The Stacks project

Remark 46.3.15. Let $A$ be a ring. The proof of Lemma 46.3.14 shows that any extension $0 \to \underline{M} \to E \to L \to 0$ of module-valued functors on $\textit{Alg}_ A$ with $L$ linearly adequate splits. It uses only the following properties of the module-valued functor $F = \underline{M}$:

1. $F(B) \otimes _ B B' \to F(B')$ is an isomorphism for a flat ring map $B \to B'$, and

2. $F(C)^{(1)} = F(p_1)(F(B)^{(1)}) \oplus F(p_2)(F(B)^{(1)})$ where $B = A[x_1, \ldots , x_ n]/(\sum a_{ij}x_ j)$ and $C = A[x_1, \ldots , x_ n, y_1, \ldots , y_ n]/ (\sum a_{ij}x_ j, \sum a_{ij}y_ j)$.

These two properties hold for any adequate functor $F$; details omitted. Hence we see that $L$ is a projective object of the abelian category of adequate functors.