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
for M \subset U a subobject, the dotted arrow exists.
To check that an object is injective, one only needs to check that lifting holds for subobjects of a generator.
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
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
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
can be extended to a morphism \chi : U \to I. Since \chi |_{\mathop{\mathrm{Ker}}(\psi )} = 0 we see that \chi factors as
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
Comments (1)
Comment #1091 by Alex Youcis on
There are also: