The Stacks project

Proof. We may write $R' = R[x_1, \ldots , x_ n]/I$ for some ideal $I$. For an element $g \in R[x_1, \ldots , x_ n]$ denote $\tilde g \in R[X_0, \ldots , X_ n]$ the element homogeneous of minimal degree such that $g = \tilde g(1, x_1, \ldots , x_ n)$. Let $\tilde I \subset R[X_0, \ldots , X_ n]$ generated by all elements $\tilde g$, $g \in I$. Set $S = R[X_0, \ldots , X_ n]/\tilde I$ and denote $f$ the image of $X_0$ in $S$. By construction we have an isomorphism

To do the same thing with the module $M$ we choose a presentation

with $k_ j = (k_{1j}, \ldots , k_{rj})$. Let $d_{ij} = \deg (\tilde k_{ij})$. Set $d_ j = \max \{ d_{ij}\} $. Set $K_{ij} = X_0^{d_ j - d_{ij}}\tilde k_{ij}$ which is homogeneous of degree $d_ j$. With this notation we set

which works. Some details omitted. $\square$