The Stacks project

Example 46.4.6. Let $F$ be a module-valued functor as in Lemma 46.4.5. It is not always the case that the two module structures on $TF(B, N)$ agree. Here is an example. Suppose $A = \mathbf{F}_ p$ where $p$ is a prime. Set $F(B) = B$ but with $B$-module structure given by $b \cdot x = b^ px$. Then $TF(B, N) = N$ with $B$-module structure given by $b \cdot x = b^ px$ for $x \in N$. However, the second $B$-module structure is given by $x \cdot b = bx$. Note that in this case the canonical map $N \otimes _ B F(B) \to TF(B, N)$ is zero as raising an element $n \in B[N]$ to the $p$th power is zero.