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homotopically injective object

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Definition

Let π’œ\mathcal{A} be an abelian category with translation.

An object in the category of chain complexes modulo chain homotopy, K(π’œ)K(\mathcal{A}), is homotopically injective if for every X∈K(π’œ)X \in K(\mathcal{A}) that is quasi-isomorphic to 00 we have

Hom K(π’œ)(X,I)≃0. Hom_{K(\mathcal{A})}(X,I) \simeq 0 \,.

Let

QuasiIsoMono={f∈Mor(A c)∣fmonoandquasiio} QuasiIsoMono = \{f \in Mor(A_c) | f mono and quasiio\}

be the set of morphisms in the category of chain complexes Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) which are both quasi-isomorphisms as well as monomorphisms.

Then

A complex II is an injective object with respect to monomorphic quasi-isomorphisms precisely if

  • it is homotopically injective in the sense of complexes in π’œ\mathcal{A};

  • it is injective as an object of π’œ\mathcal{A} (with respect to morphisms f:Xβ†’Yf : X \to Y such that 0β†’Xβ†’fY0 \to X \stackrel{f}{\to} Y is exact).

Properties

In complexes in a Grothendieck category

Proposition For π’œ\mathcal{A} a Grothendieck category with translation T:π’œβ†’π’œT : \mathcal{A} \to \mathcal{A}, every complex XX in Ch β€’(π’œ)Ch_\bullet(\mathcal{A}) is quasi-isomorphic to a complex II which is injective and homotopically injective (i.e. QuasiIsoMono-injective).

Relation to derived categories

For π’œ\mathcal{A} an abelian Grothendieck category with translation the full subcategory K hi(π’œ)βŠ‚K(π’œ)K_{hi}(\mathcal{A}) \subset K(\mathcal{A}) of homotopically injective complexes realizes the derived category D(π’œ)D(\mathcal{A}) of π’œ\mathcal{A}:

Q∣ K hi(A):K hi(A)→≃D(A), Q|_{K_{hi}(A)} : K_{hi}(A) \stackrel{\simeq}{\to} D(A) \,,

where Q:K(A)β†’D(A)Q : K(A) \to D(A) and Q∣ K hi(A)Q|_{K_{hi}(A)} has a right adjoint.

It follows that for DD any other triangulated category, every triangulated functor F:K(π’œ)β†’DF : K(\mathcal{A}) \to D has a right derived functor RF:D(π’œ)β†’DR F : D(\mathcal{A}) \to D which is computed by evaluating FF on injective replacements: for R:D(π’œ)→≃K hi(π’œ)R : D(\mathcal{A}) \stackrel{\simeq}{\to} K_{hi}(\mathcal{A}) a weak inverse to QQ, we have

RF≃D(A)β†’RK hi(A)β†ͺK(A)β†’FD. R F \simeq D(A) \stackrel{R}{\to} K_{hi}(A) \hookrightarrow K(A) \stackrel{F}{\to} D \,.

References

Much of this discussion can be found in

The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.

Revised on September 23, 2012 15:43:21 by Urs Schreiber (89.204.137.161)