Exercise 3, § 6

Let us take up again the notations of the preceding exercise. Let $M$ be an A-module, endowed with a filtration $(K_i)_{i \in \mathbf{Z}}$ such that $K_i = M$ for $i \leq 0$. a) Show that the graded A-module $\mathscr{M} = \sum_{i \in \mathbf{Z}} K_i v^{-i}$ possesses a natural structure of graded $\mathscr{A}$-module. b) Show that the graded A-algebra $\mathscr{A}$ is generated by its elements of degree 1 and −1 if and only if there exists an ideal q of A such that $\mathscr{F}_i = q^i$ ($i \in \mathbf{Z}$). c) Assuming this last condition is realized, show that the filtration $(K_i)_{i \in \mathbf{Z}}$ is q-good if and only if the graded $\mathscr{A}$-module $\mathscr{M}$ is of finite type.

d) Show that if the filtration $(K_i)_{i\in\mathbf{Z}}$ is q-good, the $A$-module $M$ is without $v$-torsion, and that conversely, given a finite type $A$-module $M$ without $v$-torsion, one can associate with it a finite type $A$-module $M$ endowed with a q-good filtration such that the $A$-module associated with $M$ is $M$.