Exercise 2, § 6

Let $A$ be a ring, and $\mathscr{F} = (\mathscr{F}_i)_{i \in \mathbf{Z}}$ a decreasing filtration of A such that $\mathscr{F}_i = A$ for $i \leq 0$. Let $\mathscr{A}$ be the subring $\bigoplus_{i \in \mathbf{Z}} \mathscr{F}_i . v^{-i}$ of $A[v, v^{-1}]$ and let $\mathrm{gr}_{\mathscr{F}}(A)$ be the graded ring associated with the filtered ring (A, $\mathscr{F}$). (a) Prove that the homomorphism $f : \mathscr{A} \to \mathrm{gr}_{\mathscr{F}}(A)$ defined by $f(\sum_i a_i v^{-i}) = \sum_i \overline{a}_i$, where $\overline{a}_i$ is the class of $a_i$ in $\mathscr{F}_i / \mathscr{F}_{i+1}$, induces an isomorphism of $\mathscr{A}/v.\mathscr{A}$ with $\mathrm{gr}_{\mathscr{F}}(A)$. (b) Prove that for every invertible element $v_0$ in A, the homomorphism $e_{v_0} : \mathscr{A} \to A$ defined by

induces an isomorphism of $\mathscr{A}/(v - v_0) \mathscr{A}$ with A. (c) Prove that $\mathscr{A}/A[v]$ is a $A[v]$-module of torsion. (d) Assuming now that A contains a field k, prove that the composed morphism makes $\mathscr{A}$ a $k[v]$-module without torsion. Deduce that $\mathscr{A}$ is a $k[v]$-module flat. (e) Suppose that A is local, with maximal ideal $m$, and contains a field $k$ such that A is direct sum of $k$ and $m$ (as additive group). Prove that there exists an ideal S of $\mathscr{A}$ such that $\mathscr{A}$ is direct sum of $k[v]$ and S (we have set $\mathscr{F}_i = m^i$ for every $i \in \mathbf{Z}$).