The Stacks project

Example 15.95.1. Let $A$ be a ring. Let $f \in A$ be a nonzerodivisor. Consider the functor $\eta _ f : K(f\text{-torsion free }A\text{-modules}) \to K(f\text{-torsion free }A\text{-modules})$. Let $M^\bullet $ be a complex of $f$-torsion free $A$-modules. Multiplication by $f$ defines an isomorphism $\eta _ f(M^\bullet [1]) \to (\eta _ fM^\bullet )[1]$, so in this sense $\eta _ f$ is compatible with shifts. However, consider the diagram

\[ \xymatrix{ A \ar[r]_ f & A \ar[r]_1 & A \ar[r] & 0 \\ 0 \ar[r] \ar[u] & 0 \ar[r] \ar[u] & A \ar[r]^{-1} \ar[u]^ f & A \ar[u] } \]

Think of each column as a complex of $f$-torsion free $A$-modules with the module on top in degree $1$ and the module under it in degree $0$. Then this diagram provides us with a distinguished triangle in $K(f\text{-torsion free }A\text{-modules})$ with triangulated structure as given in Derived Categories, Section 13.10. Namely the third complex is the cone of the map between the first two complexes. However, applying $\eta _ f$ to each column we obtain

\[ \xymatrix{ fA \ar[r]_ f & fA \ar[r]_1 & fA \ar[r] & 0 \\ 0 \ar[r] \ar[u] & 0 \ar[r] \ar[u] & A \ar[r]^{-1} \ar[u]^ f & A \ar[u] } \]

However, the third complex is acyclic and even homotopic to zero. Hence if this were a distinguished triangle, then the first arrow would have to be an isomorphism in the homotopy category, which is not true unless $f$ is a unit.

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