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.
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3$\begingroup$ You may want to google "quasi-homogeneous" and "weighted projective spaces". $\endgroup$– YoungsuJan 14, 2020 at 19:25
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$\begingroup$ 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 $\pi_2 THH(k).$ $\endgroup$– StahlJan 15, 2020 at 5:34
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$\begingroup$ 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 $H^2(\Bbb{C}\Bbb{P}^n,\Bbb{Z})\cong\Bbb{Z}.$ $\endgroup$– StahlJan 15, 2020 at 5:41
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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.
Wanting to understand certain sheaves like invertible sheaves definitely is something you want to do since many proofs in algebraic geometry are cohomological ones using exact sequences of sheaves and in particular also invertible sheaves/line bundles.
