The Stacks project

Lemma 15.73.11. Let $R$ be a ring. If $K$ and $L$ are perfect objects of $D(R)$, then $K \otimes _ R^\mathbf {L} L$ is a perfect object too.

Proof. We can prove this using the definition as follows. We may represent $K$, resp. $L$ by a bounded complex $K^\bullet $, resp. $L^\bullet $ of finite projective $R$-modules. Then $K \otimes _ R^\mathbf {L} L$ is represented by the bounded complex $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$. The terms of this complex are direct sums of the modules $M^ a \otimes _ R L^ b$. Since $M^ a$ and $L^ b$ are direct summands of finite free $R$-modules, so is $M^ a \otimes _ R L^ b$. Hence we conclude the terms of the complex $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$ are finite projective.

Another proof can be given using the characterization of perfect complexes in Lemma 15.73.2 and the corresponding lemmas for pseudo-coherent complexes (Lemma 15.63.16) and for tor amplitude (Lemma 15.65.10 used with $A = B = R$). $\square$


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