## Tag `07LU`

Chapter 15: More on Algebra > Section 15.69: Characterizing perfect complexes

Lemma 15.69.4. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that

- $K \otimes_R^\mathbf{L} R/I$ is perfect in $D(R/I)$, and
- $I$ 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 15.66.1. By Lemma 15.67.3 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 17001–17010 (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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