Exercise 3, § 5

Let $M$ be an a-module, $n$ an integer $\geqslant 1$. Now prove that the following conditions are equivalent:

α) $M$ admits a finite $n$-presentation (X, p. 180, exercise 6).

β) For every filtered inductive system $(N_\alpha)_{\alpha \in I}$, the canonical homomorphism

is bijective for $i < n$.

γ) For every family $(P_\beta)_{\beta \in J}$ of right A-modules, the canonical homomorphism

is bijective for $i < n$.

δ) For every set I, the canonical homomorphism $A_d^! \otimes_A M \to M^!$ is bijective, and one has

(Reason by induction on $n$, using exercise 18, p. 170.)