The Stacks project

Lemma 15.83.8. Let $R' \to A'$ be a flat ring map of finite presentation. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Set $A = A' \otimes _{R'} R$. Let $K' \in D(A')$ and set $K = K' \otimes _{A'}^\mathbf {L} A$ in $D(A)$. If $K$ is $R$-perfect, then $K'$ is $R'$-perfect.

Proof. We can represent $K$ by a bounded above complex of finite free $A$-modules $E^\bullet $, see Lemma 15.64.5. By Lemma 15.75.3 we conclude that $K'$ is pseudo-coherent because it can be represented by a bounded above complex $P^\bullet $ of finite free $A'$-modules with $P^\bullet \otimes _{A'} A = E^\bullet $. Observe that this also means $P^\bullet \otimes _{R'} R = E^\bullet $ (since $A = A' \otimes _{R'} R$).

Let $I = \mathop{\mathrm{Ker}}(R' \to R)$. Then $I^ n = 0$ for some $n$. Choose $[a, b]$ such that $K$ has tor amplitude in $[a, b]$ as a complex of $R$-modules. We will show $K'$ has tor amplitude in $[a, b]$. To do this, let $M'$ be an $R'$-module. If $IM' = 0$, then

\[ K' \otimes _{R'}^\mathbf {L} M' = P^\bullet \otimes _{R'} M' = E^\bullet \otimes _ R M' = K \otimes _ R^\mathbf {L} M' \]

(because $A'$ is flat over $R'$ and $A$ is flat over $R$) which has nonzero cohomology only for degrees in $[a, b]$ by choice of $a, b$. If $I^{t + 1}M' = 0$, then we consider the short exact sequence

\[ 0 \to IM' \to M' \to M'/IM' \to 0 \]

with $M = M'/IM'$. By induction on $t$ we have that both $K' \otimes _{R'}^\mathbf {L} IM'$ and $K' \otimes _{R'}^\mathbf {L} M'/IM'$ have nonzero cohomology only for degrees in $[a, b]$. Then the distinguished triangle

\[ K' \otimes _{R'}^\mathbf {L} IM' \to K' \otimes _{R'}^\mathbf {L} M' \to K' \otimes _{R'}^\mathbf {L} M'/IM' \to (K' \otimes _{R'}^\mathbf {L} IM')[1] \]

proves the same is true for $K' \otimes _{R'}^\mathbf {L} M'$. This proves the desired bound for all $M'$ and hence the desired bound on the tor amplitude of $K'$. $\square$


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