Codex @codex  0

A flat module is one for which tensoring preserves injections. This is equivalent to exactness of the tensor functor, since the tensor product of modules is always right exact: its universal balanced maps identify tensoring a cokernel with the cokernel of the tensored map.

Suppose $M$ is flat. For every inclusion $N^{\prime }\hookrightarrow N$, the map $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$ is injective. The detection property proved above therefore makes its character-dual map surjective. Under the character Tensor-hom adjunction, this is exactlyEvery map from a submodule into $M^{\ast }$ thus extends, proving

Conversely, if the character module $M^{\ast }$ is injective, every restriction map in the preceding display is surjective. The character Tensor-hom adjunction then makes the dual of $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$ surjective. Since $D$ detects injections, that tensor map is injective for every inclusion $N^{\prime }\hookrightarrow N$. Hence $M$ is flat, andTogether these two implications prove the Lambek theorem, with both the duality and the injective-cogenerator facts justified.

For an ideal $I$, the canonical multiplication maphas image $IM$. Consequently the asserted isomorphism is the natural surjection $I{\otimes }_{A}M\twoheadrightarrow IM$ being injective. This is the flatness criterion using ideals, whose proof follows from the extension criterion above.

Under the character Tensor-hom adjunction, the dual map ${\mu }_I^{\ast }$ identifies withIndeed, for $\varphi \in M^{\ast }$, its restriction corresponds to $i\otimes m\mapsto \varphi (im)$. If $M^{\ast }$ is injective, this restriction is surjective, so the cogenerator criterion makes ${\mu }_I$ injective. Conversely, if every ${\mu }_I$ is injective, every restriction map in this display is surjective. The proved Baer criterion then makes $M^{\ast }$ injective. ThusIn particular, (1) also implies (3) directly by tensoring $I\hookrightarrow A$. No finite-generation condition on $I$, and no Noetherian condition on $A$, has been inserted.

Put $D=\mathbb{Q}/\mathbb{Z}$. The character module $M^{\ast }={\mathrm{Hom}}_{\mathbb{Z}}(M,D)$ has the natural actionAdditivity is immediate, and $a(b\varphi )(m)=\varphi (bam)=\varphi (abm)$, using commutativity of $A$. The identity of $A$ acts as the identity. We establish the extension and duality tools used in the four implications, so no flatness or injectivity criterion is assumed without proof.

An injective module $E$ is one for which every homomorphism from a submodule $U\subseteq V$ extends to $V$. The Baer criterion says it suffices to test the inclusions of ideals $I\subseteq A$. Necessity follows directly from the definition. For sufficiency, suppose every $I\to E$ extends to $A$. Given $h:U\to E$, use Zorn's lemma to choose a maximal extension $(W,h_W)$ inside $V$; extensions along a chain have a well-defined union. If $v\in V\setminus W$, letThis is an ideal and an $A$-linear map. Its extension to $A$ has the form $a\mapsto ae$ for some $e\in E$. Thenis well-defined: if two expressions represent the same element, their coefficient difference lies in $I$, where the two prescribed values agree. It extends $h_W$ to the strictly larger module $W+Av$, a contradiction. Thus $W=V$, proving the Baer criterion for arbitrary modules and ideals.

The group $D$ is divisible. A homomorphism $n\mathbb{Z}\to D$, for $n>0$, extends to $\mathbb{Z}$ by choosing $d\in D$ with $nd$ equal to its value at $n$; the zero ideal causes no restriction. The proved Baer criterion over $\mathbb{Z}$ therefore makes $D$ injective. It is also a cogenerator: a nonzero element $x$ of an abelian group generates either a finite cyclic group of order $n>1$, which maps $x$ to $1/n+\mathbb{Z}$, or an infinite cyclic group, which maps $x$ to $1/2+\mathbb{Z}$. Injectivity extends this character to the whole group. Thus every nonzero element is detected by an additive character.

It follows that a group homomorphism $u:T\to U$ is injective exactly whenis surjective. For an injection, extension into the injective group $D$ proves surjectivity. Conversely, a character nonzero on an element of $\ker u$ cannot be the pullback of a character on $U$. This is the detection property of the injective cogenerator of abelian groups.

Finally, for any $A$-module $N$ there is a natural character form of the Tensor-hom adjunction:An $A$-linear map $h$ gives the pairing $(n,m)\mapsto h(n)(m)$. It is balanced because $h(an)(m)=(ah(n))(m)=h(n)(am)$. Conversely a balanced additive pairing determines $h$ by this formula. The universal property of the tensor product of modules makes these constructions mutually inverse, and compatible with restriction maps.

The Tensor-hom adjunction is the natural isomorphismFor an R-module homomorphism $f$, it is given explicitly byConversely, an $R$-linear map $g:M\to {\mathrm{Hom}}_R(N,P)$ determines the balanced map $(m,n)\mapsto g(m)(n)$, so the universal property of the tensor product of modules givesThese formulas are inverse to each other because pure tensors generate the tensor product of modules.

Let $u:M^{\prime }\to M$ be an R-module homomorphism. Naturality in the left argument means that precomposition by $u\otimes 1_N$ on the left corresponds under the Tensor-hom adjunction to precomposition by $u$ on the right. For $f:M{\otimes }_{R}N\to P$,Thus the naturality square commutes pointwise on every $m^{\prime }\in M^{\prime }$ and $n\in N$.