ Jan 15, 2020 at 5:41
One reason is the following: If you for example want to understand all line bundles on the projective space $$\mathbb{P}^n$$, you will see that they will be given by twists of the structure sheaf, i.e. by $$\mathscr{O}_{\mathbb{P}^n}(m)$$. These twists are basically defined by shifting the grading and hence by having a different grading on the polynomial ring.