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Tag 052N

Chapter 10: Commutative Algebra > Section 10.56: Proj of a graded ring

Lemma 10.56.10. Let $R$ be a ring. Let $R'$ be a finite type $R$-algebra, and let $M$ be a finite $R'$-module. There exists a graded $R$-algebra $S$, a graded $S$-module $N$ and an element $f \in S$ homogeneous of degree $1$ such that

  1. $R' \cong S_{(f)}$ and $M \cong N_{(f)}$ (as modules),
  2. $S_0 = R$ and $S$ is generated by finitely many elements of degree $1$ over $R$, and
  3. $N$ is a finite $S$-module.

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 $$ S_{(f)} \longrightarrow R', \quad X_i/X_0 \longmapsto x_i. $$ To do the same thing with the module $M$ we choose a presentation $$ M = (R')^{\oplus r}/\sum\nolimits_{j \in J} R'k_j $$ 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 $$ N = \mathop{\rm Coker}\Big( \bigoplus\nolimits_{j \in J} S(-d_j) \xrightarrow{(K_{ij})} S^{\oplus r} \Big) $$ which works. Some details omitted. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 13108–13120 (see updates for more information).

    \begin{lemma}
    \label{lemma-homogenize}
    Let $R$ be a ring.
    Let $R'$ be a finite type $R$-algebra, and let $M$ be a finite $R'$-module.
    There exists a graded $R$-algebra $S$, a graded $S$-module $N$ and
    an element $f \in S$ homogeneous of degree $1$ such that
    \begin{enumerate}
    \item $R' \cong S_{(f)}$ and $M \cong N_{(f)}$ (as modules),
    \item $S_0 = R$ and $S$ is generated by finitely many elements
    of degree $1$ over $R$, and
    \item $N$ is a finite $S$-module.
    \end{enumerate}
    \end{lemma}
    
    \begin{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
    $$
    S_{(f)} \longrightarrow R', \quad
    X_i/X_0 \longmapsto x_i.
    $$
    To do the same thing with the module $M$ we choose a presentation
    $$
    M = (R')^{\oplus r}/\sum\nolimits_{j \in J} R'k_j
    $$
    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
    $$
    N = \Coker\Big(
    \bigoplus\nolimits_{j \in J} S(-d_j) \xrightarrow{(K_{ij})} S^{\oplus r}
    \Big)
    $$
    which works. Some details omitted.
    \end{proof}

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