Example 39.12.3. Let $A$ be a $\mathbf{Z}$-graded ring, i.e., $A$ comes with a direct sum decomposition $A = \bigoplus _{n \in \mathbf{Z}} A_ n$ and $A_ n \cdot A_ m \subset A_{n + m}$. Set $X = \mathop{\mathrm{Spec}}(A)$. Then we obtain a $\mathbf{G}_ m$-action

$a : \mathbf{G}_ m \times X \longrightarrow X$

by the ring map $\mu : A \to A \otimes \mathbf{Z}[x, x^{-1}]$, $f \mapsto f \otimes x^{\deg (f)}$. Namely, to check this we have to verify that

$\xymatrix{ A \ar[r]_\mu \ar[d]_\mu & A \otimes \mathbf{Z}[x, x^{-1}] \ar[d]^{\mu \otimes 1} \\ A \otimes \mathbf{Z}[x, x^{-1}] \ar[r]^-{1 \otimes m} & A \otimes \mathbf{Z}[x, x^{-1}] \otimes \mathbf{Z}[x, x^{-1}] }$

where $m(x) = x \otimes x$, see Example 39.5.1. This is immediately clear when evaluating on a homogeneous element. Suppose that $M$ is a graded $A$-module. Then we obtain a $\mathbf{G}_ m$-equivariant quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F} = \widetilde{M}$ by using $\alpha$ as in Definition 39.12.1 corresponding to the $A \otimes \mathbf{Z}[x, x^{-1}]$-module map

$M \otimes _{A, \mu } (A \otimes \mathbf{Z}[x, x^{-1}]) \longrightarrow M \otimes _{A, \text{id}_ A \otimes 1} (A \otimes \mathbf{Z}[x, x^{-1}])$

sending $m \otimes 1 \otimes 1$ to $m \otimes 1 \otimes x^{\deg (m)}$ for $m \in M$ homogeneous.

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