The Stacks project

Lemma 15.81.3. Let $R \to A$ be a ring map of finite type. Let $K^\bullet $ be a complex of $A$-modules. Let $m \in \mathbf{Z}$. The following are equivalent

  1. for some presentation $\alpha : R[x_1, \ldots , x_ n] \to A$ the complex $K^\bullet $ is an $m$-pseudo-coherent complex of $R[x_1, \ldots , x_ n]$-modules,

  2. for all presentations $\alpha : R[x_1, \ldots , x_ n] \to A$ the complex $K^\bullet $ is an $m$-pseudo-coherent complex of $R[x_1, \ldots , x_ n]$-modules.

In particular the same equivalence holds for pseudo-coherence.

Proof. If $\alpha : R[x_1, \ldots , x_ n] \to A$ and $\beta : R[y_1, \ldots , y_ m] \to A$ are presentations. Choose $f_ j \in R[x_1, \ldots , x_ n]$ with $\alpha (f_ j) = \beta (y_ j)$ and $g_ i \in R[y_1, \ldots , y_ m]$ with $\beta (g_ i) = \alpha (x_ i)$. Then we get a commutative diagram

\[ \xymatrix{ R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \ar[d]^{x_ i \mapsto g_ i} \ar[rr]_-{y_ j \mapsto f_ j} & & R[x_1, \ldots , x_ n] \ar[d] \\ R[y_1, \ldots , y_ m] \ar[rr] & & A } \]

After a change of coordinates the ring homomorphism $R[x_1, \ldots , x_ n, y_1, \ldots , y_ m] \to R[x_1, \ldots , x_ n]$ is isomorphic to the ring homomorphism which maps each $y_ i$ to zero. Similarly for the left vertical map in the diagram. Hence, by induction on the number of variables this lemma follows from Lemma 15.81.2. The pseudo-coherent case follows from this and Lemma 15.64.5. $\square$

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Your email address will not be published.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 065H. Beware of the difference between the letter 'O' and the digit '0'.