The Homotopy Category

We always assume that A comes equipped with a specific choice of the zero-object, of a kernel and cokernel for each map, and of a direct sum for any two objects. Nevertheless we will often abuse notation by allowing the symbol 0 to stand for any initial object in A thus for T G. 4, T = 0 means only that A is isomorphic to the zero-object. 

For a complex C as above, since d od = 0 therefore induces a natural map 

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Exercise 2.3.9. Show that if is a Grothendieck category then D(X) is equivalent to the homotopy category of q-injective complexes. Hence if A has inverse limits then so does D(yl). 

The homotopy category 3 (T,l) of a site with interval (1,1) is the localization ofA°f Shv(T) with respect to the class ofl-weak equivalences. 

More generally, any ringed site (T, < ) defines a site with interval. In particular we may consider the homotopy category associated with any subcategory in the category of schemes (over a base) which contains affine line. 

Note that the realization functor — Iaj A SfwJT) A Shv(T) takes values in the fuU subcategory of simplicial sheaves of simplicial dimension zero, i.e. factors through a functor — a A ShifT) Sfw(Tj which is left adjoint to the restriction of C, to Shv(T). Together with Lemma 3.12 this fact can be used to obtain an alternative description of the homotopy category, 9(0 (T, I) as follows. 

In this section we study the basic properties of A -homotopy category of smooth schemes over a base. Modulo the conventions of the previous section the definition of the A -homotopy category, % (S) of smooth schemes over a base scheme S takes one line - (S) is the homotopy category of the site with interval ((5m/S)%, A ), where... 

As was shown in Section 3 the classes of A -weak equivalences, monomorphisms and A -fibrations form a proper simplicial model structure on the category of simplidal sheaves on (5m/S)jvu. The corresponding homotopy category, i.e. the localization of the category of simplicial sheaves on Sm/ xu with respect to the class of A -weak equivalences is called the homotopy category of smooth schemes over S. We denote this category by (S). 

The category A need only be additive for us to define the homotopy invariant of a... 

In Section 3 we apply this localization theorem to define a model category structure on the category of simplicial sheaves on a site with interval (see [31, 2.2]). We show that this model category structure is always proper (in the sense of [2, Definition 1.2]) and give examples of how some known homotopy categories can be obtained using this construction. 

The following simple result describes the basic functoriality of the simplicial homotopy categories for morphisms of sites. 

Proposition 1.47. — iMf Ti —T2 be a morphism of sites. Then the Junctor f preserves weak equivalences and the corresponding Jiinctor between homotopy categories is lejt adjoint to If Ti T2- Ts is a composable pair of morphisms of sites dm the canonical morphism offunctors... 

Let us say that a morphism of pointed simplicial sheaves is a fibration, cofibration or weak equivalence (simplicial) if it belongs to the corresponding class as a morphism of sheaves without base points. This definition clearly provides us with a model category structures which we will call the simplicial model category structures on A Shvjf Sm/S)f We denote the corresponding homotopy categories by... 

Moreover there is a morphism (of simplicial sheaves of groups) G(B(G)) G which is a weak equivalence and the induced morphism (in the pointed homotopy category) G Ri2](Di< 0(B(G)))) is the previous one, as required. 

Let T be the category Sm/S of smooth schemes over a base S considered with the Nisnevich topology (see Definition 1.2) and I be the sheaf represented by the affine line over S. The corresponding homotopy category 3 ... 

Proof — Let f 36- be a strict I-homotopy equivalence and be a I-homotopy inverse to f We have to show that the compositions fog and gof are equal to the corresponding identity morphisms in the I-homotopy category. By definition these compositions are I-homotopic to identity and it remains to show that two elementary I-homotopic morphisms coincide in the I-homotopy category which follows immediately from definitions. 

We consider the functoriality of homotopy categories of sites with intervals only in the case of reasonable continuous maps of sites cf 1.55). We have the following obvious lemma. 

Same argument as in the proof of Proposition 1.20 implies that the continuous map of sites (/) Sm/S2)Ms Sm/ )jfis associated to a smooth morphism of schemes / Si —> S2 is reasonable cf 1.55). Therefore, the functor of inverse image / = (<(>(/)). between the corresponding homotopy categories of simplicial sheaves has a left adjoint which we denote by Yf. Note that the continuous map f) is not a morphism of sites unless/is an isomorphism. The following example shows that the functor f does not have to preserve weak equivalence. 

Let Si 2 be a morphism of base schemes. For any smooth scheme U over S2 we have/ (U x A )=/ (U) x A. Therefore the functor If preserves A -weak equivalences and induces a functor on ALhomotopy categories which we again denote I/. We also know that the functor R/i, preserves A -local objects and we denote the induced functor on A -homotopy categories by f . Proposition 3.17 gives us the following result. 

Proposition 2.8. — For any morphism y Si S2 /Af fimetor R y is right adjoint to L/. For any composable pair f, g of morphisms of base schemes there is a canonical isomorphism of Junctors bewteen A -homotopy categories of the form... 

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