Resolutions of unbounded chain complexes Jérôme Scherer (joint work with Wojciech Chachólski and Wolfgang Pitsch)

Here is the setup of the problem. Let A denote a bicomplete abelian category and Ch(A) the category of unbounded chain complexes in A. We fix a class W of objects, which we wish to consider as injectives in A (the dual case where one works with projective objects can be handled in the same way). A morphism f : X_{•} → Y_{•} is a W -equivalence if, for any object W ∈ W, the morphism f^{∗} : A(Y_{•}, W) → A(X_{•}, W) is a quasi-isomorphism in Ch(Z).

In [2], Christensen and Hovey work in the projective case. In order to do “relative homological algebra”, in particular to construct projective resolutions in this setting, they analyze when Ch(A) can be given a Quillen model structure, [3], where of course the weak equivalences should be the W-equivalences. They succeed to do so in many cases, for example when A is the category of bimodules over a k-algebra A and W is the class of free A-bimodules. In this case they show that the class of homotopy classes [Σ^{n}A, N] ≅ HH^{n}(A; N) for any (chain complex of) A-bimodule N.

Our point of view is that for most applications one does not really need to know Ch(A) is a model category. We will instead provide a “model approximation”, roughly speaking a model category which is easier to deal with, and in which Ch(A) can be embedded. In this way important characteristics of model categories still hold:

(1) One can form the homotopy category Ho(Ch(A)) and deduce that homotopy classes [X_{•}, Y_{•}] form a set.

(2) One can construct injective (fibrant) resolutions and thus construct right

derived functors in the relative setting. Definition. Let C be a category with a distinguished class of weak equivalences. A model category M together with a pair of adjoint functors

l : C ⇄ M : r

forms a right model approximation if (1) the left adjoint l sends weak equivalences to weak equivalences,

(2) the right adjoint r sends weak equivalences between fibrant objects to weak equivalences,

(3) for a fibrant object X ∈ M, if a morphism lA → X is a weak equivalence, then so is its adjoint A → rX.

The algorithm to build a (fake) fibrant replacement for an object A ∈ C is then as follows. Push A into the model category M with the functor l. Using the model category structure in M construct a fibrant replacement lA → R(lA). Take the adjoint A → r(R(lA)). This is not really a fibrant replacement in C, not only because there is no model structure on C, but more seriously because the lifting property with respect to acyclic cofibrations might fail, for some obvious choice of cofibrations.

Example. When D is a model category and I is a small category, it is not known whether the category C = F un(I, D) of D-valued diagrams indexed by I form a model category. There is however an obvious choice of weak equivalences, i.e. the natural transformations of diagrams F → G such that F (i) → G(i) is a weak equivalence in D for all i ∈ I. We proved in [1] that there always exist a model approximation. Consider the simplex category of the nerve N(I), where the morphisms are generated by the face and degeneracy maps and define M to be the full subcategory F un^{b}(N(I)^{op}, D) of functors indexed by N(I)^{op} which are bounded, in the sense that degeneracies induce isomorphisms.

The projection functor ε : N(I)^{op} → I yields a pair of adjoint functor

ε : F un(I, D) ⇄ F un^{b}(N(I)^{op}, D) : ε^{k} ,

where ε^{k} denotes the right Kan extension. This forms a model approximation. In particular the homotopy limit of any functor F : I → D can be computed by taking the limit of the (fake) fibrant replacement:

holim_{I}F = lim_{I}ε^{k}(R(ϵF)).

Let us now go back to chain complexes. To do relative homological algebra, we will exhibit a model approximation for Ch(A). The only standing assumption is that there are “enough injective objects”. More precisely we will assume that any object A ∈ A admits a W-monomorphism A → W to some object of the class W. Whereas unbounded chain complexes are difficult to handle, bounded ones are tame. Let us denote by Ch(A)_{≤0} the category of bounded above chain complexes (differentials have degree −1 and the chain complexes are zero in positive degrees).

Proposition. The category Ch(A)_{≤0} admits a model category structure where the weak equivalences are the W-equivalences.

The model approximation for Ch(A) will be constructed out of bounded chain complexes by taking towers. The objects of the category Tow(Ch(A)_{≤0}) of towers are sequences X_{•}^{0}, X_{•}^{1}, X_{•}^{2}, … of chain complexes in Ch(A)_{≤0} together with structure maps ΣX_{•}^{i+1} → X_{•}^{i} for all i > 0, which should remind the reader of spectra. Weak equivalences are levelwise W-equivalences.

Proposition. The category Tow(Ch(A)_{≤0}) is a model category.

The structure maps provide towers in A of the form

X_{n}^{m} ← X_{n−1}^{m+1} ← X_{n−2}^{m+2} ← …

for all n ≤ 0 and m ≥ 0. Taking the limit yields thus a functor

lim : Tow(Ch(A)_{≤0}) → Ch(A)

to unbounded chain complex, which is right adjoint to the truncation functor. We are now ready to state our main result.

Theorem. The pair of adjoint functors Trunc : Ch(A) ⇄ Tow(Ch(A)_{≤0}) : lim forms a model approximation.

To conclude, let us remark that when A is the category of R-modules and the class W consists of all injective R-modules, then the fibrant resolution we construct coincides with the “right injective resolutions” Spaltenstein constructs in [4]. In his work there is no mention of the word model category.

References

[1] Chachólski, W. and Scherer, J., Homotopy Theory of Diagrams, Mem. Amer. Math. Soc. 155 no. 736 (2002), ix+90 pp.

[2] Christensen, D. and Hovey, M., Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002), 261–293.

[3] Quillen, D. G., Homotopical algebra, Lecture Notes in Mathematics 43 (1967), iv+156 pp.

[4] Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), 121– 154.