Lemma 15.81.16. Let $R \to A$ be a finite type ring map. Let $K^\bullet $ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. Let $f_1, \ldots , f_ r \in A$ generate the unit ideal. The following are equivalent
each $K^\bullet \otimes _ A A_{f_ i}$ is $m$-pseudo-coherent relative to $R$, and
$K^\bullet $ is $m$-pseudo-coherent relative to $R$.
The same equivalence holds for pseudo-coherence relative to $R$.
Proof. The implication (2) $\Rightarrow $ (1) is in Lemma 15.81.11. Assume (1). Write $1 = \sum f_ ig_ i$ in $A$. Choose a surjection $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r] \to A$. such that $y_ i$ maps to $f_ i$ and $z_ i$ maps to $g_ i$. Then we see that there exists a surjection
Note that $P$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-module and that $P[1/y_ i]$ is pseudo-coherent as an $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r, 1/y_ i]$-module. Hence by Lemma 15.64.11 we see that $K^\bullet \otimes _ A A_{f_ i}$ is an $m$-pseudo-coherent complex of $P[1/y_ i]$-modules for each $i$. Thus by Lemma 15.64.14 we see that $K^\bullet $ is pseudo-coherent as a complex of $P$-modules, and Lemma 15.64.11 shows that $K^\bullet $ is pseudo-coherent as a complex of $R[x_1, \ldots , x_ n, y_1, \ldots , y_ r, z_1, \ldots , z_ r]$-modules. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)