The Stacks project

Example 93.4.1 (Finite projective modules). Let $\mathcal{F}$ be the category defined as follows

1. an object is a pair $(A, M)$ consisting of an object $A$ of $\mathcal{C}_\Lambda $ and a finite projective $A$-module $M$, and

2. a morphism $(f, g) : (B, N) \to (A, M)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a map $g : N \to M$ which is $f$-linear and induces an isomorpism $N \otimes _{B, f} A \cong M$.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, M)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a finite dimensional $k$-vector space $V$, let $x_0 = (k, V)$ be the corresponding object of $\mathcal{F}(k)$. We set