The Stacks project

Lemma 15.78.5. Let $R$ be a ring. Let $I, J \subset R$ be ideals. Let $K$ be an object of $D(R)$. Assume that

  1. $K \otimes _ R^\mathbf {L} R/I$ is perfect in $D(R/I)$, and

  2. $K \otimes _ R^\mathbf {L} R/J$ is perfect in $D(R/J)$.

Then $K \otimes _ R^\mathbf {L} R/IJ$ is perfect in $D(R/IJ)$.

Proof. It is clear that we may assume replace $R$ by $R/IJ$ and $K$ by $K \otimes _ R^\mathbf {L} R/IJ$. Then $R \to R/(I \cap J)$ is a surjection whose kernel has square zero. Hence by Lemma 15.78.4 it suffices to prove that $K \otimes _ R^\mathbf {L} R/(I \cap J)$ is perfect. Thus we may assume that $I \cap J = 0$.

We prove the lemma in case $I \cap J = 0$. First, we may represent $K$ by a K-flat complex $K^\bullet $ with all $K^ n$ flat, see Lemma 15.59.10. Then we see that we have a short exact sequence of complexes

\[ 0 \to K^\bullet \to K^\bullet /IK^\bullet \oplus K^\bullet /JK^\bullet \to K^\bullet /(I + J)K^\bullet \to 0 \]

Note that $K^\bullet /IK^\bullet $ represents $K \otimes ^\mathbf {L}_ R R/I$ by construction of the derived tensor product. Similarly for $K^\bullet /JK^\bullet $ and $K^\bullet /(I + J)K^\bullet $. Note that $K^\bullet /(I + J)K^\bullet $ is a perfect complex of $R/(I + J)$-modules, see Lemma 15.74.9. Hence the complexes $K^\bullet /IK^\bullet $, and $K^\bullet /JK^\bullet $ and $K^\bullet /(I + J)K^\bullet $ have finitely many nonzero cohomology groups (since a perfect complex has finite Tor-amplitude, see Lemma 15.74.2). We conclude that $K \in D^ b(R)$ by the long exact cohomology sequence associated to short exact sequence of complexes displayed above. In particular we assume $K^\bullet $ is a bounded above complex of free $R$-modules (see Derived Categories, Lemma 13.15.4).

We will now show that $K$ is perfect using the criterion of Proposition 15.78.3. Thus we let $E_ j \in D(R)$ be a family of objects parametrized by a set $J$. We choose complexes $E_ j^\bullet $ with flat terms representing $E_ j$, see for example Lemma 15.59.10. It is clear that

\[ 0 \to E_ j^\bullet \to E_ j^\bullet /IE_ j^\bullet \oplus E_ j^\bullet /JE_ j^\bullet \to E_ j^\bullet /(I + J)E_ j^\bullet \to 0 \]

is a short exact sequence of complexes. Taking direct sums we obtain a similar short exact sequence

\[ 0 \to \bigoplus E_ j^\bullet \to \bigoplus E_ j^\bullet /IE_ j^\bullet \oplus E_ j^\bullet /JE_ j^\bullet \to \bigoplus E_ j^\bullet /(I + J)E_ j^\bullet \to 0 \]

(Note that $- \otimes _ R R/I$ commutes with direct sums.) This short exact sequence determines a distinguished triangle in $D(R)$, see Derived Categories, Lemma 13.12.1. Apply the homological functor $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, -)$ (see Derived Categories, Lemma 13.4.2) to get a commutative diagram

\[ \xymatrix{ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E_ j^\bullet /(I + J))[-1] \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , \bigoplus E_ j^\bullet /(I + J))[-1] \ar[d] \\ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E_ j^\bullet /I \oplus E_ j^\bullet /J)[-1] \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , \bigoplus E_ j^\bullet /I \oplus E_ j^\bullet /J)[-1] \ar[d] \\ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E_ j^\bullet ) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , \bigoplus E_ j^\bullet ) \ar[d] \\ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E_ j^\bullet /I \oplus E_ j^\bullet /J) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , \bigoplus E_ j^\bullet /I \oplus E_ j^\bullet /J) \ar[d] \\ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E_ j^\bullet /(I + J)) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , \bigoplus E_ j^\bullet /(I + J)) } \]

with exact columns. It is clear that, for any complex $E^\bullet $ of $R$-modules we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K^\bullet , E^\bullet /I) & = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(K^\bullet , E^\bullet /I) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(R/I)}(K^\bullet /IK^\bullet , E^\bullet /I) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(R/I)}(K^\bullet /IK^\bullet , E^\bullet /I) \end{align*}

and similarly for when dividing by $J$ or $I + J$, see Derived Categories, Lemma 13.19.8. Derived Categories. Thus all the horizontal arrows, except for possibly the middle one, are isomorphisms as the complexes $K^\bullet /IK^\bullet $, $K^\bullet /JK^\bullet $, $K^\bullet /(I + J)K^\bullet $ are perfect complexes of $R/I$, $R/J$, $R/(I + J)$-modules, see Proposition 15.78.3. It follows from the $5$-lemma (Homology, Lemma 12.5.20) that the middle map is an isomorphism and the lemma follows by Proposition 15.78.3. $\square$


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