The Stacks project

Lemma 15.81.13. Let $R \to A \to B$ be finite type ring maps. Let $m \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $A$-modules. Assume $B$ as a $B$-module is pseudo-coherent relative to $A$. If $K^\bullet $ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$, then $K^\bullet \otimes _ A^{\mathbf{L}} B$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$.

Proof. Choose a surjection $A[y_1, \ldots , y_ m] \to B$. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Combined we get a surjection $R[x_1, \ldots , x_ n, y_1, \ldots y_ m] \to B$. Choose a resolution $E^\bullet \to B$ of $B$ by a complex of finite free $A[y_1, \ldots , y_ n]$-modules (which is possible by our assumption on the ring map $A \to B$). We may assume that $K^\bullet $ is a bounded above complex of flat $A$-modules. Then

\begin{align*} K^\bullet \otimes _ A^{\mathbf{L}} B & = \text{Tot}(K^\bullet \otimes _ A B[0]) \\ & = \text{Tot}(K^\bullet \otimes _ A A[y_1, \ldots , y_ m] \otimes _{A[y_1, \ldots , y_ m]} B[0]) \\ & \cong \text{Tot}\left( (K^\bullet \otimes _ A A[y_1, \ldots , y_ m]) \otimes _{A[y_1, \ldots , y_ m]} E^\bullet \right) \\ & = \text{Tot}(K^\bullet \otimes _ A E^\bullet ) \end{align*}

in $D(A[y_1, \ldots , y_ m])$. The quasi-isomorphism $\cong $ comes from an application of Lemma 15.59.7. Thus we have to show that $\text{Tot}(K^\bullet \otimes _ A E^\bullet )$ is $m$-pseudo-coherent as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots y_ m]$-modules. Note that $\text{Tot}(K^\bullet \otimes _ A E^\bullet )$ has a filtration by subcomplexes with successive quotients the complexes $K^\bullet \otimes _ A E^ i[-i]$. Note that for $i \ll 0$ the complexes $K^\bullet \otimes _ A E^ i[-i]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent (over any ring). Hence, applying Lemma 15.81.6 and induction, it suffices to show that $K^\bullet \otimes _ A E^ i[-i]$ is pseudo-coherent relative to $R$ for all $i$. Note that $E^ i = 0$ for $i > 0$. Since also $E^ i$ is finite free this reduces to proving that $K^\bullet \otimes _ A A[y_1, \ldots , y_ m]$ is $m$-pseudo-coherent relative to $R$ which follows from Lemma 15.81.12 for instance. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 067B. Beware of the difference between the letter 'O' and the digit '0'.