Exercise 19, § 2

Let $\mathscr{F}$ be a set of left ideals of a ring $A$; suppose that $\mathscr{F}$ is topologizing and idempotent (Exercises 16 and 18).

(a) Show that, if $m \in \mathscr{F}$, $n \in \mathscr{F}$ and $u \in \mathrm{Hom}_A(n, A_s)$, then $\overline{u}^1(m) \in \mathscr{F}$ (consider the exact sequence $0 \to n/\overline{u}^1(m) \to A_s/\overline{u}^1(m) \to A_s/n \to 0$). For every $A$-module $M$ and all $v \in \mathrm{Hom}_A(m, M)$, let $u.v$ denote the canonical image in $M_{(\mathscr{F})}$ (Exercise 17 (c)) of the composite homomorphism $\overline{u}^1(m) \xrightarrow{u} m \xrightarrow{v} M$. Show that the mapping $(u, v) \mapsto u.v$ is $\mathbf{Z}$-bilinear; there corresponds to it a $\mathbf{Z}$-linear mapping

These mappings form a direct system and their direct limit therefore corresponds canonically to a $\mathbf{Z}$-bilinear mapping

Show that, if $M = A_0$, $\psi$ is an internal law of composition which makes $A_{(\mathscr{F})}$ into a ring and the canonical mapping $A \to A_{(\mathscr{F})}$ is a ring homomorphism for this structure. For any left $A$-module $M$, $\psi$ is an external law defining on $M_{(\mathscr{F})}$ a left $A_{(\mathscr{F})}$-module structure; with this structure (and the canonical mapping $A \to A_{(\mathscr{F})}$) the canonical mapping $M \to M_{(\mathscr{F})}$ is an $A$-module homomorphism.

(b) If $i : M \to M_{(\mathscr{F})}$ is the canonical homomorphism, show that

and that the $A$-module $\operatorname{Coker}\,(i) = M_{(\mathscr{F})}/i(M)$ is 9-negligible.

(c) For every left $A$-module $M$ let $M_{\mathscr{F}}$ denote the left $A$-module $(M/\mathscr{F}M)_{(\mathscr{F})}$ and $j_M$ the composite canonical mapping

then $\operatorname{Ker}(j_M) = \mathscr{F}M$ and the $A$-module $\operatorname{Coker}(j_M) = M_{\mathscr{F}}/j_M(M)$ is $\mathscr{F}$-negligible.

Let $u : P \to M$, $h : P \to Q$ be $A$-module homomorphisms such that $\operatorname{Ker}(h)$ and $\operatorname{Coker}(h)$ are $\mathscr{F}$-negligible; show that there exists a unique $A$-homomorphism $v : Q \to M_{\mathscr{F}}$ which makes the following diagram commutative

(First reduce it to the case where $h$ is injective; then, identifying $P$ with a submodule of $Q$, note that, for $x \in Q$, there exists $m \in \mathscr{F}$ such that $mx \in P$ and map $x$ to the canonical image in $M_{\mathscr{F}}$ of the composite homomorphism

(d) For every $A$-module homomorphism $u : M \to N$ show that there exists a unique homomorphism $u_{\mathscr{F}} : M_{\mathscr{F}} \to N_{\mathscr{F}}$ which makes the following diagram commutative

(use (a), (b) and (c)). Show that, if

is an exact sequence of A-modules, the sequence

is exact (note that the mapping $M/\mathscr{F}M \to N/\mathscr{F}N$ derived from $u$ by taking quotients is injective).

Show that the mappings $j_{M_{\mathscr{F}}}$ and $(j_M)_{\mathscr{F}}$ coincide and are isomorphisms (and therefore $\mathscr{F}M_{\mathscr{F}} = 0$). If $N'$ is a sub-A-module of $M_{\mathscr{F}}$, $(j_{M_{\mathscr{F}}}^{-1}(N'))_{\mathscr{F}}$ is canonically identified with $N'_{\mathscr{F}}$ (observe that, if we write $N = j_M(N')$, $N'/j_M(N)$ is 9-negligible).

(e) Let M be a left A-module and $\phi : A \times M \to M$ the Z-bilinear mapping $(a, m) \mapsto am$. Show that there exists a unique Z-bilinear mapping $\phi_{\mathscr{F}} : A_{\mathscr{F}} \times M_{\mathscr{F}} \to M_{\mathscr{F}}$ which makes the following diagram commutative

We also define on $A_{\mathscr{F}}$ a ring structure and on $M_{\mathscr{F}}$ a left $A_{\mathscr{F}}$-module structure. If $u : M \to N$ is an A-module homomorphism, $u_{\mathscr{F}} : M_{\mathscr{F}} \to N_{\mathscr{F}}$ is an $A_{\mathscr{F}}$-module homomorphism. In order that, for every A-module M, the mapping $u \mapsto u_{\mathscr{F}}$ of $\mathrm{Hom}_A(M, N)$ to $\mathrm{Hom}_{A_{\mathscr{F}}}(M_{\mathscr{F}}, N_{\mathscr{F}})$ be bijective, it is necessary and sufficient that $j_N$ be bijective.

(f) Take A to be the polynomial ring $K[X, Y]$ over a commutative field K and $\mathscr{F}$ to be the (topologizing and idempotent) set of ideals of A containing a power of the ideal $m = (X) + (Y)$. If $M = A/AX$ and $u : A \to M$ is the canonical mapping, show that $u_{\mathscr{F}} : A_{\mathscr{F}} \to M_{\mathscr{F}}$ is not surjective. (Show that $A_{\mathscr{F}}$ is identified with A and $M_{\mathscr{F}}$ with $K[\overline{Y}, 1/\overline{Y}]$, where $\overline{Y}$ is the class of Y in M, so that $X\overline{Y} = 0$ and $X(1/\overline{Y}) = 0$.)

(g) If we set $\mathscr{F}^1M = M_{\mathscr{F}}/j_M(M)$, show that, for every exact sequence $0 \to M \to N \to P \to 0$ of A-modules, there is an exact sequence

(use the snake diagram).

(h) Let $\mathscr{F}'$ be the set of left ideals $l'$ of $A_{\mathscr{F}}$ such that $(A_{\mathscr{F}})_s/l'$ is an $\mathscr{F}$-negligible A-module: show that $\mathscr{F}'$ is the set of left ideals of $A_{\mathscr{F}}$ containing the $j_A(m)$, where $m \in \mathscr{F}$ (observe that $m_{\mathscr{F}} = A_{\mathscr{F}}$ for all $m \in \mathscr{F}$, using the exact sequence of (d); consequently $A_{\mathscr{F}}/j_A(m)$ is 9-negligible). For every $A_{\mathscr{F}}$-module $M'$, $\mathscr{F}M' = \mathscr{F}'M'$ ($M'$ being considered as an $A$-module by means of $j_A$), $\mathscr{F}'$ is topologizing and idempotent and $M'_{\mathscr{F}'}$ is canonically identified with $M'_{\mathscr{F}}$.