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Tag 066R

Chapter 15: More on Algebra > Section 15.67: Perfect complexes

Lemma 15.67.4. 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 15.67.2, 15.60.7, and 15.61.5. $\square$

    The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 15960–15966 (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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