Processing math: 100%

The Stacks project

Lemma 13.24.1. Let \mathcal{A} be an abelian category. Assume \mathcal{A} has functorial injective embeddings, see Homology, Definition 12.27.5.

  1. There exists a functor inj : \text{Comp}^{+}(\mathcal{A}) \to \text{InjRes}(\mathcal{A}) such that s \circ inj = \text{id}.

  2. For any functor inj : \text{Comp}^{+}(\mathcal{A}) \to \text{InjRes}(\mathcal{A}) such that s \circ inj = \text{id} we obtain a resolution functor, see Definition 13.23.2.

Proof. Let A \mapsto (A \to J(A)) be a functorial injective embedding, see Homology, Definition 12.27.5. We first note that we may assume J(0) = 0. Namely, if not then for any object A we have 0 \to A \to 0 which gives a direct sum decomposition J(A) = J(0) \oplus \mathop{\mathrm{Ker}}(J(A) \to J(0)). Note that the functorial morphism A \to J(A) has to map into the second summand. Hence we can replace our functor by J'(A) = \mathop{\mathrm{Ker}}(J(A) \to J(0)) if needed.

Let K^\bullet be a bounded below complex of \mathcal{A}. Say K^ p = 0 if p < B. We are going to construct a double complex I^{\bullet , \bullet } of injectives, together with a map \alpha : K^\bullet \to I^{\bullet , 0} such that \alpha induces a quasi-isomorphism of K^\bullet with the associated total complex of I^{\bullet , \bullet }. First we set I^{p, q} = 0 whenever q < 0. Next, we set I^{p, 0} = J(K^ p) and \alpha ^ p : K^ p \to I^{p, 0} the functorial embedding. Since J is a functor we see that I^{\bullet , 0} is a complex and that \alpha is a morphism of complexes. Each \alpha ^ p is injective. And I^{p, 0} = 0 for p < B because J(0) = 0. Next, we set I^{p, 1} = J(\mathop{\mathrm{Coker}}(K^ p \to I^{p, 0})). Again by functoriality we see that I^{\bullet , 1} is a complex. And again we get that I^{p, 1} = 0 for p < B. It is also clear that K^ p maps isomorphically onto \mathop{\mathrm{Ker}}(I^{p, 0} \to I^{p, 1}). As our third step we take I^{p, 2} = J(\mathop{\mathrm{Coker}}(I^{p, 0} \to I^{p, 1})). And so on and so forth.

At this point we can apply Homology, Lemma 12.25.4 to get that the map

\alpha : K^\bullet \longrightarrow \text{Tot}(I^{\bullet , \bullet })

is a quasi-isomorphism. To prove we get a functor inj it rests to show that the construction above is functorial. This verification is omitted.

Suppose we have a functor inj such that s \circ inj = \text{id}. For every object K^\bullet of \text{Comp}^{+}(\mathcal{A}) we can write

inj(K^\bullet ) = (i_{K^\bullet } : K^\bullet \to j(K^\bullet ))

This provides us with a resolution functor as in Definition 13.23.2. \square


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.