## Tag `0629`

Chapter 15: More on Algebra > Section 15.26: The Koszul complex

Lemma 15.26.8. Let $R$ be a ring. Let $f_1, \ldots, f_r$ be a sequence of elements of $R$. The complex $K_\bullet(f_1, \ldots, f_r)$ is isomorphic to the cone of the map of complexes $$ f_r : K_\bullet(f_1, \ldots, f_{r - 1}) \longrightarrow K_\bullet(f_1, \ldots, f_{r - 1}). $$

Proof.Special case of Lemma 15.26.7. $\square$

The code snippet corresponding to this tag is a part of the file `more-algebra.tex` and is located in lines 5653–5664 (see updates for more information).

```
\begin{lemma}
\label{lemma-cone-koszul}
Let $R$ be a ring. Let $f_1, \ldots, f_r$ be a sequence of elements
of $R$. The complex $K_\bullet(f_1, \ldots, f_r)$ is isomorphic to the
cone of the map of complexes
$$
f_r :
K_\bullet(f_1, \ldots, f_{r - 1})
\longrightarrow
K_\bullet(f_1, \ldots, f_{r - 1}).
$$
\end{lemma}
\begin{proof}
Special case of
Lemma \ref{lemma-cone-koszul-abstract}.
\end{proof}
```

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