The Stacks project

Lemma 46.3.9. Let $A$ be a ring. Let $\varphi : F \to \underline{M}$ be a map of module-valued functors on $\textit{Alg}_ A$ with $F$ adequate. Then $\mathop{\mathrm{Coker}}(\varphi )$ is adequate.

Proof. By Lemma 46.3.6 we may assume that $F = \bigoplus L_ i$ is a direct sum of linearly adequate functors. Choose exact sequences $0 \to L_ i \to \underline{A^{\oplus n_ i}} \to \underline{A^{\oplus m_ i}}$. For each $i$ choose a map $A^{\oplus n_ i} \to M$ as in Lemma 46.3.8. Consider the diagram

\[ \xymatrix{ 0 \ar[r] & \bigoplus L_ i \ar[r] \ar[d] & \bigoplus \underline{A^{\oplus n_ i}} \ar[r] \ar[ld] & \bigoplus \underline{A^{\oplus m_ i}} \\ & \underline{M} } \]

Consider the $A$-modules

\[ Q = \mathop{\mathrm{Coker}}(\bigoplus A^{\oplus n_ i} \to M \oplus \bigoplus A^{\oplus m_ i}) \quad \text{and}\quad P = \mathop{\mathrm{Coker}}(\bigoplus A^{\oplus n_ i} \to \bigoplus A^{\oplus m_ i}). \]

Then we see that $\mathop{\mathrm{Coker}}(\varphi )$ is isomorphic to the kernel of $\underline{Q} \to \underline{P}$. $\square$

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Your email address will not be published.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 06V1. Beware of the difference between the letter 'O' and the digit '0'.