Lemma 46.3.11. Let A be a ring. Let \varphi : F \to G be a map of adequate functors on \textit{Alg}_ A. Then \mathop{\mathrm{Ker}}(\varphi ) is adequate.
Proof. Choose an injection F \to \underline{M} and an injection G \to \underline{N}. Denote F \to \underline{M \oplus N} the diagonal map so that
\xymatrix{ F \ar[d] \ar[r] & G \ar[d] \\ \underline{M \oplus N} \ar[r] & \underline{N} }
commutes. By Lemma 46.3.10 we can find a module map M \oplus N \to K such that F is the kernel of \underline{M \oplus N} \to \underline{K}. Then \mathop{\mathrm{Ker}}(\varphi ) is the kernel of \underline{M \oplus N} \to \underline{K \oplus N}. \square
Comments (0)