The Stacks project

Remark 15.60.4 (Warning). Let $R \to A$ be a ring map, and let $N$ and $N'$ be $A$-modules. Denote $N_ R$ and $N'_ R$ the restriction of $N$ and $N'$ to $R$-modules, see Algebra, Section 10.14. In this situation, the objects $N_ R \otimes _ R^\mathbf {L} N'$ and $N \otimes _ R^\mathbf {L} N'_ R$ of $D(A)$ are in general not isomorphic! In other words, one has to pay careful attention as to which of the two sides is being used to provide the $A$-module structure.

For a specific example, set $R = k[x, y]$, $A = R/(xy)$, $N = R/(x)$ and $N' = A = R/(xy)$. The resolution $0 \to R \xrightarrow {xy} R \to N'_ R \to 0$ shows that $N \otimes _ R^\mathbf {L} N'_ R = N[1] \oplus N$ in $D(A)$. The resolution $0 \to R \xrightarrow {x} R \to N_ R \to 0$ shows that $N_ R \otimes _ R^\mathbf {L} N'$ is represented by the complex $A \xrightarrow {x} A$. To see these two complexes are not isomorphic, one can show that the second complex is not isomorphic in $D(A)$ to the direct sum of its cohomology groups, or one can show that the first complex is not a perfect object of $D(A)$ whereas the second one is. Some details omitted.

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