The Stacks project

Lemma 19.2.7. Let $R$ be a ring.

  1. The construction $M \mapsto (M \to \mathbf{M}(M))$ is functorial in $M$.

  2. The map $M \to \mathbf{M}(M)$ is injective.

  3. For any ideal $\mathfrak {a}$ and any $R$-module map $\varphi : \mathfrak a \to M$ there is an $R$-module map $\varphi ' : R \to \mathbf{M}(M)$ such that

    \[ \xymatrix{ \mathfrak {a} \ar[d] \ar[r]_\varphi & M \ar[d] \\ R \ar[r]^{\varphi '} & \mathbf{M}(M) } \]

    commutes.

Proof. Parts (2) and (3) are immediate from the construction. To see (1), let $\chi : M \to N$ be an $R$-module map. We claim there exists a canonical commutative diagram

\[ \xymatrix{ \bigoplus _{\mathfrak a} \bigoplus _{\varphi \in \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak a, M)} \mathfrak {a} \ar[r] \ar[d] \ar[rrd] & M \ar[rrd]^\chi \\ \bigoplus _{\mathfrak a} \bigoplus _{\varphi \in \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak a, M)} R \ar[rrd] & & \bigoplus _{\mathfrak a} \bigoplus _{\psi \in \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak a, N)} \mathfrak {a} \ar[r] \ar[d] & N \\ & & \bigoplus _{\mathfrak a} \bigoplus _{\psi \in \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak a, N)} R } \]

which induces the desired map $\mathbf{M}(M) \to \mathbf{M}(N)$. The middle east-south-east arrow maps the summand $\mathfrak a$ corresponding to $\varphi $ via $\text{id}_{\mathfrak a}$ to the summand $\mathfrak a$ corresponding to $\psi = \chi \circ \varphi $. Similarly for the lower east-south-east arrow. Details omitted. $\square$

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