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Tag 07LU

Chapter 15: More on Algebra > Section 15.70: Characterizing perfect complexes

Lemma 15.70.4. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that

  1. $K \otimes_R^\mathbf{L} R/I$ is perfect in $D(R/I)$, and
  2. $I$ is a nilpotent ideal.

Then $K$ is perfect in $D(R)$.

Proof. Choose a finite complex $\overline{P}^\bullet$ of finite projective $R/I$-modules representing $K \otimes_R^\mathbf{L} R/I$, see Definition 15.67.1. By Lemma 15.68.3 there exists a complex $P^\bullet$ of projective $R$-modules representing $K$ such that $\overline{P}^\bullet = P^\bullet/IP^\bullet$. It follows from Nakayama's lemma (Algebra, Lemma 10.19.1) that $P^\bullet$ is a finite complex of finite projective $R$-modules. $\square$

    The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 17303–17312 (see updates for more information).

    \begin{lemma}
    \label{lemma-perfect-modulo-nilpotent-ideal}
    Let $R$ be a ring. Let $I \subset R$ be an ideal.
    Let $K$ be an object of $D(R)$. Assume that
    \begin{enumerate}
    \item $K \otimes_R^\mathbf{L} R/I$ is perfect in $D(R/I)$, and
    \item $I$ is a nilpotent ideal.
    \end{enumerate}
    Then $K$ is perfect in $D(R)$.
    \end{lemma}
    
    \begin{proof}
    Choose a finite complex $\overline{P}^\bullet$ of finite projective
    $R/I$-modules representing $K \otimes_R^\mathbf{L} R/I$, see
    Definition \ref{definition-perfect}. By
    Lemma \ref{lemma-lift-complex-projectives}
    there exists a complex $P^\bullet$ of projective $R$-modules
    representing $K$ such that $\overline{P}^\bullet = P^\bullet/IP^\bullet$.
    It follows from Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK})
    that $P^\bullet$ is a finite complex of finite projective
    $R$-modules.
    \end{proof}

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