# The Stacks Project

## Tag 066R

Lemma 15.66.4. Let $R$ be a ring. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$ be a distinguished triangle in $D(R)$. If two out of three of $K^\bullet, L^\bullet, M^\bullet$ are perfect then the third is also perfect.

Proof. Combine Lemmas 15.66.2, 15.59.7, and 15.60.5. $\square$

The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 15658–15664 (see updates for more information).

\begin{lemma}
\label{lemma-two-out-of-three-perfect}
Let $R$ be a ring. Let $(K^\bullet, L^\bullet, M^\bullet, f, g, h)$
be a distinguished triangle in $D(R)$. If two out of three of
$K^\bullet, L^\bullet, M^\bullet$ are
perfect then the third is also perfect.
\end{lemma}

\begin{proof}
Combine
Lemmas \ref{lemma-perfect}, \ref{lemma-two-out-of-three-pseudo-coherent}, and
\ref{lemma-cone-tor-amplitude}.
\end{proof}

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