Remark 35.4.4. Any functor F : \mathcal{A} \to \mathcal{B} of abelian categories which is exact and takes nonzero objects to nonzero objects reflects injections and surjections. Namely, exactness implies that F preserves kernels and cokernels (compare with Homology, Section 12.7). For example, if f : R \to S is a faithfully flat ring homomorphism, then \bullet \otimes _ R S: \text{Mod}_ R \to \text{Mod}_ S has these properties.
35.4.3 Universally injective morphisms
Recall that \textit{Rings} denotes the category of commutative rings with 1. For an object R of \textit{Rings} we denote \text{Mod}_ R the category of R-modules.
Let R be a ring. Recall that a morphism f : M \to N in \text{Mod}_ R is universally injective if for all P \in \text{Mod}_ R, the morphism f \otimes 1_ P: M \otimes _ R P \to N \otimes _ R P is injective. See Algebra, Definition 10.82.1.
Definition 35.4.5. A ring map f: R \to S is universally injective if it is universally injective as a morphism in \text{Mod}_ R.
Example 35.4.6. Any split injection in \text{Mod}_ R is universally injective. In particular, any split injection in \textit{Rings} is universally injective.
Example 35.4.7. For a ring R and f_1, \ldots , f_ n \in R generating the unit ideal, the morphism R \to R_{f_1} \oplus \ldots \oplus R_{f_ n} is universally injective. Although this is immediate from Lemma 35.4.8, it is instructive to check it directly: we immediately reduce to the case where R is local, in which case some f_ i must be a unit and so the map R \to R_{f_ i} is an isomorphism.
Lemma 35.4.8. Any faithfully flat ring map is universally injective.
Proof. This is a reformulation of Algebra, Lemma 10.82.11. \square
The key observation from [mesablishvili1] is that universal injectivity can be usefully reformulated in terms of a splitting, using the usual construction of an injective cogenerator in \text{Mod}_ R.
Definition 35.4.9. Let R be a ring. Define the contravariant functor C : \text{Mod}_ R \to \text{Mod}_ R by setting
C(M) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),
with the R-action on C(M) given by rf(s) = f(rs).
This functor was denoted M \mapsto M^\vee in More on Algebra, Section 15.55.
Lemma 35.4.10. For a ring R, the functor C : \text{Mod}_ R \to \text{Mod}_ R is exact and reflects injections and surjections.
Proof. Exactness is More on Algebra, Lemma 15.55.6 and the other properties follow from this, see Remark 35.4.4. \square
Remark 35.4.11. We will use frequently the standard adjunction between \mathop{\mathrm{Hom}}\nolimits and tensor product, in the form of the natural isomorphism of contravariant functors
35.4.11.1
\begin{equation} \label{descent-equation-adjunction} C(\bullet _1 \otimes _ R \bullet _2) \cong \mathop{\mathrm{Hom}}\nolimits _ R(\bullet _1, C(\bullet _2)): \text{Mod}_ R \times \text{Mod}_ R \to \text{Mod}_ R \end{equation}
taking f: M_1 \otimes _ R M_2 \to \mathbf{Q}/\mathbf{Z} to the map m_1 \mapsto (m_2 \mapsto f(m_1 \otimes m_2)). See Algebra, Lemma 10.14.5. A corollary of this observation is that if
\xymatrix@C=9pc{ C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P) }
is a split coequalizer diagram in \text{Mod}_ R, then so is
\xymatrix@C=9pc{ C(M \otimes _ R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes _ R Q) \ar[r] & C(P \otimes _ R Q) }
for any Q \in \text{Mod}_ R.
Lemma 35.4.12. Let R be a ring. A morphism f: M \to N in \text{Mod}_ R is universally injective if and only if C(f): C(N) \to C(M) is a split surjection.
Proof. By (35.4.11.1), for any P \in \text{Mod}_ R we have a commutative diagram
\xymatrix@C=9pc{ \mathop{\mathrm{Hom}}\nolimits _ R( P, C(N)) \ar[r]_{\mathop{\mathrm{Hom}}\nolimits _ R(P,C(f))} \ar[d]^{\cong } & \mathop{\mathrm{Hom}}\nolimits _ R(P,C(M)) \ar[d]^{\cong } \\ C(P \otimes _ R N ) \ar[r]^{C(1_{P} \otimes f)} & C(P \otimes _ R M ). }
If f is universally injective, then 1_{C(M)} \otimes f: C(M) \otimes _ R M \to C(M) \otimes _ R N is injective, so both rows in the above diagram are surjective for P = C(M). We may thus lift 1_{C(M)} \in \mathop{\mathrm{Hom}}\nolimits _ R(C(M), C(M)) to some g \in \mathop{\mathrm{Hom}}\nolimits _ R(C(N), C(M)) splitting C(f). Conversely, if C(f) is a split surjection, then both rows in the above diagram are surjective, so by Lemma 35.4.10, 1_{P} \otimes f is injective. \square
Remark 35.4.13. Let f: M \to N be a universally injective morphism in \text{Mod}_ R. By choosing a splitting g of C(f), we may construct a functorial splitting of C(1_ P \otimes f) for each P \in \text{Mod}_ R. Namely, by (35.4.11.1) this amounts to splitting \mathop{\mathrm{Hom}}\nolimits _ R(P, C(f)) functorially in P, and this is achieved by the map g \circ \bullet .
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