Cauchy modules

A module $\relax\modmap M \from R \to S$ gives rise to a module $\modmap M^* = \Hom_R (M, R) \from S \to R$ called the left dual of M. There is a canonical module morphism

$$\map \rho_L^M \from M^* \xR L \to {\Hom_R (M, L)}$$

given by $\rho^M_L (u \xO l)(m) = u(m)l$, for each left R-module L.

Call an $\modmap M \from R \to S$ cauchy when $\rho^M_L$ is an isomorphism for all left R-modules L. Our goal in this section is to characterize cauchy modules more intrinsically.

A module P is called projective when, for all surjective module morphisms $\map e \from L \to {L'}$ and all module morphisms $\map f \from P \to {L'}$, there exists some module morphism $\map g \from P \to L$ with $f = e \circ g$.

$$\spreaddiagramcolumns {.5pc}\diagram & P \dldash_-{g} \drto^-{f} & \\ L \rronto^-{e} & & L' \\ \enddiagram$$

A morphism $\map r \from M \to N$ is said to be a retraction when there exists a morphism $\map i \from N \to M$ with $r \circ i = 1_N^{}$. When a retraction exists from M to N, we call N a retract of M.

Proposition 6.1  

A module P is projective iff P is a retract of some free module F.

Exercise 6.2  

Show that a module P is finitely generated and projective if and only if P is a retract of a free module on a finite set.

Hint: In (3) we did not need $\Free(M)$; only $\Free(X)$ for any X generating M.

This brings us to the fundamental theorem of ``Morita theory".

Theorem 6.3  

The following conditions on a module $\modmap M \from R \to S$ are equivalent:

(i) M is cauchy.

(ii) there exists a morphism $\twomodmap d \from S \to\to M^* \xR M \from S \to S$ such that both the following two composites are identity morphisms

$$\spreaddiagramcolumns{1.75pc}\spreaddiagramrows{-2pc}\diagra... ...} \xo \eval_{\!M}} & M^* \xR R \,\isom\, M^*\;. \\ \enddiagram$$

(iii) there exists a module $\modmap N \from S \to R$ and morphisms

$$\map e \from M \xS N \to R\quad, \;\quad \map d \from S \to {N \xR M}$$

such that the following composite is the identity morphism

$$\spreaddiagramcolumns{1.5pc}\diagram M \,\isom\, M \xS S \rt... ...to^-{e \xo 1_{\!M}} & R \xR M \,\isom\, M \;. \\ \enddiagram$$

(iv) M is a finitely generated projective left R-module.

Given rings R and S, from any ring morphism $\map f \from S \to R$ we obtain two modules $\modmap \fR \from S \to R$ and $\modmap R_f^{} \from R \to S$, which have R as underlying abelian group. They have scalar multiplicatons

$$\begin{eqnarray*} {\map \from {S \times \fR}\; \to {\;\fR}} & \quad,\quad & {\... ...\;\; r\,f(s) & \quad,\quad & (r',r) \;\;\mapsto\;\; r'\,r \;\;. \end{eqnarray*}$$

For any module $\modmap L\from R \to T$ we have canonical isomorphisms

$$\begin{eqnarray*} \fR\, \xR L & \;\;\isom\;\;\;\; L \;\;\;\;\isom\;\; & \Hom_R (... ...} & l & \\ & u(1) & \llap{$\xylongmapsfrom<2pc>$}\quad u \;\;. \end{eqnarray*}$$

It follows easily from this that

$$(R_f^{})^* \;\;\isom\;\; \fR$$

and that R_f is cauchy.

A module $\modmap M \from R \to S$ is called convergent when there exists a ring morphism $\map f \from S \to R$ and a module isomorphism $M \isom R_f^{}$.

The product $\modmap \prod_{i \in I}^{} M_i^{} \from R \to S$ of a family of modules $\modmap M_i^{} \from R \to S$ with $i \in I$, has as elements the families ${\mbf m} = (m_i^{})_{i \in I}^{}$ with $m_i^{} \in M_i^{}$; addition and scalar multiplication are given by

$${\mbf m}+ {\mbf m}' \;=\; (m_i^{} + m_i')_{i \in I}^{}\quad, \quad r\,{\mbf m}\,s = (r\,m_i^{}\,s)_{i \in I}^{} \;\;.$$

There are projections

$$\map \prj_j \from \prod_{i \in I} M_i^{} \to {M_j^{}} \qquad\hbox{for each } j \in I$$

given by $\prj_j ({\mbf m}) = m_j^{}$. There are also injective module morphisms

$$\map \inj_j \from M_j^{} \to {\prod_{i \in I} M_i^{}} \qquad\hbox{for each } j \in I$$

given by $\inj_j (m) = {\mbf m}$ where m_j = m and m_i = 0 for all $i \neq j$; we can use these to identify each M_j with the submodule of $\prod_{i \in I} M_i^{}$ consisting of those ${\mbf m}$ with m_i = 0 for all $i \neq j$.

The direct sum $\modmap \sum_{i \in I}^{} M_i^{} \from R \to S$ is the submodule of $\prod_{i \in I}^{} M_i^{}$ which consists of those ${\mbf m}$ for which m_i = 0 for all but finitly many $i \in I$. This is the submodule generated by the union $\cup_{i \in I}^{} M_i^{}\,$, hence we can write $\sum_{i \in I}^{} m_i^{}$ instead of ${\mbf m} \in \sum_{i \in I}^{} M_i^{}$. Of course the injections $\inj_j$ actually land in $\sum_{i \in I}^{} M_i^{}$.

Proposition 6.4  

There are module isomorphisms

$$\begin{eqnarray*} \noalign{\vskip-\baselineskip} \smash{\lower1.25\baselineskip\... ...uad & \moddecomp & \quad {\textstyle\sum}_i (m_i^{} \xO n) \;. \end{eqnarray*}$$

When I is finite, notice that $\sum_{i \in I} M_i^{} = \prod_{i \in I} M_i^{}$. This is also frequently written $\oplus_{i \in I} M_i^{}$. So $M \oplus N = M \times N = M + N$.