19.11 Injectives in Grothendieck categories
The existence of a generator implies that given an object $M$ of a Grothendieck abelian category $\mathcal{A}$ there is a set of subobjects. (This may not be true for a general “big” abelian category.)
Lemma 19.11.1. Let $\mathcal{A}$ be an abelian category with a generator $U$ and $X$ and object of $\mathcal{A}$. If $\kappa $ is the cardinality of $\mathop{\mathrm{Mor}}\nolimits (U, X)$ then
There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of $X$ indexed by a cardinal bigger than $\kappa $.
If $\alpha $ is an ordinal of cofinality $> \kappa $ then any increasing (or decreasing) sequence of subobjects of $X$ indexed by $\alpha $ is eventually constant.
The cardinality of the set of subobjects of $X$ is $\leq 2^\kappa $.
Proof.
For (1) assume $\kappa ' > \kappa $ is a cardinal and assume $X_ i$, $i \in \kappa '$ is strictly increasing. Then take for each $i$ a $\phi _ i \in \mathop{\mathrm{Mor}}\nolimits (U, X)$ such that $\phi _ i$ factors through $X_{i + 1}$ but not through $X_ i$. Then the morphisms $\phi _ i$ are distinct, which contradicts the definition of $\kappa $.
Part (2) follows from the definition of cofinality and (1).
Proof of (3). For any subobject $Y \subset X$ define $S_ Y \in \mathcal{P}(\mathop{\mathrm{Mor}}\nolimits (U, X))$ (power set) as $S_ Y = \{ \phi \in \mathop{\mathrm{Mor}}\nolimits (U,X) : \phi )\text{ factors through }Y\} $. Then $Y = Y'$ if and only if $S_ Y = S_{Y'}$. Hence the cardinality of the set of subobjects is at most the cardinality of this power set.
$\square$
By Lemma 19.11.1 the following definition makes sense.
Definition 19.11.2. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\mathcal{A}$. The size $|M|$ of $M$ is the cardinality of the set of subobjects of $M$.
Lemma 19.11.3. Let $\mathcal{A}$ be a Grothendieck abelian category. If $0 \to M' \to M \to M'' \to 0$ is a short exact sequence of $\mathcal{A}$, then $|M'|, |M''| \leq |M|$.
Proof.
Immediate from the definitions.
$\square$
Lemma 19.11.4. Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$.
If $|M| \leq \kappa $, then $M$ is the quotient of a direct sum of at most $\kappa $ copies of $U$.
For every cardinal $\kappa $ there exists a set of isomorphism classes of objects $M$ with $|M| \leq \kappa $.
Proof.
For (1) choose for every proper subobject $M' \subset M$ a morphism $\varphi _{M'} : U \to M$ whose image is not contained in $M'$. Then $\bigoplus _{M' \subset M} \varphi _{M'} : \bigoplus _{M' \subset M} U \to M$ is surjective. It is clear that (1) implies (2).
$\square$
Proposition 19.11.5. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an object of $\mathcal{A}$. Let $\kappa = |M|$. If $\alpha $ is an ordinal whose cofinality is bigger than $\kappa $, then $M$ is $\alpha $-small with respect to injections.
Proof.
Please compare with Proposition 19.2.5. We need only show that the map (19.2.0.1) is a surjection. Let $f : M \to \mathop{\mathrm{colim}}\nolimits B_\beta $ be a map. Consider the subobjects $\{ f^{-1}(B_\beta )\} $ of $M$, where $B_\beta $ is considered as a subobject of the colimit $B = \bigcup _\beta B_\beta $. If one of these, say $f^{-1}(B_\beta )$, fills $M$, then the map factors through $B_\beta $.
So suppose to the contrary that all of the $f^{-1}(B_\beta )$ were proper subobjects of $M$. However, because $\mathcal{A}$ has AB5 we have
\[ \mathop{\mathrm{colim}}\nolimits f^{-1}(B_\beta ) = f^{-1}\left(\mathop{\mathrm{colim}}\nolimits B_\beta \right) = M. \]
Now there are at most $\kappa $ different subobjects of $M$ that occur among the $f^{-1}(B_\alpha )$, by hypothesis. Thus we can find a subset $S \subset \alpha $ of cardinality at most $\kappa $ such that as $\beta '$ ranges over $S$, the $f^{-1}(B_{\beta '})$ range over all the $f^{-1}(B_\alpha )$.
However, $S$ has an upper bound $\widetilde{\alpha } < \alpha $ as $\alpha $ has cofinality bigger than $\kappa $. In particular, all the $f^{-1}(B_{\beta '})$, $\beta ' \in S$ are contained in $f^{-1}(B_{\widetilde{\alpha }})$. It follows that $f^{-1}(B_{\widetilde{\alpha }}) = M$. In particular, the map $f$ factors through $B_{\widetilde{\alpha }}$.
$\square$
slogan
Lemma 19.11.6. Let $\mathcal{A}$ be a Grothendieck abelian category with generator $U$. An object $I$ of $\mathcal{A}$ is injective if and only if in every commutative diagram
\[ \xymatrix{ M \ar[d] \ar[r] & I \\ U \ar@{-->}[ru] } \]
for $M \subset U$ a subobject, the dotted arrow exists.
Proof.
Please see Lemma 19.2.6 for the case of modules. Choose an injection $A \subset B$ and a morphism $\varphi : A \to I$. Consider the set $S$ of pairs $(A', \varphi ')$ consisting of subobjects $A \subset A' \subset B$ and a morphism $\varphi ' : A' \to I$ extending $\varphi $. Define a partial ordering on this set in the obvious manner. Choose a totally ordered subset $T \subset S$. Then
\[ A' = \mathop{\mathrm{colim}}\nolimits _{t \in T} A_ t \xrightarrow {\mathop{\mathrm{colim}}\nolimits _{t \in T} \varphi _ t} I \]
is an upper bound. Hence by Zorn's lemma the set $S$ has a maximal element $(A', \varphi ')$. We claim that $A' = B$. If not, then choose a morphism $\psi : U \to B$ which does not factor through $A'$. Set $N = A' \cap \psi (U)$. Set $M = \psi ^{-1}(N)$. Then the map
\[ M \to N \to A' \xrightarrow {\varphi '} I \]
can be extended to a morphism $\chi : U \to I$. Since $\chi |_{\mathop{\mathrm{Ker}}(\psi )} = 0$ we see that $\chi $ factors as
\[ U \to \mathop{\mathrm{Im}}(\psi ) \xrightarrow {\varphi ''} I \]
Since $\varphi '$ and $\varphi ''$ agree on $N = A' \cap \mathop{\mathrm{Im}}(\psi )$ we see that combined the define a morphism $A' + \mathop{\mathrm{Im}}(\psi ) \to I$ contradicting the assumed maximality of $A'$.
$\square$
Theorem 19.11.7. Let $\mathcal{A}$ be a Grothendieck abelian category. Then $\mathcal{A}$ has functorial injective embeddings.
Proof.
Please compare with the proof of Theorem 19.2.8. Choose a generator $U$ of $\mathcal{A}$. For an object $M$ we define $\mathbf{M}(M)$ by the following pushout diagram
\[ \xymatrix{ \bigoplus _{N \subset U} \bigoplus _{\varphi \in \mathop{\mathrm{Hom}}\nolimits (N, M)} N \ar[r] \ar[d] & M \ar[d] \\ \bigoplus _{N \subset U} \bigoplus _{\varphi \in \mathop{\mathrm{Hom}}\nolimits (N, M)} U \ar[r] & \mathbf{M}(M). } \]
Note that $M \to \mathbf{M}(N)$ is a functor and that there exist functorial injective maps $M \to \mathbf{M}(M)$. By transfinite induction we define functors $\mathbf{M}_\alpha (M)$ for every ordinal $\alpha $. Namely, set $\mathbf{M}_0(M) = M$. Given $\mathbf{M}_\alpha (M)$ set $\mathbf{M}_{\alpha + 1}(M) = \mathbf{M}(\mathbf{M}_\alpha (M))$. For a limit ordinal $\beta $ set
\[ \mathbf{M}_\beta (M) = \mathop{\mathrm{colim}}\nolimits _{\alpha < \beta } \mathbf{M}_\alpha (M). \]
Finally, pick any ordinal $\alpha $ whose cofinality is greater than $|U|$. Such an ordinal exists by Sets, Proposition 3.7.2. We claim that $M \to \mathbf{M}_\alpha (M)$ is the desired functorial injective embedding. Namely, if $N \subset U$ is a subobject and $\varphi : N \to \mathbf{M}_\alpha (M)$ is a morphism, then we see that $\varphi $ factors through $\mathbf{M}_{\alpha '}(M)$ for some $\alpha ' < \alpha $ by Proposition 19.11.5. By construction of $\mathbf{M}(-)$ we see that $\varphi $ extends to a morphism from $U$ into $\mathbf{M}_{\alpha ' + 1}(M)$ and hence into $\mathbf{M}_\alpha (M)$. By Lemma 19.11.6 we conclude that $\mathbf{M}_\alpha (M)$ is injective.
$\square$
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