                                      Source file: src/hypercovering.tex

HYPERCOVERINGS

Contents

Section 1. Introduction 1 Section 2. Definitions 1 Subsection 2.1. Goals 2 Subsection 2.2. Making simplicial objects 2 Subsection 2.3. Doubly simplicial stuff 3 Section 3. The general case 3 References 3

Section 1. Introduction

Hypercoverings can be used to compute cohomology of abelian sheaves on sites without recourse to injective resolutions. See [MA71, Expose V, Sec. 7]. A nice manuscript on cohomological descent is the text by Brian Conrad, see http://www.

math.lsa.umich.edu/^{∼}bdconrad/papers/hypercover.pdf. Probably it is useless

to try to improve on Brian’s article, so we look at the question a little differently (more naively).

Section 2. Definitions

Let C be a category. Let Δ be the category of finite ordered sets with objects [0] = {0}, [1] = {0, 1}, [2] = {0, 1, 2}, … and order preserving maps. A simplicial object U_{•} of C is a contravariant functor U_{•} : Δ → C. This means there are objects U_{0}, U_{1}, U_{2}, … and morphisms U_{•}(φ) : U_{n} → U_{m}, where φ is any order preserving map φ : [m] → [n]. In particular there is a unique morphism U_{0} → U_{n} and there are exactly n + 1 morphisms U_{n} → U_{0} corresponding to the n+1 maps [0] → [n]. Obviously we need some more notation to be able to talk intelligently about these simplicial objects.

Definition 2.0.1. For any integer n ≥ 1, and any 0 ≤ j ≤ n we let d_{j}^{n} : [n−1] → [n] denote the injective order preserving map skipping j. For any integer n ≥ 0, and any 0 ≤ j ≤ n we denote s_{j}^{n} : [n + 1] → [n] the surjective order preserving map with (s_{j}^{n})^{−1}({j}) = {j, j + 1}.

We get a unique morphism U_{•}(s_{0}^{0}) : U_{0} → U_{1} and two morphisms U_{•}(d_{0}^{1}), U_{•}(d_{1}^{1}) : U_{1} → U_{0}. There are two morphisms U_{•}(s_{0}^{1}), U_{•}(s_{1}^{1}) : U_{1} → U_{2} and three morphisms U_{•}(d_{0}^{2}), U_{•}(d_{1}^{2}), U_{•}(d_{2}^{2}) : U_{3} → U_{2}. And so on. FIXME: This notation... FIXME: Much more.

Example 2.0.2. (1) The simplest example is the constant simplicial object with value X ∈ Ob(C). In other words, U_{n} = X and all maps are id_{X}.

(2) Suppose that Y → X is a morphism of C such that all the fibred products Y_{/X}^{n}Y = ×_{X}Y ×_{X} … Y exist. Then we set U_{n} = Y_{/X}^{n+1}, and we let s : [n] → [m] correspond to the map (on “coordinates”) (y_{0}, …, y_{m}) ↦ (y_{s(0)}, …, y_{s(n)}).

Subsection 2.1. Goals. Assume that C is a site with the property that the set of coverings consisting of 1 morphism is cofinal. Let F be a sheaf of abelian groups on the site C which is assumed to have the property that the set of coverings consisting of 1 morphism is cofinal. Choose an injective resolution F → J^{•} (for example a canonical one, see Injectives, Subsection 2.3). Let X be an object of C. We want to compute RΓ(X, F) = Γ(X, J^{•}) or at least the cohomology groups H^{j}(X, F). The idea is to construct simplicial objects U_{•} augmented towards X, so U_{•} → X, such that

(∗)  RΓ(X, F) = Tot(RΓ(U_{•}, J^{•}))

is a quasi-isomorphism (for any F). On the right hand side this is the total complex associated to the double complex. (The maps are always canonical since we have the resolution over all of C.) The complex Γ(U_{•}, F) maps into the complex on the right. We will show that for any element η ∈ H^{j}(X, F) there exists a choice of U_{•} → X such that η comes from an element in H^{j}(U_{•}, F). This is a first step and it already allows us to define cup products for example. The starting point is the following.

Lemma 2.1.1. Suppose that {Y → X} is a covering in the topology of C. Let U_{n} = Y_{/X}^{n} be the simplicial object defined in Example 2.0.2. The augmentation U_{•} → X has the property that (∗) is a quasi-isomorphism for all F.

Proof. FIXME. □

Subsection 2.2. Making simplicial objects. Suppose that U_{•} is a simplicial object of C. Now let n ≥ 0 and let V → U_{n} be a representable morphism of C. This means that the fibre products V ×_{U_{n}} W exist for all morphisms W → U_{n}. For any m consider the fibre product (over U_{m})

U′_{m} = ∏_{φ∈Mor_{Δ}([n],[m])} V ×_{U_{n},U_{•}(φ)} U_{m}.

By our assumption on the morphism V → U_{n} this fibre product exists. For any ψ : [m1] → [m2] there is a canonical morphism U′_{m2} → U′_{m1} coming from the map Mor_{Δ}([n], [m1]) → Mor_{Δ}([n], [m2]), φ ↦ φ ∘ ψ, the identity map on V and the canonical map U_{•}(ψ) : U_{m2} → U_{m1} Clearly, these data give rise to a simplicial object U′_{•} in C. The natural morphisms U′_{m} → U_{m} give rise to a morphism of simplicial objects U′_{•} → U_{•}. Note that the morphism U′_{n} → U_{n} factors throught the morphism V → U_{n} by projection onto the factor corresponding to φ = id_{[n]}. Also, note that if C is a site and if {V → U_{n}} is a covering in the site then for any m it is true that {U′_{m} → U_{m}} is a covering. This proves the following lemma.

Lemma 2.2.1. Suppose that U_{•} and V → U_{n} are as above such that {V → U_{n}} is a covering for the topology on the site C. The morphism of simplicial objects U′_{•} → U_{•} constructed above has the following properties: (1) The morphism U′_{n} → U_{n} factors trough V → U_{n}. (2) For any m the set {U′_{m} → U_{m}} is a covering in the topology of C.

Subsection 2.3. Doubly simplicial stuff. A doubly simplicial object of C is a functor U_{•,•} : (Δ × Δ)^{∘} → C. By subdividing we can make this into a simplicial object W (U_{•,•}) with the same cohomology. FIXME: Explain this. Suppose that U′_{•} → U_{•} is a morphism of simplicial objects of C such that each of the morphisms U′_{n} → U_{n} is representable. Then we can construct a doubly-simplicial object U′_{•,•} by setting U′_{n,0} = U′_{n},

U′_{n,1} = U′_{n} ×_{U_{n}} U′_{n},

etc. Compare Example 2.0.2. Out of this object we can construct a single simplicial object W (U′_{•,•}) as explained above. Construct the natural morphism of simplicial objects W (U′_{•,•}) → U_{•}.

Lemma 2.3.1. Suppose that every {U′_{n} → U_{n}} is a covering for the topology of C. Suppose that F is a sheaf on C. Then there is a natural morphism of complexes

RΓ(U_{•}, F) → RΓ(W (U_{•,•}), F)

which is a quasi-isomorphism. FIXME: Something like this in any case.

Section 3. The general case

Mention how things work more generally, for example if C does not have the property that coverings consisting of a single map are cofinal. State the theorem in the correct generality. To continue reading,

(1) visit the next section: Stacks, Section 1, or

(2) go back to the table of contents: index.html#contents.

References

[MA71] J.L. Verdier M. Artin, A. Grothendieck. Theorie de Topos et Cohomologie Etale des Schemas I, II, III, volume 269, 270, 305 of Lecture Notes in Mathematics. Springer, 1971.