HOMOLOGICAL ALGEBRA (M2 UPMC, 2007/08)

Jan Nekovář

0. Introduction

0.0. What is this course about? Homological algebra studies algebraic properties of complexes and their (co)homology. Its origins lie in algebraic topology, but it has proved to be extremely important in many other areas of mathematics. The most useful references for this course will be the following texts:

[GM] S.I. Gelfand, Yu.I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Math., Springer, Berlin, 2003.

[W] C. Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994.

[S1] P. Schapira, Categories and Homological Algebra, lecture notes available at

http://www.institut.math.jussieu.fr/∼schapira/polycopies/

[S2] P. Schapira, An Introduction to Sheaves, lecture notes available at

http://www.institut.math.jussieu.fr/∼schapira/polycopies/

[KS] M. Kashiwara, P. Schapira, Sheaves on manifolds, Grund. Math. Wiss. 292, Springer, Berlin, 1994.

[McL] S. MacLane, Homology, Classics in Math., Springer, Berlin, 1995. We shall use the standard language of category theory in this course; see [S1, ch. 2,3] for the details.

0.1. Example: the space C^{∗} = C − {0} has a hole (at the origin). This can be seen geomerically, by noting that the unit circle γ = {z ∈ C | |z| = 1} ⊂ C^{∗} (with its positive orientation) is a closed curve (i.e., its boundary ∂γ = ∅ is empty) which is not a boundary of a relatively compact region U ⊂ C^{∗} (or a combination of such boundaries): γ ≠ ∂U. Analytically, we can argue that there exists a differential 1-form ω = a(x, y)dx + b(x, y)dy on C^{∗} which is closed, i.e., for which

0 = dω = d(a dx+b dy) = da∧dx+db∧dy = (\frac{∂a}{∂x} dx+ \frac{∂a}{∂y} dy)∧dx+(\frac{∂b}{∂x} dx+ \frac{∂b}{∂y} dy)∧dy = (\frac{∂b}{∂x} − \frac{∂a}{∂y}) dx∧dy,

but which is not exact, i.e.,

ω ≠ df = (\frac{∂f}{∂x} dx + \frac{∂f}{∂y} dy)

for any smooth function f : C^{∗} ⟶ R (note that dd f = 0 for any smooth function). For example, the differential form

ω = Im(\frac{dz}{z} ) = \frac{x dy − y dx}{x^{2} + y^{2}}  (z = x + iy)  (0.1.1)

has a non-trivial period integral along the unit circle γ

∫_{γ} ω = Im ∫_{γ} \frac{dz}{z} = 2π ≠ 0,

which implies both of the assertions above, as we have (by the Stokes theorem)

∫_{∂U} ω = ∫_{U} dω = ∫_{U} 0 = 0, ∫_{γ} df = ∫_{∂γ} f = ∫_{∅} f = 0.
\begin{matrix}c \\ ◯ \end{matrix} Jan Nekovář 2007

0.2. The de Rham complex The previous example is a special case of the following construction. Let X ⊂ R^{n} be an non-empty open subset (resp., a smooth (= C^{∞}) manifold of dimension n). The de Rham complex of X (with real coefficients) is the following sequence of real vector spaces and linear maps:

A^{•}(X) = [A^{0}(X) ⟶^{d} A^{1}(X) ⟶^{d} ⋯ ⟶^{d} A^{n}(X)] ,

where A^{p}(X) is the vector space of real-valued differential p-forms on X:

A^{0}(X) = C^{∞}(X, R) = {smooth functions f : X ⟶ R},
A^{p}(X) = { ∑_{|I|=p} f_{I} dx_{I} | f_{I} ∈ C^{∞}(X, R)},  I = {i_{1} < ⋯ < i_{p}} ⊆ {1, …, n},  dx_{I} = dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}},

where x_{1}, …, x_{n} are the standard coordinates in R^{n} (resp., local coordinates on X), and

df = ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} dx_{i},  d(f_{I} dx_{I}) = (df_{I}) ∧ dx_{I},  dx_{i} ∧ dx_{i} = 0,  dx_{j} ∧ dx_{i} = −dx_{i} ∧ dx_{j}.

The commutation rule

\frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} = \frac{∂}{∂x_{j}} \frac{∂}{∂x_{i}}

implies that

∀ω ∈ A^{p}(X) d^{2}ω = ddω = 0.

In particular, the spaces of exact (resp., closed) p-forms

B^{p}(X) = {exact p − forms on X} := {dφ | φ ∈ A^{p−1}(X)}
Z^{p}(X) = {closed p − forms on X} := {ω ∈ A^{p}(X) | dω = 0}

satisfy

B^{p}(X) ⊆ Z^{p}(X).

The degree p de Rham cohomology of X (with real coefficients) is defined as

H_{dR}^{p}(X, R) := Z^{p}(X)/B^{p}(X).

This is the space of obstructions for closed p-forms to be exact: a closed p-form ω ∈ Z^{p}(X) is exact iff its cohomology class [ω] = ω + B^{p}(X) ∈ H_{dR}^{p}(X, R) is equal to zero. 0.3. Examples of de Rham cohomology groups: (1) H_{dR}^{0}(X, R) consists of the functions f : X ⟶ R that are constant on each connected component of X.

(2) H^{p}(C^{∗}, R) is generated by the class of the form ω from (0.1.1) (resp., is equal to zero) for p = 1 (resp., for p > 1).

1. Complexes of R-modules

1.1. Modules

1.1.1. Let R be a ring (associative, but not necessarily with a unit). Recall that a left (resp., a right) R-module is an abelian group (M, +) equipped with an operation R × M ⟶ M, (r, m) ↦ rm (resp., M × R ⟶ M, (m, r) ↦ mr) satisfying the usual rules

(r + r′)m = rm + r′m,  r(m + m′) = rm + rm′,  (rr′)m = r(r′m)

resp.,

m(r + r′) = mr + mr′,  (m + m′)r = mr + m′r,  m(rr′) = (mr)r′

(which imply that 1m = m (resp., m1 = m) if R has a unit 1 ∈ R). We write M ∈ _{R}Mod (resp., M ∈ Mod_{R}). 1.1.2. (1) The opposite ring to R is the ring R^{op} equal to R as an abelian group, but with new multiplication r ·_{op} r′ = r′r.

(2) If M ∈ Mod_{R}, then M ∈ _{R^{op}}Mod, with the operation rm = mr (as (r ·_{op} r′)m = (r′r)m = m(r′r) = (mr′)r = r(mr′) = r(r′m)).

(3) If R is commutative, then R^{op} = R, hence Mod_{R} = _{R}Mod.

(4) Z-modules are just abelian groups. 1.1.3. Morphisms. A (homo)morphism f : M ⟶ N (say, for M, N ∈ _{R}Mod) is a map satisfying f(m + m′) = f(m) + f(m′), f(rm) = rf(m), for all r ∈ R and m, m′ ∈ M. 1.1.4. Examples of morphisms. (1) The zero map f = 0 (f(m) = 0 for all m ∈ M).

(2) The identity map id_{M} : M ⟶ M (id_{M}(m) = m for all m ∈ M).

(3) The inclusion M′ ↪ M of an R-submodule (i.e., of an abelian subgroup of (M, +) satisfying rm′ ∈ M′ for all r ∈ R and m′ ∈ M′).

(4) The projection p : M ⟶ M/M′ (p(m) = m + M′).

(5) If f : M ⟶ N is a morphism and M′ ⊂ M and N′ ⊂ N are submodules satisfying f(M′) ⊆ N′, then f induces a morphism \overline{f} : M/M′ ⟶ N/N′, \overline{f}(m + M′) = f(m) + N′.

(6) If f : M ⟶ N and g : N ⟶ P are morphisms, so is g ∘ f : M ⟶ P. 1.1.5. Products and sums of modules. Let I be a set; assume that we are given, for each i ∈ I, an R-module M_{i} ∈ _{R}Mod. The direct product of these modules is

∏_{i∈I} M_{i} = {(m_{i})_{i∈I} | m_{i} ∈ M_{i}}

with component-wise operations

(m_{i}) + (m′_{i}) = (m + m′_{i}), r(m_{i}) = (rm_{i}).

The direct sum of the modules M_{i} is the following submodule of their direct product:

⨁_{i∈I} M_{i} = {(m_{i})_{i∈I} | m_{i} ∈ M_{i}, only finitely many components m_{i} are notzero} ⊂ ∏_{i∈I} M_{i}.

If all modules M_{i} are equal to the same R-module M, we write

M^{I} = ∏_{i∈I} M = {maps I ⟶ M} ⊃ M^{(I)} = ⨁_{i∈I} M = {maps I ⟶ M with finite support}

(the support of a map f : I ⟶ M is the set supp(f) = {i ∈ I | f(i) ≠ 0}). In the special case when M = R ∈ _{R}Mod (with operation (r, m) = rm given by the product in R), then

R^{(I)} = {maps I ⟶ R with finite support}

is the free R-module on the set I. In particular, R^{(∅)} = 0 = {0}.

1.2. Complexes of (left) R-modules

1.2.1. Definition. A homological (resp., a cohomological) complex (C_{•}, ∂) (resp., (C^{•}, d)) of (left) R-modules is a sequence of R-modules and morphisms (“differentials”) such that

C_{•} :  ⋯ ⟶ C_{n+1} ⟶^{∂_{n+1}} C_{n} ⟶^{∂_{n}} C_{n−1} ⟶ ⋯,  ∀n ∈ Z ∂_{n} ∘ ∂_{n+1} = 0

resp.,

C^{•} :  ⋯ ⟶ C^{n−1} ⟶^{d^{n−1}} C^{n} ⟶^{d^{n}} C^{n+1} ⟶ ⋯,  ∀n ∈ Z d^{n} ∘ d^{n−1} = 0

It is customary to suppress the indices and write the defining conditions as ∂^{2} = ∂ ∘ ∂ = 0 (resp., d^{2} = d ∘ d = 0). Any homological complex can be transformed into a cohomological one (and vice versa), using the renumbering C^{n} = C_{−n}, d^{n} = ∂_{−n}. However, complexes naturally occurring in geometry usually satisfy ∀n < 0 C_{n} = 0 (resp., ∀n < 0 C^{n} = 0). 1.2.2. Example. The de Rham complex A^{•}(X) of a smooth manifold X of dimension n (completed by zeros in degrees outside {0, 1, …, n})

⋯ ⟶ 0 ⟶ 0 ⟶ A^{0}(X) ⟶^{d} A^{1}(X) ⟶^{d} ⋯ ⟶^{d} A^{n}(X) ⟶ 0 ⟶ 0 ⟶ ⋯

is a cohomological complex of real vector spaces. 1.2.3. Definition. Let (C^{•}, d) be a cohomological complex of R-modules. For each n ∈ Z, define

B^{n}(C^{•}) = {the n−coboundaries of C^{•}} := {d^{n−1}y | y ∈ C^{n−1}}
Z^{n}(C^{•}) = {the n−cocycles of C^{•}} := {x ∈ C^{n} | d^{n}(x) = 0}
H^{n}(C^{•}) = {the n−th cohomology module of C^{•}} := Z^{n}(C^{•})/B^{n}(C^{•})

(note that B^{n}(C^{•}) is a sub-module of Z^{n}(C^{•}), as d^{n} ∘ d^{n−1} = 0). For x ∈ Z^{n}(C^{•}), we denote by [x] = x + B^{n}(C^{•}) ∈ H^{n}(C^{•}) the cohomology class of x; we have [x] = [x′] iff x′ = x + dy for some y ∈ C^{n−1}. We say that the complex C^{•} is acyclic if ∀n ∈ Z H^{n}(C^{•}) = 0. 1.2.4. Analogously, for any homological complex C_{•} we have submodules

B_{n}(C_{•}) = ∂_{n+1}C_{n+1} ⊆ Z_{n}(C_{•}) = {x ∈ C_{n} | ∂_{n}(x) = 0} ⊆ C_{n}

and the homology modules

H_{n}(C_{•}) = Z_{n}(C_{•})/B_{n}(C_{•}).

1.2.5. Definition. A morphism of (cohomological) complexes α : (C^{•}, d) ⟶ (C^{′•}, d′) is a collection of morphisms of R-modules α_{n} : C^{n} ⟶ C^{′n} that are compatible with the differentials in the sense that

∀n ∈ Z α_{n+1} ∘ d^{n} = d^{′n} ∘ α_{n}, \begin{matrix}C^{n} & ⟶^{α_{n}} & C^{′n} \\ ↓_{d^{n}} & & ↓_{d^{′n}} \\ C^{n+1} & ⟶^{α_{n+1}} & C^{′n+1} \end{matrix}

(or, suppressing the indices, αd = d′α). In this case

α_{n}(Z^{n}(C^{•})) ⊆ Z^{n}(C^{′•}), α_{n}(B^{n}(C^{•})) ⊆ B^{n}(C^{′•}),

which implies that α_{n} induces a morphism of R-modules

H^{n}(α) : H^{n}(C^{•}) ⟶ H^{n}(C^{′•}), [x] ↦ [α_{n}(x)].

1.2.6. Example. If, in the situation of 1.2.5, each map α_{n} is the inclusion into C^{′n} of a sub-module C^{n}, we say that C^{•} is a subcomplex of C^{′•}. The modules ⋯ ⟶ C^{′n}/C^{n} ⟶ C^{′n+1}/C^{n+1} ⟶ ⋯ equipped with the differentials induced by d′ then form the corresponding quotient complex C^{′•}/C^{•} of C^{′•}. 1.2.7. Example. If g : X ⟶ Y is a smooth map between two smooth manifolds, then the pull-back maps g^{∗} : A^{p}(Y) ⟶ A^{p}(X) (e.g., g^{∗}(f) = f ∘ g for f ∈ A^{0}(Y)) define a morphism of complexes g^{∗} : A^{•}(Y) ⟶ A^{•}(X), hence morphisms of real vector spaces

H^{p}(g^{∗}) : H_{dR}^{p}(Y, R) ⟶ H_{dR}^{p}(X, R).

1.2.8. Definition. A morphism of complexes α : C^{•} ⟶ C^{′•} is a quasi-isomorphism (notation: Qis) if H^{n}(α) is an isomorphism, for all n ∈ Z.

1.3. Simplicial homology

1.3.1. Many natural complexes are of simplicial origin: the elements of C_{n} (resp., C^{n}) are indexed by a certain set of n-simplices^{1}; the differentials ∂_{n} (resp., d^{n}) are deduced from the boundary operators on the simplices. 1.3.2. Definition. Let v_{0}, …, v_{n} ∈ V (n ≥ 0) be a set of n + 1 affinely independent points in a real affine space V (i.e., the dimension of the smallest affine subspace of V containing v_{0}, …, v_{n} is equal to n). The n-simplex with vertices v_{0}, …, v_{n} is the convex hull of {v_{0}, …, v_{n}}, namely the subset

σ = [v_{0}, …, v_{n}] = Conv(v_{0}, …, v_{n}) = {∑_{i=0}^{n} λ_{i}v_{i} | λ_{i} ∈ R, λ_{i} ≥ 0, ∑_{i=0}^{n} λ_{i} = 1} ⊂ V.

The set of vertices {v_{0}, …, v_{n}} depends only on σ. For any non-empty subset {w_{0}, …, w_{m}} ⊂ {v_{0}, …, v_{n}}, the m-simplex Conv(w_{0}, …, w_{m}) is called a face of σ (of codimension n − m). In particular, each n-simplex has n + 1 faces of codimension 1, each of which is an (n − 1)-simplex. 1.3.3. Definition. A (finite) simplicial complex in V is a finite set K of simplices in V such that (a) ∀σ ∈ K each face of σ is an element of K;

(b) ∀σ, τ ∈ K the intersection σ ∩ τ is a face of σ and of τ. The set K is a disjoint union K = K_{0} ∪ K_{1} ∪ ⋯ ∪ K_{N} (for suitable N ≥ 0), where K_{n} consists of the n-simplices in K. We say that K is ordered if its set of vertices K_{0} is totally ordered. 1.3.4. Examples. (1) All faces of a given n-simplex σ (including σ).

(2) All faces of a given n-simplex σ except σ itself. 1.3.5. Face maps. Let K be an ordered simplicial complex. For each n ≥ 1 there are n + 1 face maps (in codimension 1)

\begin{aligned}& d_{0}, …, d_{n} : K_{n} ⟶ K_{n−1} \\ d_{i}[v_{0}, …, v_{n}] = & [v_{0}, …, v̂_{i}, …, v_{n}] = [v_{0}, …, v_{i−1}, v_{i+1}, …, v_{n}], v_{0} < ⋯ < v_{n} \end{aligned}

(the notation “… x̂ …” means that the term “x” is omitted). The face maps satisfy the following fundamental relation:

∀i < j d_{i}d_{j} = d_{j−1}d_{i}.  (1.3.5.1)

1.3.6. Example. If v_{0} < v_{1} < v_{2} < v_{3} and σ = [v_{0}, v_{1}, v_{3}], then

d_{2}(σ) = [v_{0}, v_{1}],  d_{0}d_{2}(σ) = [v_{1}],  d_{0}(σ) = [v_{1}, v_{3}],  d_{1}d_{0}(σ) = [v_{1}] = d_{0}d_{2}(σ).

1 0-simplex = point, 1-simplex = segment, 2-simplex = triangle, 3-simplex = tetrahedron, … 1.3.7. Definition. Let K = K_{0} ∪ K_{1} ∪ ⋯ ∪ K_{N} be an ordered (finite) simplicial complex; let A be an abelian group. The chain complex of K with coefficients in A is the following homological complex C_{•}(K, A) of abelian groups:

C_{n}(K, A) = A^{(K_{n})}, ∂_{n} = ∑_{i=0}^{n}(−1)^{i}d_{i} : C_{n}(K, A) = A^{(K_{n})} ⟶ C_{n−1}(K, A) = A^{(K_{n−1})}

(C_{n}(K, A) = 0 for n > N and n < 0).

1.3.8. Example. If K consists of a 2-simplex [v_{0}, v_{1}, v_{2}] (v_{0} < v_{1} < v_{2}) and all its faces, then

C_{2} = A · [v_{0}, v_{1}, v_{2}] ⟶^{∂_{2}} C_{1} = A · [v_{0}, v_{1}] ⊕ A · [v_{0}, v_{2}] ⊕ A · [v_{1}, v_{2}] ⟶^{∂_{1}} C_{0} = A · [v_{0}] ⊕ A · [v_{1}] ⊕ A · [v_{2}],

∂_{1}([v_{0}, v_{1}]) = [v_{1}] − [v_{0}], ∂_{1}([v_{0}, v_{2}]) = [v_{2}] − [v_{0}], ∂_{1}([v_{1}, v_{2}]) = [v_{2}] − [v_{1}],

∂_{2}([v_{0}, v_{1}, v_{2}]) = [v_{1}, v_{2}]−[v_{0}, v_{2}]+[v_{0}, v_{1}], ∂_{1}∘∂_{2}([v_{0}, v_{1}, v_{2}]) = ([v_{2}]−[v_{1}])−([v_{2}]−[v_{0}])+([v_{1}]−[v_{0}]) = 0.

1.3.9. Proposition-Definition. C_{•}(K, A) is, indeed, a complex (i.e., ∂^{2} = 0). Denote by H_{n}(K, A) := H_{n}(C_{•}(K, A)) its homology (n ∈ Z).

Proof. It follows from the relation (1.3.5.1) that

∂_{n−1} ∘ ∂_{n} = ∑_{i=0}^{n−1} ∑_{j=0}^{n}(−1)^{i+j} d_{i}d_{j} = ∑_{n>i≥j≥0}(−1)^{i+j} d_{i}d_{j} + ∑_{0≤i<j≤n}(−1)^{i+j} d_{j−1}d_{i};

replacing i (resp., j − 1) in the second sum by j (resp., by i), we obtain

∂_{n−1} ∘ ∂_{n} = ∑_{n>i≥j≥0}((−1)^{i+j} + (−1)^{i+j+1}) d_{i}d_{j} = 0.

1.3.10. Denote by

|K| = ⋃_{σ∈K} σ ⊂ V

the subset of V (with the induced topology) consisting of the “physical points” of K. The main result of the classical simplicial homology theory states that the homology groups H_{n}(K, A) depend only on |K|, in a functorial way:

(1) if K, K′ are (finite, ordered) simplicial complexes, then any continuous map f : |K| ⟶ |K′| defines a morphism of abelian groups

f_{n∗} : H_{n}(K, A) ⟶ H_{n}(K′, A),

which depends only on the homotopy class of f (see 1.5.1 below);

(2) if K, K′, K′′ are (finite, ordered) simplicial complexes and f : |K| ⟶ |K′|, g : |K′| ⟶ |K′′| continuous maps, then (g ∘ f)_{n∗} = g_{n∗} ∘ f_{n∗}.

The definition of the maps f_{n∗} (in fact, of a morphism of complexes C_{•}(K, A) ⟶ C_{•}(K′, A)) is straightforward when f is a simplicial map (which means that f maps simplices of K to simplices of K′ – possibly of a smaller dimension – and the restriction f to each simplex σ ∈ K is an affine map): a simplex [v_{0} < ⋯ < v_{n}] ∈ C_{n}(K, A) is mapped to 0 if f(v_{i}) = f(v_{j}) for some i ≠ j (resp., to ε[v′_{0} < ⋯ < v′_{n}] ∈ C_{n}(K′, A) if the set {f(v_{0}), …, f(v_{n})} = {v′_{0} < ⋯ < v′_{n}} has n + 1 elements; the sign ε = ±1 is equal to the sign of the permutation transforming (f(v_{0}), …, f(v_{n})) into (v′_{0}, …, v′_{n})).

In order to define f_{n∗} for a general continuous map f one has to approximate f by a map that is simplicial on a sufficiently fine subdivision of K (see, for example, M.A. Armstrong, Basic Topology, Springer, 1983).

1.4. Singular homology

1.4.1. Unlike simplicial homology, singular homology is defined for arbitrary topological spaces in a manifestly functorial manner. 1.4.2. Definition. The standard n-simplex

Δ_{n} = Conv(e_{0}, …, e_{n}) = {( λ_{0} ; ⋮ ; λ_{n} ) ∈ R^{n+1}| ∑_{i=0}^{n} λ_{i} = 1, λ_{i} ≥ 0} ⊂ R^{n+1}

is the convex hull of the endpoints of the vectors of the canonical basis e_{0}, …, e_{n} of R^{n+1}. It is naturally ordered, as its vertices correspond to the elements of [n] := {0, …, n} (e_{i} ⟷ i). 1.4.3. Face maps. Any non-decreasing map f : [m] ⟶ [n] defines an affine map

f : Δ_{m} ⟶ Δ_{n},  ∑_{i=0}^{m} λ_{i}e_{i} ↦ ∑_{i=0}^{m} λ_{i}e_{f(i)};

the faces of Δ_{n} correspond to increasing maps f. In particular, the codimension one faces

d_{0}, …, d_{n} : Δ_{n−1} ⟶ Δ_{n}

are given by the increasing maps d_{i} : [n − 1] ⟶ [n], i ∉ Im(d_{i}). Recall that these maps satisfy the relations (1.3.5.1). 1.4.4. Definition (singular simplices). Let X be a topological space. A singular n-simplex of X (n ≥ 0) is a continuous map σ : Δ_{n} ⟶ X. Denote by X(Δ_{n}) the set of all singular n-simplices of X. For n > 0 there are face maps

X(Δ_{n}) ⟶ X(Δ_{n−1}),  σ ↦ σ ∘ d_{i}  (i = 0, …, n).

1.4.5. Definition. Let X be a topological space and A an abelian group. The singular chain complex of X is the following (homological) complex C_{•} = Csing_{•}(X, A) (with C_{n} = 0 for n < 0):

⋯ ⟶ C_{n} = A^{(X(Δ_{n}))} ⟶^{∂_{n}} C_{n−1} = A^{(X(Δ_{n−1}))} ⟶ ⋯, ∂_{n}(σ) = ∑_{i=0}^{n}(−1)^{i} (σ ∘ d_{i}).

The homology groups of this complex H_{n}^{sing}(X, A) := H_{n}(Csing_{•}(X, A)) are called the singular homology groups of X with coefficients in A.

1.4.6. In concrete terms,

C_{n}^{sing}(X, A) = { ∑_{σ:Δ_{n}⟶X} a_{σ} · σ | σ continuous, a_{σ} ∈ A, the sum is finite},

∂_{n}( ∑_{σ:Δ_{n}⟶X} a_{σ} · σ) = ∑_{σ:Δ_{n}⟶X} ∑_{i=0}^{n}(−1)^{i} a_{σ} · (σ ∘ d_{i}).

The fact that ∂^{2} = 0 follows from (1.3.5.1), as in 1.3.9. 1.4.7. Example (homology of a point). If X = pt, then each X(Δ_{n}) consists of one element, C_{n}^{sing}(X, A) = A and ∂_{n} is given by multiplication by ∑_{i=0}^{n}(−1)^{i}; thus

Csing_{•}(pt, A) : ⋯ ⟶ A ⟶^{id} A ⟶^{0} A ⟶^{id} A ⟶^{0} A (= C_{0}) ⟶ 0 ⟶ 0 ⟶ ⋯,

which implies that

H_{n}^{sing}(pt, A) = \{ \begin{matrix}A, & n = 0, \\ 0, & n > 0. \end{matrix}

1.4.8. Exercise. Show that H_{0}^{sing}(X) = A^{(π_{0}(X))}, where π_{0}(X) is the set of path components of X:

π_{0}(X) = X/ ∼ x ∼ x′ := ∃σ : [0, 1] ⟶ X continuous, σ(0) = x, σ(1) = x′.

1.4.9. Functoriality. Let f : X ⟶ Y be a continuous map between two topological spaces. If σ : Δ_{n} ⟶ X is a singular n-simplex of X, then f ∘ σ : Δ_{n} ⟶ X ⟶ Y is a singular n-simplex of Y. As (f ∘ σ) ∘ d_{i} = f ∘ (σ ∘ d_{i}) for all i = 0, …, n, the composition with f defines a morphism of complexes

Csing_{•}(f) : Csing_{•}(X, A) ⟶ Csing_{•}(Y, A),

hence morphisms of abelian groups

H_{n}^{sing}(f) : H_{n}^{sing}(X, A) ⟶ H_{n}^{sing}(Y, A).

If g : Y ⟶ Z is continuous, then

H_{n}^{sing}(g) ∘ H_{n}^{sing}(f) = H_{n}^{sing}(g ∘ f) : H_{n}^{sing}(X, A) ⟶ H_{n}^{sing}(Z, A).

If Y = X and f = id_{X}, then H_{n}^{sing}(id_{X}) = id.

1.5. Homotopy invariance of singular homology

1.5.1. Definition. Let f, g : X ⟶ Y be continuous maps between topological spaces X and Y. A homotopy H : f ≃ g between f and g is a continuous map H : X × I ⟶ Y (I = [0, 1]) such that ∀x ∈ X H(x, 0) = f(x), H(x, 1) = g(x). For t ∈ I, denote by i_{t} : X ↪ X × I the map i_{t}(x) = (x, t). The family of continuous maps H ∘ i_{t} : X ⟶ Y is then a “continuous deformation” of f = H ∘ i_{0} into g = H ∘ i_{1}.

1.5.2. Definition. A continuous map f : X ⟶ Y is a homotopy equivalence if there exists a continuous map g : Y ⟶ X and homotopies gf ≃ id_{X}, fg ≃ id_{Y} (we say that g is a homotopy inverse of f).

1.5.3. Definition. A topological space X is contractible if there exists a homotopy equivalence X ⟶ pt.

1.5.4. Examples. (1) The inclusion i : S^{n} ↪ R^{n+1} − {0} is a homotopy equivalence, as the retraction r : R^{n+1}−{0} ⟶ S^{n}, r(x) = x/∥x∥ satisfies ri = id, ir ≃ id (via the homotopy H(x, t) = x/(t+(1−t)∥x∥)).

(2) Any star-shaped subset X ⊂ R^{n} (i.e., such that there exists a point x_{0} ∈ X with the property that, for each point x ∈ X, the segment [x_{0}, x] is contained in X) is contractible: if i : {x_{0}} ↪ X (resp., r : X ⟶ {x_{0}}) is the inclusion (resp., the constant map), then ri = id_{{x_{0}}} and ir ≃ id_{X}, via the homotopy H(x, t) = tx + (1 − t)x_{0}.

1.5.5. Theorem. Let f, g : X ⟶ Y be continuous maps. If there exists a homotopy between f and g, then

∀n ≥ 0 H_{n}^{sing}(f) = H_{n}^{sing}(g) : H_{n}^{sing}(X, A) ⟶ H_{n}^{sing}(Y, A).

1.5.6. Corollary. If f : X ⟶ Y is a homotopy equivalence, then Csing_{•}(f) : Csing_{•}(X, A) ⟶ Csing_{•}(Y, A) is a quasi-isomorphism, i.e., the maps H_{n}^{sing}(f) : H_{n}^{sing}(X, A) ⟶ H_{n}^{sing}(Y, A) are isomorphisms (for all n ≥ 0).

Proof. Let g : Y ⟶ X be a homotopy inverse of f. Then

H_{n}(f) ∘ H_{n}(g) = H_{n}(f ∘ g) =^{1.5.5} H_{n}(id_{Y}) = id_{H_{n}(Y)}

H_{n}(g) ∘ H_{n}(f) = H_{n}(g ∘ f) =^{1.5.5} H_{n}(id_{X}) = id_{H_{n}(X)},

which implies that H_{n}^{sing}(g) : H_{n}^{sing}(Y, A) ⟶ H_{n}^{sing}(X, A) is an inverse of H_{n}^{sing}(f).

1.5.7. Corollary. If X is contractible, then

H_{n}^{sing}(X, A) = \{ \begin{matrix}A, & n = 0, \\ 0, & n > 0. \end{matrix}

Proof of Theorem 1.5.5. The statement is a consequence of the following two Lemmas, as the composite morphisms of abelian groups

h_{n} = H ∘ t_{n} : C_{n}(X, A) ⟶^{t_{n}} C_{n+1}(X × I, A) ⟶^{H} C_{n+1}(Y, A)

(where we have denoted, for simplicity, the morphism C_{n+1}^{sing}(H) by H) satisfy

∂h + h∂ = ∂Ht + H∂t = H(∂t + t∂) = h(i_{1} − i_{0}) = g − f,

hence give rise to a homotopy between Csing_{•}(f) and Csing_{•}(g) : Csing_{•}(X, A) ⟶ Csing_{•}(Y, A).

1.5.8. Lemma-Definition. Let α, β : (C_{•}, ∂) ⟶ (C′_{•}, ∂′) be morphisms of (homological) complexes of R-modules. A homotopy between α and β is a collection h = (h_{n}) of morphisms of R-modules h_{n} : C_{n} ⟶ C′_{n+1} such that ∂′_{n+1} ∘h_{n} +h_{n−1} ∘∂_{n} = β_{n} −α_{n}, for all n ∈ Z (or, suppressing the indices, ∂′h+h∂ = β −α). If such a homotopy exists, then

∀n ≥ 0 H_{n}(α) = H_{n}(β) : H_{n}(C_{•}) ⟶ H_{n}(C′_{•}).

[Similarly, a homotopy between morphisms α, β : (C^{•}, d) ⟶ (C^{′•}, d′) of cohomological complexes of Rmodules is a collection h = (h_{n}) of morphisms of R-modules h_{n} : C^{n} ⟶ C^{′n−1} such that d′h + hd = β − α. If h exists, then H^{n}(α) = H^{n}(β) : H^{n}(C^{•}) ⟶ H^{n}(C^{′•}) for all n ∈ Z.]

Proof. An element of H_{n}(C_{•}) is of the form [x], for some x ∈ C_{n} such that ∂_{n}(x) = 0; then

H_{n}(α)([x]) − H_{n}(β)([x]) = [α_{n}(x)] − [β_{n}(x)] = [∂′_{n+1}(h_{n}(x))] + [h_{n−1}(∂_{n}(x))] = [∂′_{n+1}(h_{n}(x))] + 0 = 0.

1.5.9. Lemma. There exist morphisms of abelian groups t_{n} : C_{n}(X, A) ⟶ C_{n+1}(X × I, A) satisfying ∂t + t∂ = i_{1} − i_{0}.

Proof. It is enough to define t_{n}(σ) for any singular n-simplex σ : Δ_{n} ⟶ X and then extend it by linearity: t_{n}(∑_{σ} a_{σ} · σ) = ∑_{σ} a_{σ} · t_{n}(σ). Geometrically, t_{n} is the “prism operator” given by

t_{n} : σ ↦ a natural subdivision of the prism σ × I into singular (n + 1)−simplices.

In fact, it is enough to subdivide the standard prism Δ_{n} ×I into n+1 (n+1)-simplices; the required formula will then be a consequence of the fact that the boundary of Δ_{n} × I is a sum of its vertical part −(∂Δ_{n}) × I and its horizontal part Δ_{n} × {1} − Δ_{n} × {0}.

The formal definition is the following: the vertices of Δ_{n} ×I correspond to the elements of [n]×[1]. This set has a natural partial order (i, j) ≤ (i′, j′) ⟺ i ≤ i′, j ≤ j′. For i ∈ [n], let t_{n,i} : [n + 1] ⟶ [n] × [1] be the unique monotonous injective map whose image contains {i} × [1]; as in 1.4.3, this defines an affine map t_{n,i} : Δ_{n+1} ⟶ Δ_{n} × I which maps the (ordered) vertices [e_{0}, …, e_{n+1}] of Δ_{n+1} into the vertices [(e_{0}, 0), …, (e_{i}, 0), (e_{i}, 1), …, (e_{n}, 1)] of a certain (n+1)-simplex inside Δ_{n} ×I. The alternating sum of these simplices

∑_{i=0}^{n}(−1)^{i} [(e_{0}, 0), …, (e_{i}, 0), (e_{i}, 1), …, (e_{n}, 1)]

then gives the desired decomposition of Δ_{n} × I. This suggests that we should define t_{n} as

t_{n}(σ) := ∑_{i=0}^{n}(−1)^{i} (σ × I) ∘ t_{n,i} ∈ C_{n+1}(X × I, A) (σ : Δ_{n} ⟶ X),

where σ × I : Δ_{n} × I ⟶ X × I is given by (σ × I)(a, t) = (σ(a), t).

We check the formula ∂t_{n} + t_{n−1}∂ = i_{1} − i_{0} in the case n = 1 and leave the general case as an exercise to the reader:

t_{1}([e_{0}, e_{1}]) = [(e_{0}, 0), (e_{0}, 1), (e_{1}, 1)] − [(e_{0}, 0), (e_{1}, 0), (e_{1}, 1)],
∂t_{1}([e_{0}, e_{1}]) = [(e_{0}, 1), (e_{1}, 1)] − [(e_{0}, 0), (e_{1}, 0)] + [(e_{0}, 0), (e_{0}, 1)] − [(e_{1}, 0), (e_{1}, 1)]
t_{0}∂([e_{0}, e_{1}]) = t_{0}([e_{1}] − [e_{0}]) = [(e_{1}, 0), (e_{1}, 1)] − [(e_{0}, 0), (e_{0}, 1)]
(∂t_{1} + t_{0}∂)([e_{0}, e_{1}]) = [e_{0}, e_{1}] × {1} − [e_{0}, e_{1}] × {0}.

1.5.10. Exercise. Describe a natural decomposition of Δ_{n} × Δ_{m} into (m + n)-simplices. 1.5.11. Exercise (homotopy invariance of de Rham cohomology). Let X be a smooth manifold. As the product X × I is a manifold with boundary, we denote by A^{p}(X × I) the space of (smooth) differential p-forms on X × (0, 1) that extend to a smooth p-form on some open subset of X × R containing X × I. With this definition, the de Rham complex A^{•}(X × I) : ⋯ ⟶ A^{p}(X × I) ⟶^{d} A^{p+1}(X × I) ⟶ ⋯ of X × I makes sense. If f, g : X ⟶ Y are smooth maps to another smooth manifold Y, a smooth homotopy between f and g is a map H : X × I ⟶ Y that extends to a smooth map U ⟶ Y on some open subset U ⊂ X × R containing X × I.

The aim of this exercise is to show that such a smooth homotopy gives rise to a homotopy between the morphisms f^{∗}, g^{∗} : A^{•}(Y) ⟶ A^{•}(X); as a result, H^{n}(f^{∗}) = H^{n}(g^{∗}) : H_{dR}^{n}(Y, R) ⟶ H_{dR}^{n}(X, R) for all n ∈ Z.

The following construction is dual to that of Lemma 1.5.9: if {x_{i}} are local coordinates on X (and t is the standard coordinate on R), define maps

k_{n} : A^{n}(X × I) ⟶ A^{n−1}(X)

by the formula

k_{n}( ∑_{|J|=n} α_{J}(x, t) dx_{J} + ∑_{|J|=n−1} β_{J}(x, t) dt ∧ dx_{J}) = ∑_{|J|=n−1}(∫_{0}^{1} β_{J}(x, t) dt) dx_{J}.

(1) Show that

∀ω ∈ A^{n}(X × I) (dk + kd)(ω) = i_{1}^{∗}(ω) − i_{0}^{∗}(ω) (i_{t}(x) = (x, t))

(2) Deduce from (1) that any smooth homotopy H between f, g : X ⟶ Y gives rise to a homotopy k ∘ H^{∗} = (k_{n} ∘ H^{∗} : A^{n}(Y) ⟶ A^{n−1}(X)) between f^{∗} and g^{∗}.

1.6. Other constructions: relative (singular) homology, cohomology

1.6.1. Definition (relative homology). If Y is a subset of a topological space X (equipped with the induced topology), then Csing_{•}(Y, A) is a subcomplex of Csing_{•}(X, A). Denote by

Csing_{•}(X, Y; A) = Csing_{•}(X, A)/Csing_{•}(Y, A)

the corresponding quotient complex; its homology groups H_{n}^{sing}(X, Y; A) = H_{n}(Csing_{•}(X, Y; A)) are the relative (singular) homology groups of the pair (X, Y) (with coefficients in the abelian group A). 1.6.2. Definition (cohomology). The cohomological singular complex Csing^{•}(X, A) of a topological space X (with coefficients in an abelian group A) is defined as

C_{sing}^{n}(X, A) = A^{X(Δ_{n})} = {maps α : X(Δ_{n}) ⟶ A}, (d^{n}α)(σ) = ∑_{i=0}^{n}(−1)^{i} α(σ ∘ d_{i}).

The cohomology groups H_{sing}^{n}(X, A) = H^{n}(Csing^{•}(X, A)) are the (singular) cohomology groups of X with coefficients in A. If f : X ⟶ Y is a continuous map and (α : Y (Δ_{n}) ⟶ A) ∈ C_{sing}^{n}(Y, A), then the map α ∘ f : (σ : Δ_{n} ⟶ X) ↦ α(f ∘ σ) is an element of C_{sing}^{n}(X, A); this defines a morphism of complexes Csing^{•}(Y, A) ⟶ Csing^{•}(X, A).

1.6.3. Exercise. Show that H_{sing}^{0}(X, A) = A^{π_{0}(X)}. 1.6.4. Definition (relative cohomology). In the situation of 1.6.1, the cohomology groups of the subcomplex

Csing^{•}(X, Y; A) = Ker (restriction : Csing^{•}(X, A) ⟶ Csing^{•}(Y, A))

of Csing^{•}(X, A) are the relative (singular) cohomology groups H_{sing}^{n}(X, Y; A) = H^{n}(Csing^{•}(X, Y; A)) of the pair (X, Y) (with coefficients in A). 1.6.5. Functoriality. If f : X ⟶ X′ is a continuous map between topological spaces and Y ⊂ X, Y′ ⊂ X′ are subsets such that f(Y) ⊆ Y′, then f induces morphisms of complexes

Csing_{•}(X, Y; A) ⟶ Csing_{•}(X′, Y′; A),  Csing^{•}(X′, Y′; A) ⟶ Csing^{•}(X, Y; A).  (1.6.5.1)

If g : X ⟶ X′ is another continuous map and H : X × I ⟶ X′ is a homotopy between f and g satisfying H(Y × I) ⊆ Y′, then the morphisms (1.6.5.1) induced by f and g are homotopic (the proof of Theorem 1.5.5 works with minor modifications). 1.6.6. These complexes are related as follows:

Csing_{•}(X, Y; A) = Csing_{•}(X, A)/Csing_{•}(Y, A)
Csing^{•}(Y, A) = Csing^{•}(X, A)/Csing^{•}(X, Y; A)
Csing_{•}(X, A) = Csing_{•}(X, Z) ⊗_{Z} A
Csing^{•}(X, A) = HomZ^{•}(Csing_{•}(X, Z), A)

(strictly speaking, the last equality holds only if we change the signs of differentials; see ?? below for the sign conventions). 1.6.7. As we shall see in ?? below, the relations 1.6.6 yield the following exact sequences (the last two of which are split):

⋯ ⟶ H_{n}^{sing}(Y, A) ⟶ H_{n}^{sing}(X, A) ⟶ H_{n}^{sing}(X, Y; A) ⟶ H_{n−1}^{sing}(Y, A) ⟶ H_{n−1}^{sing}(X, A) ⟶ ⋯
⋯ ⟶ H_{sing}^{n−1}(X, A) ⟶ H_{sing}^{n−1}(Y, A) ⟶ H_{sing}^{n}(X, Y; A) ⟶ H_{sing}^{n}(X, A) ⟶ H_{sing}^{n}(Y, A) ⟶ ⋯
0 ⟶ H_{n}^{sing}(X, Z) ⊗_{Z} A ⟶ H_{n}^{sing}(X, A) ⟶ Tor_{1}^{Z}(H_{n−1}^{sing}(X, Z), A) ⟶ 0
0 ⟶ Ext_{Z}^{1}(H_{n+1}^{sing}(X, Z), A) ⟶ H_{sing}^{n}(X, A) ⟶ Hom_{Z}(H_{n}^{sing}(X, Z), A) ⟶ 0

2. Exactness

2.1. Exactness in the category of R-modules

In this section, R is a ring, a “module” is a left R-module, a “morphism” is a morphism of left R-modules. 2.1.1. Definition. Let f : M ⟶ N be a morphism. The kernel of f is Ker(f) = {m ∈ M | f(m) = 0} (a submodule of M); the image of f is Im(f) = {f(m) | m ∈ M} (a submodule of N); the cokernel of f is Coker(f) = N/Im(f) (a quotient module of N). We say that f is a monomorphism if Ker(f) = 0 ( ⟺ f is injective); an epimorphism if Coker(f) = 0 ( ⟺ f is surjective); an isomorphism if there is a morphism g : N ⟶ M such that fg = id_{N} and gf = id_{M} ( ⟺ f is bijective ⟺ f is both a mono and an epimorphism). 2.1.2. Definition. A sequence of morphisms

M^{i_{0}} ⟶ ⋯ ⟶ M^{i−1} ⟶^{f_{i−1}} M^{i} ⟶^{f_{i}} M^{i+1} ⟶ ⋯ ⟶ M^{i_{1}}  (2.1.2.1)

is exact at the term M^{i} if Im(f_{i−1}) = Ker(f_{i}) (which implies that f_{i}f_{i−1} = 0 and Coker(f_{i−1}) ⟶^{∼} Im(f_{i})); it is exact if it is exact at M^{i} for all i_{0} < i < i_{1}. In particular, an acyclic complex is the same thing as an exact sequence indexed by i ∈ Z. 2.1.3. Examples. (1) 0 ⟶ M ⟶^{f} N is exact ⟺ Ker(f) = 0 ⟺ f is a monomorphism.

(2) M ⟶^{f} N ⟶ 0 is exact ⟺ Im(f) = N ⟺ f is an epimorphism.

(3) 0 ⟶ M ⟶^{f} N ⟶ 0 is exact ⟺ f is both mono end epi ⟺ f is an isomorphism.

(4) If M is a submodule of N, then 0 ⟶ M ⟶^{i} N ⟶^{p} N/M ⟶ 0 is exact, where i and p denote the inclusion and the canonical projection, respectively.

(5) Any “short exact sequence” 0 ⟶ M ⟶^{f} N ⟶^{g} P ⟶ 0 is naturally isomorphic to a sequence of the form considered in (4).

(6) For any morphism f : M ⟶ N, the sequence 0 ⟶ Ker(f) ⟶ M ⟶^{f} N ⟶ Coker(f) ⟶ 0 is exact.

(7) For any short exact sequence 0 ⟶ M ⟶^{f} N ⟶^{g} P ⟶ 0 the following conditions are equivalent (if they are satisfied, we say that the sequence is split):

(a) there exists a morphism s : P ⟶ N such that gs = id_{P};

(b) there exists a morphism r : N ⟶ M such that rf = id_{M};

(c) the 7-tuple (M, N, P, f, g, r, s) is isomorphic to (X_{1}, X_{1} ⊕ X_{2}, X_{2}, i_{1}, p_{2}, p_{1}, i_{2}), where p_{i}(x_{1}, x_{2}) = x_{i}, i_{1}(x_{1}) = (x_{1}, 0) and i_{2}(x_{2}) = (0, x_{2}).

(8) Any exact sequence (2.1.2.1) can be cut into short exact sequences

0 ⟶ N^{i} ⟶^{α_{i}} M^{i} ⟶^{β_{i}} N^{i+1} ⟶ 0  (i_{0} < i < i_{1} − 1),  N^{i} = Im(f_{i−1}) = Ker(f_{i}),  α_{i+1}β_{i} = f_{i},

and vice versa.

2.1.4. The Snake Lemma. Let

(0 ⟶) A ⟶^{f} B ⟶^{g} C ⟶ 0
↓^{α}  ↓^{β}  ↓^{γ}
0 ⟶ A′ ⟶^{f′} B′ ⟶^{g′} C′ (⟶ 0)

be a commutative diagram with exact rows. Then there is an exact sequence

(0 ⟶) Ker(α) ⟶^{f} Ker(β) ⟶^{g} Ker(γ) ⟶^{Δ} Coker(α) ⟶^{f′} Coker(β) ⟶^{g′} Coker(γ) (⟶ 0),

in which the non-obvious morphism Δ : Ker(γ) ⟶ Coker(α) is given by “Δ = f^{′−1}βg^{−1}”.

Proof. We only give a definition of Δ and leave the verification of exactness as an exercise. Let c ∈ Ker(γ). As g is an epimorphism, there exists b ∈ B such that g(b) = c. As g′β(b) = γg(b) = γ(c) = 0, there exists a′ ∈ A′ (unique, since f′ is a monomorphism) such that f′(a′) = β(b). We wish to define Δ(c) := a′ + α(A) ∈ Coker(α). In order to check that this definition makes sense we must analyse what happens if we take two different elements b_{1}, b_{2} ∈ B satisfying g(b_{i}) = c. As b_{1} − b_{2} ∈ Ker(g) = Im(f), we have b_{1} − b_{2} = f(a) for some a ∈ A; thus the corresponding elements a′_{i} ∈ A′ (where f′(a′_{i}) = β(b_{i})) satisfy f′(a′_{1} − a′_{2}) = β(b_{1} − b_{2}) = βf(a) = f′α(a), hence a′_{1} − a′_{2} = α(a) ∈ α(A) (since f′ is a monomorphism), which implies that Δ(c) := a′_{1} + α(A) = a′_{2} + α(A) ∈ Coker(α) is, indeed, independent of the choice of b. As Δ is a composition of possibly multivalued R-linear maps, it is also R-linear. 2.1.5. Example. Let 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 be an exact sequence of abelian groups. For any integer m ≥ 1, the Snake Lemma applied to the diagram

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0
↓^{m}  ↓^{m}  ↓^{m}
0 ⟶ X ⟶ Y ⟶ Z ⟶ 0

(in which each vertical map is given by multiplication by m) gives an exact sequence

0 ⟶ _{m}X ⟶ _{m}Y ⟶ _{m}Z ⟶ X/mX ⟶ Y/mY ⟶ Z/mZ ⟶ 0,  (2.1.5.1)

where we have put, for any abelian group A,

mA = {a ∈ A | ma = 0}.

2.1.6. Exercise. Let A ⟶^{f} B ⟶^{g} C be morphisms; then there is an exact sequence

0 ⟶ Ker(f) ⟶ Ker(gf) ⟶ Ker(g) ⟶ Coker(f) ⟶ Coker(gf) ⟶ Coker(g) ⟶ 0.

2.1.7. Proposition. Any exact sequence of (cohomological) complexes 0 ⟶ A^{•} ⟶^{f} B^{•} ⟶^{g} C^{•} ⟶ 0 (in the sense of 3.1.2 below) gives rise to the following cohomology exact sequence

⋯ ⟶ H^{n}(A^{•}) ⟶^{H^{n}(f)} H^{n}(B^{•}) ⟶^{H^{n}(g)} H^{n}(C^{•}) ⟶^{δ} H^{n+1}(A^{•}) ⟶^{H^{n+1}(f)} H^{n+1}(B^{•}) ⟶^{H^{n+1}(g)} H^{n+1}(C^{•}) ⟶ ⋯,

in which δ[c] = [f^{−1}dg^{−1}(c)] (more precisely, if c ∈ C^{n} and dc = 0, then there exist b ∈ B^{n} and a ∈ A^{n+1} such that g(b) = c and f(a) = db; then da = 0 and δ[c] = [a]).

Proof. The Snake Lemma applied to the diagram

0 ⟶ A^{m}  ⟶^{f_{m}} B^{m}  ⟶^{g_{m}} C^{m} ⟶ 0
↓^{d}  ↓^{d}  ↓^{d}
0 ⟶ A^{m+1} ⟶^{f_{m+1}} B^{m+1} ⟶^{g_{m+1}} C^{m+1} ⟶ 0

for m = n ± 1 implies that the rows in the commutative diagram

A^{n}/dA^{n−1} ⟶ B^{n}/dB^{n−1} ⟶ C^{n}/dC^{n−1} ⟶ 0
↓^{d}  ↓^{d}  ↓^{d}
0 ⟶ Z^{n+1}(A^{•}) ⟶ Z^{n+1}(B^{•}) ⟶ Z^{n+1}(C^{•})

are exact. Applying the Snake Lemma to the latter diagram we obtain the desired result. 2.1.8. Example. Let C^{•} be a complex of torsion-free abelian groups. For any integer m ≥ 1, the exact sequence of complexes

0 ⟶ C^{•} ⟶^{m} C^{•} ⟶ C^{•}/mC^{•} ⟶ 0

gives rise to a long exact sequence

⋯ ⟶ H^{n}(C^{•}) ⟶^{m} H^{n}(C^{•}) ⟶ H^{n}(C^{•}/mC^{•}) ⟶ H^{n+1}(C^{•}) ⟶^{m} H^{n+1}(C^{•}) ⟶ ⋯,

hence to short exact sequences

0 ⟶ H^{n}(C^{•})/mH^{n}(C^{•}) ⟶ H^{n}(C^{•}/mC^{•}) ⟶ _{m}H^{n+1}(C^{•}) ⟶ 0.

In particular, for C^{•} = Csing_{•}(X, Z) we obtain (using the homological notation) the following special case A = Z/mZ of the third exact sequence from 1.6.7:

0 ⟶ H_{n}^{sing}(X, Z)/mH_{n}^{sing}(X, Z) ⟶ H_{n}^{sing}(X, Z/mZ) ⟶ _{m}H_{n−1}^{sing}(X, Z) ⟶ 0.

2.1.9. The Five Lemma. Let

A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ A_{4} ⟶ A_{5}
↓^{f_{1}}  ↓^{f_{2}}  ↓^{f_{3}}  ↓^{f_{4}}  ↓^{f_{5}}
B_{1} ⟶ B_{2} ⟶ B_{3} ⟶ B_{4} ⟶ B_{5}

be a commutative diagram with exact rows. If f_{1}, f_{2}, f_{4} and f_{5} are isomorphisms, so is f_{3}.

Proof. Exercise.

2.1.10. Proposition. Let

0 ⟶ A^{•} ⟶ B^{•} ⟶ C^{•} ⟶ 0
↓^{α}  ↓^{β}  ↓^{γ}
0 ⟶ A^{′•} ⟶ B^{′•} ⟶ C^{′^{•}} ⟶ 0

be a commutative diagram of morphisms of (cohomological) complexes with exact rows. If two among the vertical morphisms are Qis, so is the third.

Proof. Combine the Five Lemma and the cohomology exact sequences

H^{n}(A^{•}) ⟶ H^{n}(B^{•}) ⟶ H^{n}(C^{•}) ⟶ H^{n+1}(A^{•}) ⟶ H^{n+1}(B^{•}) ⟶ H^{n+1}(C^{•})
↓  ↓  ↓  ↓  ↓  ↓
H^{n}(A^{′•}) ⟶ H^{n}(B^{′•}) ⟶ H^{n}(C^{′•}) ⟶ H^{n+1}(A^{′•}) ⟶ H^{n+1}(B^{′•}) ⟶ H^{n+1}(C^{′•}).

2.2. Limits and exactness in general categories

See [S1, §1.3, 1.5.4, 1.5.11, §3.4, §4.1, §5.1, 5.2.5, 5.2.6, §5.4] for general theory.

For example, the following facts are useful.

2.2.1. [Finite] inductive (resp., projective) limits exist in a given category C iff coequalisers (resp., equalisers) and [finite] sums (resp., products) exist in C. More precisely, such limits are obtained as coequalisers (resp., equalisers) of a pair of morphisms between two [finite] sums (resp., products). 2.2.2. In _{R}Mod, arbitrary inductive and projective limits exist; they commute with the forgetful functor RMod ⟶ Ab. Projective limits also commute with the forgetful functor _{R}Mod ⟶ Set. 2.2.3. A functor F : C ⟶ C′ that is left (resp., right) adjoint to another functor C′ ⟶ C commutes with any inductive (resp., projective) limits that exist. In particular, F is right (resp., left) exact if finite inductive (resp., projective) limits exist in C. 2.2.4. Let I be a set such that the sums (resp., products) indexed by I exist in C. It follows from 2.2.3 applied to the diagonal functor C ⟶ C^{I} that the sum functor ∐_{I} : C^{I} ⟶ C (resp., the product functor ∏_{I} : C^{I} ⟶ C) is right (resp., left) exact, provided finite inductive (resp., projective) limits exist in C.

2.2.5. More generally, if inductive (resp., projective) limits indexed by a small category I and finite inductive (resp., projective) limits exist in C, then the inductive limit \underrightarrow{lim}_{I} : C^{I} ⟶ C (resp., projective limit

\underleftarrow{lim}_{I} : C^{I^{op}} ⟶ C) functor is right (resp., left) exact.

2.2.6. It follows from 2.2.1 that an additive functor F : A ⟶ A′ between two abelian categories is right (resp., left) exact iff it commutes with cokernels (resp., kernels). 2.2.7. For any ring R and a set I, the sum and product functors ⊕_{I}, ∏_{I} : (_{R}Mod)^{I} ⟶ _{R}Mod are exact. 2.2.8. For any ring R and a filtered small category I, the inductive limit \underrightarrow{lim}_{I} : (_{R}Mod)^{I} ⟶ _{R}Mod is exact.

2.2.9. For any object X of an abelian category A, the functor Y ↦ Hom_{A}(X, Y) (resp., Y ↦ Hom_{A}(Y, X)) is a left (resp., right) exact additive functor A ⟶ Ab (resp., A^{op} ⟶ Ab). It is exact iff X is a projective (resp., an injective) object of A. Denote by P_{A} (resp., I_{A}) the full additive subcategory of A whose objects are the projective (resp., the injective) objects of A. [Note: the morphism Z ⟶^{2} Z in P_{Ab} has trivial kernel and cokernel, but is not an isomorphism; thus P_{Ab} is not an abelian category.] 2.2.10. An additive functor F : A ⟶ A′ between two abelian categories that admits an exact left adjoint preserves injectives (i.e., F (I_{A}) ⊆ I_{A′}). 2.2.11. (Freyd-Mitchell embedding theorem). Any small abelian category A admits an exact fully faithful additive functor F : A ⟶ _{R}Mod into a suitable category of modules. 2.2.12. It follows from 2.2.11 that everything in 2.1.2-2.1.10 (with the exception of 2.1.8) holds in an arbitrary abelian category, even though the proofs in §2.1 used explicit calculations with elements. See also [GM, Exercises to II.5] for an alternative approach.

2.3. Projective and injective modules

Let R be a ring. In this section we investigate basic properies of projective and injective objects in _{R}Mod (in particular, all R-modules will be left R-modules). 2.3.1. By definition, X ∈ _{R}Mod is projective iff for every epimorphism p : Y′ ⟶ Y in _{R}Mod and every morphism f : X ⟶ Y there is a morphism f′ : X ⟶ Y′ lifting f (i.e., pf′ = f):

\xymatrix{Y′ \ar[r]^{p} & Y \ar[r] & 0 \\  & X \ar@{.>}[ul]^{f′} \ar[u]_{f}}  (2.3.1.1)

Similarly, X is injective iff for every monomorphism i : Y ⟶ Y′ in _{R}Mod and every morphism f : Y ⟶ X there is a morphism f′ : Y′ ⟶ X extending f (i.e., f′i = f):

\xymatrix{0 \ar[r] & Y \ar[r]^{i} \ar[d]_{f} & Y′ \ar@{.>}[dl]^{f′} \\  & X}  (2.3.1.2)

2.3.2. Proposition. (1) A direct sum of R-modules ⨁_{I} X_{i} is projective ⟺ each X_{i} is projective.

(2) Any free R-module R^{(I)} is projective.

(3) For each R-module X there exists an epimorphism P ⟶^{p} X in which P is a projective R-module (“the category _{R}Mod has enough projectives”).

(4) Any short exact sequence 0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0 in which C is a projective R-module splits.

(5) An R-module X is projective ⟺ there is an R-module X′ such that X ⊕ X′ = R^{(I)} is free.

Proof. (1) This follows from definition (2.3.1.1) and the universal property of ⨁_{I} X_{i}.

(2) Consider the diagram (2.3.1.1) with X = R^{(I)}. If {e_{i}} (i ∈ I) is the standard basis of R^{(I)}, choose y′_{i} ∈ Y′ such that p(y′_{i}) = f(e_{i}); the morphism f′(∑_{i∈I} r_{i} e_{i}) = ∑_{i∈I} r_{i} y′_{i} (r_{i} ∈ R) then satisfies pf′ = f.

(3) If S ⊆ X is any subset that generates X as an R-module, then we can take P = R^{(S)} and p(∑_{s∈S} r_{s} e_{s}) = ∑_{s∈S} r_{s} s (r_{s} ∈ R).

(4) Consider the diagram (2.3.1.1) for the epimorphism p : B ⟶ C and the identity map f = id_{C} : C ⟶ C; the corresponding lift f′ : C ⟶ B then splits the given short exact sequence.

(5) The implication “⟸” follows from (1) and (2). Conversely, If X is a projective R-module, construct an epimorphism p : R^{(I)} ⟶ X from a suitable free module, as in the proof of (3). By (4), the exact sequence 0 ⟶ Ker(p) ⟶ R^{(I)} ⟶^{p} X ⟶ 0 splits, hence R^{(I)} = X ⊕ Ker(p). 2.3.3. Exercise. Show that Q is not a subgroup of any free abelian group (in particular, Q is not projective in Ab). 2.3.4. Remarks. (1) If R is a PID (for example, R = Z or K[x], where K is a field), then an R-module is projective iff it is free.

(2) Geometrically, projective modules of finite rank are attached to vector bundles. For example, if R is a commutative noetherian ring (with unit) and X is a finitely generated R-module, then X is projective ⟺ it is locally free on Spec(R) ⟺ its localisations X_{𝔪} at all maximal ideals 𝔪 ⊂ R are free R_{𝔪}-modules.

2.3.5. Proposition. (1) A direct product of R-modules ∏_{I} X_{i} is injective ⟺ each X_{i} is injective.

(2) Any short exact sequence 0 ⟶ A ⟶ B ⟶ C ⟶ 0 in which A is an injective R-module splits.

Proof. Exercise (cf., 2.3.2(1),(4)).

2.3.6. Proposition (Baer’s criterion). An R-module X is injective ⟺ it has the extension property (2.3.1.2) when Y′ = R (i.e., ⟺ each morphism I ⟶ X, where I ⊂ R is a left ideal, can be extended to a morphism R ⟶ X).

Proof of “⟸”. (The implication “⟹” is automatic). Assume that X has the extension property (2.3.1.2) with respect to all inclusions I ⊂ R (where I is a left ideal of R). Given a monomorphism i : Y ⟶ Y′ (which we can, and will, consider to be the inclusion of a submodule) and a morphism f : Y ⟶ X, consider the set of all pairs (Z, g), where Z ⊂ Y′ is a submodule containing Y and g : Z ⟶ X a morphism extending f (i.e., g|_{Y} = f). This set contains (Y, f) and is inductive with respect to the obvious partial order ((g, Z) ≤ (g′, Z′) iff Z ⊆ Z′ and g′|_{Z} = g), hence it contains a maximal element (Z, g), by Zorn’s Lemma. If Z = Y′, then we are done. If Z ⊊ Y′, fix y′ ∈ Y′, y′ ∉ Z. The set {r ∈ R | ry′ ∈ Z} is a left ideal of R. By assumption, the morphism I ⟶ X given by r ↦ g(ry′) can be extended to a morphism h : R ⟶ X; the formula g′(z + ry′) = g(z) + h(r) then defines a morphism g′ : Z′ = Z + Ry′ ⟶ X extending g, which contradicts the maximality of (Z, g) (as Z ⊊ Z′). This contradiction finishes the proof.

2.3.7. Corollary. If R is a PID, then an R-module X is injective ⟺ it is R-divisible (i.e., X = rX for all r ∈ R − {0}). In particular, the abelian groups Q, R, Q/Z = ⨁_{p} Q_{p}/Z_{p}, Q_{p}/Z_{p} and R/Z are injective in Ab.

Proof. By 2.3.6, it is enough to verify the extension property (2.3.1.2) with respect to monomorphisms i : R ⟶^{r} R given by multiplication by a non-zero element r ∈ R (the case of I = (0) is trivial).

2.3.8. Corollary. If R is a field (not necessarily commutative), then each R-module is both projective and injective.

2.3.9. Theorem. For each R-module X there exists a monomorphism X ⟶ I in which I is an injective R-module (“the category _{R}Mod has enough injectives”).

Proof. (1) Assume first that R = Z, hence _{R}Mod = Ab. Given X ∈ Ab, consider the set D(X) := Hom_{Z}(X, Q/Z) and the morphism

α : X ⟶ ∏_{D(X)} Q/Z = (Q/Z)^{D(X)},  α(x)(f) = f(x)  (f ∈ D(X)).

As (Q/Z)^{D(X)} is in jective by a combination of 2.3.5(1) and 2.3.7, it is enough to show that Ker(α) = 0. Assume that x ∈ X, x ≠ 0; then Zx ⊂ X is a non-zero cyclic abelian group, which implies that D(Zx) ≠ 0.

Let g : Zx ⟶ Q/Z be a non-zero element of D(Zx); as Q/Z is injective, there exists f : X ⟶ Q/Z (f ∈ D(X)) extending g; then α(x)(f) = g(x) ≠ 0, hence α(x) ≠ 0. This proves that Ker(α) = 0.

(2) Let R be arbitrary. The exact forgetful functor _{R}Mod ⟶ Ab is left adjoint to the functor

Ab ⟶ _{R}Mod, A ↦ Hom_{Z}(R_{R}, A),

via the isomorphism

Hom_{Z}(M, A) ⟶^{∼} Hom_{R}(M, Hom_{Z}(R_{R}, A))  (M ∈ _{R}Mod, A ∈ Ab)
f ↦ F : (m ↦ (r ↦ f(rm))

(above, the notation R_{R} means that we consider R as a right module over itself, which makes Hom_{Z}(R_{R}, A) into a left R-module in a natural way; see ??). By 2.2.10, the R-module I := Hom_{Z}(R_{R}, Q/Z) is injective.

(3) Given X ∈ _{R}Mod, put D_{R}(X) = Hom_{R}(X, I) and define a morphism similar to that in (1):

β : X ⟶ ∏_{D_{R}(X)} I = I^{D_{R}(X)},  β(x)(F) = F (x)  (F ∈ D_{R}(X)).

As in (1), I^{D_{R}(X)} is an injective R-module, so it remains to show that Ker(β) = 0. If x ∈ X, x ≠ 0, then there exists a morphism f ∈ Hom_{Z}(X, Q/Z) such that f(x) ≠ 0. The corresponding element

F ∈ Hom_{R}(X, Hom_{Z}(R_{R}, Q/Z)) = D_{R}(X)

then satisfies β(x)(F) = F (x) : r ↦ f(rx); in particular, F (x)(1) = f(x) ≠ 0, hence β(x) ≠ 0, as required.

3. Complexes in additive and abelian categories

3.1. Categories of complexes

3.1.1. The categories C^{∗}(A). Definition 1.2.1 of a (co)homological complex makes sense not only for R-modules, but for objects of an arbitrary additive category A.

We denote by C(A) the category whose objects are cohomological complexes (C^{•}, d) with all C^{n} ∈ A and whose morphisms are morphisms of complexes (as in 1.2.5). This is an additive category in which the finite direct sums are computed termwise ((C ⊕ C′)^{n} = C^{n} ⊕ C^{′n}, d_{C⊕C′} = d_{C} ⊕ d_{C′}).

C(A) contains as full subcategories various categories of complexes satisfying suitable boundedness conditions:

C^{≥n}(A) = {C^{•} ∈ C(A) | ∀i < n C^{i} = 0}, C^{≤n}(A) = {C^{•} ∈ C(A) | ∀i > n C^{i} = 0},
C^{+}(A) = ⋃_{n∈Z} C^{≥n}(A),  C^{−}(A) = ⋃_{n∈Z} C^{≤n}(A),  C^{b}(A) = C^{+}(A) ∩ C^{−}(A)

(the last three consisting of complexes that are, respectively, bounded below, bounded above and bounded).

An additive functor F : A ⟶ A′ induces functors C^{∗}(F) : C^{∗}(A) ⟶ C^{∗}(A′) (sometimes denoted, abusively, by F)

(⋯ ⟶ X^{n−1} ⟶^{d^{n−1}} X^{n} ⟶^{d^{n}} X^{n+1} ⟶ ⋯) ↦ (⋯ ⟶ F (X^{n−1}) ⟶^{F (d^{n−1})} F (X^{n}) ⟶^{F (d^{n})} F (X^{n+1}) ⟶ ⋯) .

3.1.2. If A is an abelian category, so is each of the categories C^{∗}(A) (∗ = ∅, +, −, b), with kernels and cokernels computed termwise. This implies that a sequence of morphisms of complexes

0 ⟶ X^{•} ⟶^{f} Y^{•} ⟶^{g} Z^{•} ⟶ 0  (3.1.2.1)

is exact in C(A) iff (= if and only if) the sequences

0 ⟶ X^{n} ⟶^{f_{n}} Y^{n} ⟶^{g_{n}} Z^{n} ⟶ 0  (3.1.2.2)

are exact in A, for all n ∈ Z. We say that the sequence (3.1.2.1) in C(A) is split exact if all sequences (3.1.2.2) are split in A ( ⟺ Y^{n} = X^{n} ⊕Z^{n}). We do not require the splittings of (3.1.2.2) to be compatible with the differentials; in other words, d_{Y} ≠ d_{X} ⊕ d_{Z}, in general.

The cohomology of X^{•} ∈ C(A) in degree n ∈ Z is defined as in 1.2.3, namely as

H^{n}(X^{•}) = Ker(Coker(X^{n−1} ⟶^{d^{n−1}} X^{n}) ⟶^{d^{n}} X^{n+1}) = Coker(X^{n−1} ⟶^{d^{n−1}} Ker(X^{n} ⟶^{d^{n}} X^{n+1})) ∈ A (3.1.2.3)

(the two expressions on the R.H.S. are canonically isomorphic to each other). The cohomology is an additive functor H^{n} : C^{∗}(A) ⟶ A. As in 1.2.8, we say that a morphism f : X^{•} ⟶ Y^{•} in C(A) is a quasiisomorphism (Qis) if H^{n}(f) : H^{n}(X^{•}) ⟶ H^{n}(Y^{•}) is an isomorphism, for all n ∈ Z.

Let F : A ⟶ A′ be an additive functor with values in another abelian category A′. If F is left exact (resp., right exact), then it induces canonical morphisms in A′

H^{n}(F (X^{•})) ⟶ F (H^{n}(X^{•})),  resp., F (H^{n}(X^{•})) ⟶ H^{n}(F (X^{•}))  (X^{•} ∈ C(A))

that give rise to a morphism of functors

H^{n} ∘ C(F) ⟶ F ∘ H^{n},  resp., F ∘ H^{n} ⟶ H^{n} ∘ C(F).  (3.1.2.4)

If F is exact, we obtain canonical isomorphisms in A′

H^{n}(F (X^{•})) ⟶^{∼} F (H^{n}(X^{•})) (X^{•} ∈ C(A)).

3.1.3. The homotopy categories K^{∗}(A). Definition 1.5.8 of a homotopy between two morphisms of cohomological complexes makes sense in C(A), for any additive category A.

It is easy to see that, if two morphisms f, g : X^{•} ⟶ Y^{•} in C(A) are homotopic to each other (f ≃ g), so are fu, gu : W^{•} ⟶ Y^{•} (resp., vf, vg : X^{•} ⟶ Z^{•}), for any morphism u : W^{•} ⟶ X^{•} (resp., v : Y^{•} ⟶ Z^{•}) in C(A), and also f + w, g + w : X^{•} ⟶ Y^{•}, for any w : X^{•} ⟶ Y^{•}.

This implies that it makes sense to define, for each ∗ = ∅, +, −, b, the corresponding homotopy category of complexes K^{∗}(A) to have the same objects as C^{∗}(A), and morphisms given by

Hom_{K^{∗}(A)}(X^{•}, Y^{•}) = Hom_{C^{∗}(A)}(X^{•}, Y^{•})/ ≃

(= the set of homotopy classes of morphisms of complexes from X^{•} to Y^{•}). These categories are additive and K^{+}(A), K^{−}(A) and K^{b}(A) are full subcategories of K(A).

There are natural functors

A ⟶ C^{∗}(A) ⟶ K^{∗}(A)

(the first of which sends X ∈ A to the complex X^{•} with X^{0} = X and X^{n} = 0 for n ≠ 0; the second sends X^{•} to X^{•}). These functors embed A as a full subcategory of both C^{∗}(A) and K^{∗}(A), as

∀X, Y ∈ A Hom_{A}(X, Y) = Hom_{C(A)}(X, Y) = Hom_{K(A)}(X, Y).

More precisely, we have, for reasons of degree,

(∀X ∈ A) (∀Y^{•} ∈ C^{≥0}(A)) Hom_{C^{+}(A)}(X, Y^{•}) = Hom_{K^{+}(A)}(X, Y^{•})
(∀X ∈ A) (∀Z^{•} ∈ C^{≤0}(A)) Hom_{C^{−}(A)}(Z^{•}, X) = Hom_{K^{−}(A)}(Z^{•}, X).

An additive functor F : A ⟶ A′ with values in another additive category A′ preserves homotopies, hence induces additive functors K^{∗}(A) ⟶ K^{∗}(A′). 3.1.4. If A is an abelian category, then the statement of 1.5.8 can be rephrased by saying that the cohomology functors H^{n} : C^{∗}(A) ⟶ A factor through

C^{∗}(A) ⟶ K^{∗}(A) ⟶^{H^{n}} A.

In particular, a homotopy equivalence of complexes (= a morphism of complexes f : X^{•} ⟶ Y^{•} that admits a homotopy inverse, i.e., a morphism of complexes g : Y^{•} ⟶ X^{•} such that there are homotopies fg ≃ id, gf ≃ id) is a Qis (see 1.5.6). This is one of the reasons why it is useful to work with the homotopy category K^{∗}(A) rather than with the category of complexes C^{∗}(A). Another reason is the fact that certain constructions are not unique, but are unique up to homotopy (see 3.2 below). 3.1.5. Warning. For most abelian categories A the corresponding homotopy categories K^{∗}(A) are not abelian. For example, if A = Ab is the category of abelian groups, then the morphism Z ⟶^{2} Z does not have cokernel in K(Ab). 3.1.6. Truncations. Let A be an abelian category. The naïve truncations of a complex

X^{•} = [⋯ ⟶ X^{n−1} ⟶ X^{n} ⟶ X^{n+1} ⟶ ⋯] ∈ C(A)

are the complexes

\begin{matrix}σ_{≥n} X^{•} = [⋯ ⟶ 0 ⟶ 0 ⟶ & X^{n} ⟶ X^{n+1} ⟶ ⋯] \\ σ_{≤n} X^{•} = [⋯ ⟶ X^{n−1} ⟶ & X^{n} ⟶ 0 ⟶ 0 ⟶ ⋯] \end{matrix}  (3.1.6.1)

(note that σ_{≥n} X^{•} is a subcomplex of X^{•} and the quotient complex X^{•}/σ_{≥n} X^{•} is canonically isomorphic to σ_{≤n−1} X^{•}). More intelligent truncations are given by the formulas

\begin{aligned}τ_{≥n} X^{•} & = [⋯ ⟶ 0 ⟶ X^{n}/dX^{n−1} ⟶ X^{n+1} ⟶ ⋯] \\ τ_{≤n} X^{•} & = [⋯ ⟶ X^{n−1} ⟶ Z^{n}(X^{•}) ⟶ 0 ⟶ 0 ⟶ ⋯] \end{aligned}  (3.1.6.2)

Note that τ_{≥n} X^{•} (resp., τ_{≤n} X^{•}) is a quotient complex (resp., a subcomplex) of X^{•} and

H^{m}(τ_{≥n} X^{•}) = \{ \begin{matrix}H^{m}(X^{•}), & m ≥ n \\ 0, & m < n, \end{matrix}  H^{m}(τ_{≤n} X^{•}) = \{ \begin{matrix}H^{m}(X^{•}), & m ≤ n \\ 0, & m > n. \end{matrix} (3.1.6.3)

3.2. Derived functors (an informal introduction)

3.2.1. Assume that F : A ⟶ A′ is a right exact additive functor between two abelian categories. It is often the case that, for any exact sequence in A

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0  (3.2.1.1)

one can naturally extend the exact sequence in A′

F (X) ⟶ F (Y) ⟶ F (Z) ⟶ 0

to a long exact sequence

⋯ ⟶ L_{2}F (Z) ⟶ L_{1}F (X) ⟶ L_{1}F (Y) ⟶ L_{1}F (Z) ⟶ F (X) ⟶ F (Y) ⟶ F (Z) ⟶ 0, (3.2.1.2)

for suitable additive functors L_{n}F : A ⟶ A′. 3.2.2. Example. Fix an integer m > 1 and consider the functor

F = • ⊗_{Z} Z/mZ : Ab ⟶ Ab, X ↦ X ⊗_{Z} Z/mZ = X/mX.

As we shall see in ?? below, the exact sequence (2.1.5.1)

0 ⟶ _{m}X ⟶ _{m}Y ⟶ _{m}Z ⟶ X/mX ⟶ Y/mY ⟶ Z/mZ ⟶ 0

is a special case of (3.2.1.2). 3.2.3. If the category A has enough projectives (in the sense that, for each X ∈ A, there exists an epimorphism P ⟶ X with P ∈ P_{A}), then the functors L_{n}F (the left derived functors of F) can be defined as follows: each object X ∈ A admits a projective resolution, namely a complex of projectives P^{•} ∈ C^{≤0}(P_{A}) equipped with a Qis P^{•} ⟶ X; then

L_{n}F (X) = H^{−n}(F (P^{•})).  (3.2.3.1)

Independence of L_{n}F (X) on the choice of P^{•} and its functoriality in X follows from the fact that two projective resolutions of X are homotopically equivalent. More precisely, the rule X ↦ P^{•} defines a “resolution functor” A ⟶ K^{−}(P_{A}) and L_{n}F can be defined as the composition of functors

L_{n}F : A ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′) ⟶^{H^{−n}} A′.  (3.2.3.2)

If X ∈ P_{A} is projective, then we can take P^{•} = X, hence L_{n}F (X) = 0 for all n > 0. As we shall see in ?? below, the left derived functors of the functor F from 3.2.2 are equal to L_{1}F (X) = _{m}X and L_{n}F = 0 for n > 1. 3.2.4. In fact, the construction from 3.2.3 can be carried out not only for objects of A, but for bounded below complexes X^{•} ∈ C^{−}(A) (still assuming that A has enough projectives): each such a complex X^{•} admits a Qis P^{•} ⟶ X^{•}, where P^{•} ∈ C^{−}(P_{A}) is a bounded above complex of projectives, and the rule X^{•} ↦ P^{•} defines a functor C^{−}(A) ⟶ K^{−}(P_{A}). The functors

L_{n}F : C^{−}(A) ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′) ⟶^{H^{−n}} A′  (3.2.4.1)

are the left hyper-derived functors of F. For any short exact sequence of complexes in C^{−}(A)

0 ⟶ X^{•} ⟶ Y^{•} ⟶ Z^{•} ⟶ 0

there is a long exact sequence in A′

⋯ ⟶ L_{n}F (X^{•}) ⟶ L_{n}F (Y^{•}) ⟶ L_{n}F (Z^{•}) ⟶ L_{n−1}F (X^{•}) ⟶ · · · .  (3.2.4.2)

Moreover,

H^{n}(X^{•}) = 0 ∀n > m ⟹ L_{n}F (X^{•}) = 0 ∀n < −m.  (3.2.4.3)

The functor

C^{−}(A) ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′)

factors through the homotopy category K^{−}(A); the corresponding functor

“LF ” : K^{−}(A) ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′)  (3.2.4.4)

is very close to the left derived functor LF of F in the sense of derived categories. 3.2.5. All of the above works thanks to the following key properties of the subcategory C^{−}(P_{A}) ⊂ C^{−}(A): 3.2.5.1. ∀X^{•} ∈ C^{−}(A) ∃P^{•} ∈ C^{−}(P_{A}) ∃ Qis P^{•} ⟶ X^{•}. 3.2.5.2. If f : P^{•} ⟶ Q^{•} is a Qis in C^{−}(P_{A}), then F (f) : F (P^{•}) ⟶ F (Q^{•}) is a Qis in C^{−}(A′). 3.2.5.3. If f : X^{•} ⟶ Y^{•} is a morphism in C^{−}(A) and u : P^{•} ⟶ X^{•}, v : Q^{•} ⟶ Y^{•} are Qis with P^{•}, Q^{•} ∈ C^{−}(P_{A}), then there exists a morphism of complexes f′ : P^{•} ⟶ Q^{•} such that vf′ is homotopic to fu. Moreover, f′ is unique up to homotopy. These properties, as well as (3.2.4.2-3), will be established in ?? below. 3.2.6. There is a dual theory of right derived functors for left exact functors F : A ⟶ A′, assuming that A has enough injectives (i.e., that for each X ∈ A there exists a monomorphism X ⟶ I into an injective object I ∈ I_{A}). The properties 3.2.5.1-3 of the opposite category A^{op} imply that 3.2.6.1. ∀X^{•} ∈ C^{+}(A) ∃I^{•} ∈ C^{+}(I_{A}) ∃ Qis X^{•} ⟶ I^{•}. 3.2.6.2. If f : I^{•} ⟶ J^{•} is a Qis in C^{+}(I_{A}), then F (f) : F (I^{•}) ⟶ F (J^{•}) is a Qis in C^{+}(A′). 3.2.6.3. If f : X^{•} ⟶ Y^{•} is a morphism in C^{+}(A) and u : X^{•} ⟶ I^{•}, v : Y^{•} ⟶ J^{•} are Qis with I^{•}, J^{•} ∈ C^{+}(I_{A}), then there exists a morphism of complexes f′ : I^{•} ⟶ J^{•} such that f′u is homotopic to vf. Moreover, f′ is unique up to homotopy. 3.2.7. Still assuming that A has enough injectives, the properties 3.2.6.1-3 imply that the rule X^{•} ↦ I^{•} defines a functor C^{+}(A) ⟶ K^{+}(I_{A}). As in 3.2.3-4 we obtain the right hyper-derived functors of F

R^{n}F : C^{+}(A) ⟶ K^{+}(I_{A}) ⟶^{F} K^{+}(A′) ⟶^{H^{n}} A′,  R^{n}F (X^{•}) = H^{n}(F (I^{•}))  (3.2.7.1)

giving rise to long exact sequences

⋯ ⟶ R^{n}F (X^{•}) ⟶ R^{n}F (Y^{•}) ⟶ R^{n}F (Z^{•}) ⟶ R^{n+1}F (X^{•}) ⟶ ⋯,  (3.2.7.2)

the right derived functors of F

R^{n}F : A ⟶ C^{+}(A) ⟶ K^{+}(I_{A}) ⟶^{F} K^{+}(A′) ⟶^{H^{n}} A′  (3.2.7.3)

giving rise to long exact sequences

0 ⟶ F (X) ⟶ F (Y) ⟶ F (Z) ⟶ R^{1}F (X) ⟶ R^{1}F (Y) ⟶ R^{1}F (Z) ⟶ R^{2}F (X) ⟶ ⋯ (3.2.7.4)

(for any exact sequence (3.2.1.1)) and the functors

“RF ” : K^{+}(A) ⟶ K^{+}(I_{A}) ⟶^{F} K^{+}(A′).  (3.2.7.5)

3.2.8. In fact, everything in 3.2.4 (resp., in 3.2.7) works for an arbitrary additive functor F : A ⟶ A′. However, if F is not right exact (resp., left exact), then L_{0}F ≠ F (resp., R^{0}F ≠ F) and F has to be replaced by L_{0}F (resp., by R^{0}F) in (3.2.1.2) (resp., in (3.2.7.4)).

3.3. The mapping cone

Let A be an additive category. In this section we explain one of the fundamental constructions in C(A), which allows to deduce properties of morphisms in C(A) from properties of objects (if A is abelian).

3.3.1. The shift of a complex. For X ∈ C(A) and n ∈ Z, we denote by X[n] ∈ C(A) the following complex: X[n]^{i} = X^{i+n} and d_{X[n]}^{i} = (−1)^{n} d_{X}^{i+n}. If f : X ⟶ Y is a morphism in C(A), we define f[n] : X[n] ⟶ Y [n] by f[n]_{i} = f_{i+n} : X[n]^{i} = X^{i+n} ⟶ Y [n]^{i} = Y^{i+n}. If A is abelian, then H^{i}(X[n]) = H^{i+n}(X). 3.3.2. The mapping cone in topology. The cone over a topological space X is defined as the quotient CX = X × I/ ∼ of the cylinder X × I (where I = [0, 1]) by the equivalence relation

(x, t) ∼ (x′, t′) ⟺ t = t′ = 1 or (x, t) = (x′, t′).

We equip CX with the quotient topology and we view X as the bottom of CX via the inclusion map i_{0} : X ↪ CX, x ↦ (x, 0).

The mapping cone Cf of a continuous map f : X ⟶ Y between topological spaces is obtained by glueing the space Y to the bottom of CX via the map f. In other words, Cf is defined as the inductive limit of the pair of arrows CX ⟵^{i_{0}} X ⟶^{f} Y in the category Top of topological spaces. This means that Cf sits in a commutative diagram

\begin{matrix}CX & ⟶ & Cf \\ ↑ ∪ & & ↑ \\ X & ⟶^{f} & Y \end{matrix}  (3.3.2.1)

and is universal among spaces having this property. In concrete terms, Cf = (CX ∐ Y)/ ∼′ is the quotient of the disjoint union of CX and Y by the equivalence relation generated by (x, 0) ∼′ f(x) (x ∈ X). 3.3.3. Examples. (1) If X ⊂ R^{N} (with the induced topology), set c = (0, …, 0, 1) ∈ R^{N} ⊕ R = R^{N+1} and denote by C_{geom}X = {tc + (1 − t)x | x ∈ X, λ ∈ [0, 1]} ⊂ R^{N+1} the geometric cone over X (with the topology induced from R^{N+1}). The map

C_{geom} ⟶ CX, tc + (1 − t)x ↦ the class of (x, t)

is a continuous bijection, which is a homeomorphism if X is compact, but not in general, for example, for X = R^{N} (exercise!).

(2) The cone (and the geometric cone) over the unit sphere S^{n} = {x ∈ R^{n+1} | ∥x∥^{2} = x_{1}^{2} + ⋯ + x_{n+1}^{2} = 1} is homeomorphic to the closed unit ball D^{n+1} = {x ∈ R^{n+1} | ∥x∥^{2} ≤ 1} and also to the upper hemisphere S_{+}^{n+1} = {x ∈ R^{n+2} | ∥x∥^{2} = 1, x_{n+2} ≥ 0}.

(3) The real projective space P^{n}(R) = S^{n}/{±1} can be obtained from the sphere by identifying each point x ∈ S^{n} with its antipode −x. Any element of P^{n}(R) can be represented by a point on the upper hemisphere x ∈ S_{+}^{n}, and x ≠ y ∈ S_{+}^{n} have the same image in P^{n}(R) iff y = −x ∈ ∂S_{+}^{n} = S^{n−1} × {0}. Denote by f_{n} : S^{n} ⟶ P^{n}(R) the canonical projection; the previous discussion implies that Cf_{n} is homeomorphic to P^{n+1}(R). 3.3.4. The mapping cone of a simplicial map. If X = |K| for some (finite) simplicial complex K in R^{N}, then CX = C_{geom}X = |CK|, where CK is the following simplicial complex in R^{N+1}:

(CK)_{n} = K_{n} ∪ {Cσ | σ ∈ K_{n−1}}, C[v_{0}, …, v_{n−1}] = [c, v_{0}, …, v_{n−1}]

(using the notation from 3.3.3(1)). If K is ordered, we order CK by c < v for any v ∈ K_{0}; then

∂(Cσ) = −C(∂σ) + σ, σ = [v_{0} < ⋯ < v_{n−1}] ∈ K_{n−1},

which implies that

C_{n}(CK, A) = C_{n−1}(K, A) ⊕ C_{n}(K, A),  ∂_{n}(σ, τ) = (−∂_{n−1}(σ), σ + ∂_{n}(τ)).  (3.3.4.1)

Assume, in addition, that Y = |L| for some (finite) simplicial complex L and f : |K| ⟶ |L| is a simplicial map (see 1.3.10 above). Even though the glued object Cf will not necessarily come from from a simplicial complex, it is natural to define C_{•}(Cf, A) as the inductive limit of the pair of morphisms of complexes C_{•}(CK, A) ⟵^{i_{0}} C_{•}(K, A) ⟶^{f} C_{•}(L, A). The formula (3.3.4.1) implies that

C_{n}(Cf, A) = C_{n−1}(K, A) ⊕ C_{n}(L, A),  ∂(σ, τ) = (−∂(σ), f(σ) + ∂(τ)),  (3.3.4.2)

where f(σ) is defined as in 1.3.10. 3.3.5. The mapping cone of a morphism of complexes. After passing to a cohomological notation, the formula (3.3.4.2) yields the following abstract algebraic definition: let f : X ⟶ Y be a morphism in C(A), where A is an additive category. The mapping cone of f is the following complex Cone(f) ∈ C(A):

Cone(f)^{n} = X^{n+1} ⊕ Y^{n} ⟶ Cone(f)^{n+1} = X^{n+2} ⊕ Y^{n+1}, d(x, y) = (−dx, f(x) + dy).

Alternatively, writing the elements of Cone(f)^{n} = X^{n+1} ⊕ Y^{n} = X[1]^{n} ⊕ Y^{n} as column vectors, we have

d_{Cone(f)} ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}−d_{X} & 0 \\ f & d_{Y} \end{matrix} ) ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}d_{X[1]} & 0 \\ f & d_{Y} \end{matrix} ) ( \begin{matrix}x \\ y \end{matrix} ) .

Note that Cone(f) is, indeed, a complex, as

dd(x, y) = d(−dx, f(x) + dy) = (ddx, f(−dx) + df(x) + ddy) = 0.

If F : A ⟶ A′ is an additive functor into another additive category, then F (Cone(f)) = Cone(F (f)).

3.3.6. Proposition. Let f : X ⟶ Y be a morphism in C(A), where A is an additive category. The formulas i(y) = (0, y) and p(x, y) = x define morphisms of complexes i : Y ⟶ Cone(f) and p : Cone(f) ⟶ X[1] satisfying pi = 0. If A is an abelian category, then i and p give rise to a split short exact sequence of complexes

0 ⟶ Y ⟶^{i} Cone(f) ⟶^{p} X[1] ⟶ 0.

For each n ∈ Z, the coboundary morphism H^{n−1}(X[1]) = H^{n}(X) ⟶^{∂} H^{n}(Y) is equal to the morphism H^{n}(f) induced by f.

Proof. The split exactness is clear, once we verify that both i and p are morphisms of complexes, which boils down to the formulas di(y) = d(0, y) = (0, dy) = id(y) and dp(x, y) = d_{X[1]}(x) = p(d_{X[1]}(x), f(x) + dy) = pd(x, y). Finally, for any x ∈ X^{n} satisfying dx = 0, we have

∂[x] = [i^{−1}d_{Cone(f)}p^{−1}(x)] = [i^{−1}d(x, 0)] = [i^{−1}(0, f(x)] = [f(x)] = H^{n}(f)[x].

3.3.7. Corollary. There is an exact sequence

⋯ ⟶ H^{n−1}(Cone(f)) ⟶^{H^{n}(p)} H^{n}(X) ⟶^{H^{n}(f)} H^{n}(Y) ⟶^{H^{n}(i)} H^{n}(Cone(f)) ⟶ ⋯

3.3.8. Corollary. f is a Qis ( ⟺ H^{n}(f) is an isomorphism for all n ∈ Z) ⟺ Cone(f) is acyclic ( ⟺ H^{n}(Cone(f)) = 0 for all n ∈ Z).

3.3.9. Functoriality of Cone(f). Let

X ⟶^{f} Y
↓^{g}  ↓^{h}
X′ ⟶^{f′} Y′

be a diagram of morphisms in C(A) (where A is an additive category) which is commutative up to homotopy. Fix a homotopy a between f′g and hf (i.e., a collection of morphisms a_{n} : X^{n} ⟶ Y^{′n−1} in A such that da + ad = hf − f′g). The formula

Cone(f)^{n} = X^{n+1} ⊕ Y^{n} ⟶ Cone(f′)^{n} = X^{′n+1} ⊕ Y^{′n}, (x, y) ↦ (g_{n+1}(x), a_{n+1}(x) + h_{n}(y))

then defines a morphism of complexes

λ = Cone(g, h, a) = ( \begin{matrix}g & 0 \\ a & h \end{matrix} ) : Cone(f) ⟶ Cone(f′)  (3.3.9.1)

which fits into the following commutative diagram:

0 ⟶ Y ⟶ Cone(f) ⟶ X[1] ⟶ 0
↓^{h}  ↓^{λ}  ↓^{g[1]}
0 ⟶ Y′ ⟶ Cone(f′) ⟶ X′[1] ⟶ 0.

(3.3.9.2)

In particular, if both g and h are Qis, so is λ (by 2.1.10). If a′ is another homotopy between f′g and hf, the morphism Cone(g, h, a′) is not necessarily homotopic to Cone(g, h, a) (the two morphisms are homotopic, for example, if there exists a “second order homotopy” between the homotopies a and a′). As a result, Cone does not define a functor from Mor(K(A)) (the category of morphisms in K(A)) to K(A), which causes problems in the theory of derived categories. 3.3.10. Morphisms Cone(f) ⟶ Z. In the topological situation of (3.3.2.1), a continuous map Cf ⟶ Z (where Z is an arbitrary topological space) is defined by a pair of continuous maps

g : Y ⟶ Z,  h : CX ⟶ Z,  gf = hi_{0}.

The latter condition simply says that h defines a homotopy between gf : X ⟶ Z and a constant map.

In the algebraic situation of 3.3.5, a morphism of complexes Cone(f) ⟶ Z (where Z ∈ C(A)) is given by a collection of maps

(h_{n+1}, g_{n}) : Cone(f)^{n} = X^{n+1} ⊕ Y^{n} ⟶ Z^{n}, (x, y) ↦ h_{n+1}(x) + g_{n}(y)

that commute with the differential, i.e., such that

(dh, dg) = d(h, g) = (h, g) ( \begin{matrix}−d & 0 \\ f & d \end{matrix} ) = (−hd + gf, gd) ⟺ dg = gd, dh + hd = gf.

In other words, g : Y ⟶ Z is a morphism of complexes and h is a homotopy between 0 and gf : X ⟶ Z. If we denote the corresponding morphism Cone(f) ⟶ Z by (h, g), then g = (h, g) ∘ i, in the notation of 3.3.6.

The previous discussion can be summed up by saying that a morphism of complexes g : Y ⟶ Z factors through i : Y ⟶ Cone(f) iff gf : X ⟶ Z is homotopic to 0. This can be reformulated in more abstract terms as follows:

3.3.11. Proposition. Let f : X ⟶ Y be a morphism in C(A), where A is an additive category. Then, for any Z ∈ C(A), the sequences of abelian groups

Hom_{C(A)}(Cone(f), Z) ⟶^{∘i} Hom_{C(A)}(Y, Z) ⟶^{∘f} Hom_{K(A)}(X, Z)

Hom_{K(A)}(Cone(f), Z) ⟶^{∘i} Hom_{K(A)}(Y, Z) ⟶^{∘f} Hom_{K(A)}(X, Z)

are exact.

3.3.12. The mapping cylinder in topology. In the situation of 3.3.2, the cylinder over a topological space X is the product X ×I, and the mapping cylinder of a continuous map f : X ⟶ Y is the inductive limit of the pair of arrows X × I ⟵^{i_{0}} X ⟶^{f} Y; it sits in a commutative diagram

X × I ⟶ Cyl(f)
↑ ∪  ↑
X  ⟶^{f}  Y

and is universal among spaces having this property. The inclusion Y ↪ Cyl(f) is a homotopy equivalence, with homotopy inverse defined by the pair fp : X × I ⟶ Y, id : Y ⟶ Y (where p : X × I ⟶ X is the projection map p(x, t) = x). Moreover, the composite map X ⟶^{i_{1}} X × I ⟶ Cyl(f) (where i_{1}(x) = (x, 1)) is injective. As the cylinder X × I differs from the cone CX only at the top level X × {1}, it is natural to translate the topological construction of Cyl(f) into the algebraic context of 3.3.5. 3.3.13. The mapping cylinder of a morphism of complexes. Let f : X ⟶ Y be a morphism in C(A), where A is an additive category. The mapping cylinder of f is the following complex Cyl(f) ∈ C(A):

Cyl(f)^{n} = X^{n}⊕X^{n+1}⊕Y^{n} ⟶ Cyl(f)^{n+1} = X^{n+1}⊕X^{n+2}⊕Y^{n+1}, d(x′, x, y) = (dx′−x, −dx, f(x)+dy).

3.3.14. Proposition. Let f : X ⟶ Y be a morphism in C(A), where A is an additive category. The formulas \overline{f}(x) = (x, 0, 0), π(x′, x, y) = (x, y), α(y) = (0, 0, y), β(x′, x, y) = f(x′) + y, h(x′, x, y) = (0, x′, 0) define a commutative diagram in C(A)

0 ⟶  Y  ⟶^{i} Cone(f) ⟶^{p} X[1] ⟶ 0
↓^{α}  ∥
0 ⟶ X ⟶^{\overline{f}} Cyl(f) ⟶^{π} Cone(f) ⟶ 0
∥  ↓^{β}
X ⟶^{f}  Y

and a homotopy h : id_{Cyl(f)} ≃ αβ; moreover, βα = id_{Y} (hence β is a homotopy inverse of α). If A is an abelian category, then the first two rows of the diagram are exact and the morphisms α, β are Qis.

Proof. An easy calculation shows that all the maps are morphisms of complexes, the diagram is commutative and h is a homotopy between id and αβ. If A is abelian, the exactness of the second row follows from the definitions; as α, β are homotopy inverse to each other, they are both Qis (see the proof of 1.5.6).

3.3.15. Proposition. Let A be an abelian category. If 0 ⟶ X ⟶^{f} Y ⟶^{g} Z ⟶ 0 is an exact sequence in C(A), then there is a commutative diagram with exact rows, in which all vertical arrows are Qis:

0 ⟶ X ⟶^{f}  Y  ⟶^{g}  Z  ⟶ 0
∥  ↑^{β}  ↑^{γ}
0 ⟶ X ⟶^{\overline{f}} Cyl(f) ⟶^{π} Cone(f) ⟶ 0

Proof. The formula γ(x, y) = g(y) defines a morphism of complexes γ : Cone(f) ⟶ Z, as γd(x, y) = γ(−dx, f(x) + dy) = gf(x) + g(dy) = g(dy) = dγ(x, y). The first (resp., the second) square is commutative, by 3.3.14 (resp., since γπ(x′, x, y) = γ(x, y) = g(y) = g(f(x′) + y) = gβ(x′, x, y)). As id_{X} and β are Qis (the latter by 3.3.14), the Five Lemma implies that γ is also a Qis. 3.3.16. Functoriality of Cyl(f). In the situation of 3.3.9, the formula

Cyl(f)^{n} = X^{n}⊕X^{n+1}⊕Y^{n} ⟶ Cyl(f′)^{n} = X^{′n}⊕X^{′n+1}⊕Y^{′n}, (x, y) ↦ (g_{n}(x), g_{n+1}(x), a_{n+1}(x)+h_{n}(y))

defines a morphism of complexes

Cyl(g, h, a) = ( \begin{matrix}g & 0 & 0 \\ 0 & g & 0 \\ 0 & a & h \end{matrix} ) : Cyl(f) ⟶ Cyl(f′)  (3.3.16.1)

which makes, together with the morphism (3.3.9.1), the following diagram commutative:

0 ⟶ X ⟶^{\overline{f}} Cyl(f) ⟶^{π} Cone(f) ⟶ 0
↓^{g}  ↓  ↓
0 ⟶ X′ ⟶^{\overline{f}′} Cyl(f′) ⟶^{π′} Cone(f′) ⟶ 0.

(3.3.16.2)

3.4. Double complexes

Let A be an additive category.

3.4.1. Definition. A double complex (= bicomplex) K^{•,•} in A is a collection of objects K^{i,j} ∈ A (i, j ∈ Z) and morphisms d_{I}^{ij} : K^{i,j} ⟶ K^{i+1,j}, d_{II}^{ij} : K^{i,j} ⟶ K^{i,j+1} such that d_{I}^{2} = 0, d_{II}^{2} = 0, d_{I}d_{II} + d_{II}d_{I} = 0. 3.4.2. The total complex of a bicomplex (two versions). (1) The small total complex of K^{•,•} is the complex L^{•} = Tot^{⊕}(K^{•,•}), where L^{n} = ⨁_{i+j=n} K^{i,j} (assuming the direct sum exists in A) and the differential is equal to d = d_{I} + d_{II}.

(2) The big total complex of K^{•,•} is the complex M^{•} = Tot^{Π}(K^{•,•}), where M^{n} = ∏_{i+j=n} K^{i,j} (assuming the direct product exists in A) and the differential is equal to d = d_{I} + d_{II}. 3.4.3. Example (the Dolbeaut bicomplex). Let X be a complex manifold of dimension n (for example, an non-empty open subset of C^{n}) with local cordinates z_{1}, …, z_{n}. The Dolbeaut bicomplex of X is formed by smooth complex-valued differential forms on X with a fixed number of holomorphic and anti-holomorphic terms:

A^{p,q}(X) = \{ ∑_{{|I|=p \atop |J|=q}} f_{IJ} dz_{I} ∧ d\overline{z}_{J} | f_{IJ} ∈ C^{∞}(X, C)\},  d_{I} = ∂,  d_{II} = \overline{∂},

where, for I = {i_{1} < ⋯ < i_{p}} ⊆ {1, …, n},

dz_{I} = dz_{i_{1}} ∧ ⋯ ∧ dz_{i_{p}}, d\overline{z}_{I} = d\overline{z}_{i_{1}} ∧ ⋯ ∧ d\overline{z}_{i_{p}},

∂(f dz_{I} ∧ d\overline{z}_{J}) = ∑_{i=1}^{n} \frac{∂f}{∂z_{i}} dz_{i} ∧ dz_{I} ∧ d\overline{z}_{J}, \overline{∂}(f dz_{I} ∧ d\overline{z}_{J}) = ∑_{i=1}^{n} \frac{∂f}{∂\overline{z}_{i}} d\overline{z}_{i} ∧ dz_{I} ∧ d\overline{z}_{J}.

The corresponding total complex Tot^{Π}(A^{•,•}(X)) = Tot^{⊕}(A^{•,•}(X)) ⟶^{∼} A^{•}(X)⊗_{R}C is canonically isomorphic to the de Rham complex of X with complex coefficients, as

A^{n}(X) ⊗_{R} C = ⨁_{p+q=n} A^{p,q}(X), d = ∂ + \overline{∂}.

3.4.4. Another sign convention. One can often find in the literature an alternative sign convention for double complexes, namely that of morphisms d̃_{I}^{ij} : K^{i,j} ⟶ K^{i+1,j}, d̃_{II}^{ij} : K^{i,j} ⟶ K^{i,j+1} satisfying d̃_{I}^{2} = 0, d̃_{II}^{2} = 0, d̃_{I} d̃_{II} = d̃_{II} d̃_{I}. One can pass between 3.4.1 and 3.4.4 by putting, for example,

d_{I} = d̃_{I},  d_{II}^{i,j} = (−1)^{i} d̃_{II}^{i,j}.  (3.4.4.1)

3.4.5. The tensor product of complexes. The tensor product X^{•} ⊗_{R} Y^{•} ∈ C(_{Z(R)}Mod) of two complexes X^{•} ∈ C(Mod_{R}) and Y^{•} ∈ C(_{R}Mod) is the small total complex associated to the double complex

(X^{i} ⊗_{R} Y^{j}, d̃_{I} = d_{X} ⊗ id, d̃_{II} = id ⊗ d_{Y})

using the sign rule (3.4.4.1):

(X^{•} ⊗_{R} Y^{•})^{n} = ⨁_{i+j=n} X^{i} ⊗_{R} Y^{j},  d(x ⊗ y) = dx ⊗ y + (−1)^{i} x ⊗ dy  (x ∈ X^{i}).  (3.4.5.1)

If f : X^{•} ⟶ X^{′•} is a morphism in C(Mod_{R}), then the formula (x, x′) ⊗ y ↦ (x ⊗ y, x′ ⊗ y) defines an isomorphism of complexes

Cone(X^{•} ⟶^{f} X^{′•}) ⊗_{R} Y^{•} ⟶^{∼} Cone(X^{•} ⊗_{R} Y^{•} ⟶^{f⊗id} X^{′•} ⊗_{R} Y^{•}).  (3.4.5.2)

If the ring R is commutative, then Mod_{R} = _{R}Mod and the formula

x ⊗ y ↦ (−1)^{ij} y ⊗ x (x ∈ X^{i}, y ∈ Y^{j})

defines an isomorphism of complexes

s_{12} : X^{•} ⊗_{R} Y^{•} ⟶^{∼} Y^{•} ⊗_{R} X^{•}.  (3.4.5.3)

3.4.6. The Hom^{•}-complex. For X^{•}, Y^{•} ∈ C(A) (where A is an additive category) we define

HomA^{•}(X^{•}, Y^{•}) = Tot^{Π}((Hom_{A}(X^{−i}, Y^{j}))_{i,j}) ∈ C(Ab)

with the following sign rules:

f ∈ Hom_{A}^{n}(X^{•}, Y^{•}) = ∏_{i∈Z} Hom_{A}(X^{i}, Y^{i+n}),  (df)(x) = d(f(x)) + (−1)^{n−1}f(dx).  (3.4.6.1)

If g : Y^{•} ⟶ Z^{•} is a morphism in C(A), then the canonical isomorphisms

Hom_{A}(X^{i}, Cone(g)^{i+n}) = Hom_{A}(X^{i}, Y^{i+n+1} ⊕ Z^{i+n}) ⟶^{∼} Hom_{A}(X^{i}, Y^{i+n+1}) ⊕ Hom_{A}(X^{i}, Z^{i+n})

define an isomorphism of complexes

HomA^{•}(X^{•}, Cone(Y^{•} ⟶^{g} Z^{•})) ⟶^{∼} Cone(HomA^{•}(X^{•}, Y^{•}) ⟶^{g∘} HomA^{•}(X^{•}, Z^{•})).  (3.4.6.2)

3.4.7. Proposition. Let X^{•}, Y^{•} ∈ C(A) be complexes in an additive category A. Then

Z^{0}(HomA^{•}(X^{•}, Y^{•})) = Hom_{C(A)}(X^{•}, Y^{•})
B^{0}(HomA^{•}(X^{•}, Y^{•})) = {f ∈ Hom_{C(A)}(X^{•}, Y^{•}) | f ≃ 0},

hence

H^{0}(HomA^{•}(X^{•}, Y^{•})) = Hom_{K(A)}(X^{•}, Y^{•}).

Proof. An element f ∈ Hom_{A}^{0}(X^{•}, Y^{•}) = ∏_{i∈Z} Hom_{A}(X^{i}, Y^{i}) satisfies df = 0 iff d(f(x)) − f(dx) = 0, for all x ∈ X^{i}, which proves the first statement. The second statement follows from the fact that, for h ∈ Hom_{A}^{−1}(X^{•}, Y^{•}) = ∏_{i∈Z} Hom_{A}(X^{i}, Y^{i−1}) and x ∈ X^{i}, we have (dh)(x) = d(h(x)) + h(dx).

3.4.8. Corollary. For n ∈ Z,

H^{n}(HomA^{•}(X^{•}, Y^{•})) = Hom_{K(A)}(X^{•}, Y^{•}[n]).

Proof. There is a canonical isomorphism HomA^{•}(X^{•}, Y^{•}[n]) ⟶^{∼} HomA^{•}(X^{•}, Y^{•})[n] (which, unlike (3.4.5.3), does not involve any signs); we apply 3.4.7 to Y^{•}[n] instead of Y^{•}.

3.5. Left derived (and hyper-derived) functors

In this section we make rigorous the discussion from 3.2, by establishing the properties 3.2.5.1-3. The key points are proved in 3.5.4 and 3.5.5, the rest is fairly straightforward. Let A be an abelian category. As in 2.2.9, we denote by P_{A} the full additive subcategory of A consisting of the projective objects of A. Recall that we say that A has enough projectives if for every X ∈ A there exists an epimorphism P ⟶ X in A with P ∈ P_{A}.

3.5.1. Proposition. If a complex P^{•} ∈ C^{−}(P_{A}) satisfies ∀m > n H^{m}(P^{•}) = 0, then τ_{≤n} P^{•} ∈ C^{−}(P_{A}) (note that the inclusion τ_{≤n} P^{•} ↪ P^{•} is a Qis, by (3.1.6.3)).

Proof. We have P^{•} ∈ C^{≤n+N}(P_{A}) for some N ≥ 1. As τ_{≤n} P^{•} = τ_{≤n} ⋯ τ_{≤n+N−1} P^{•}, we can assume, using an inductive argument, that N = 1. By definition,

\begin{aligned}P^{•} & = [⋯ ⟶ P^{n−1} ⟶ P^{n} ⟶^{d^{n}} P^{n+1} ⟶ 0 ⟶ ⋯] \\ τ_{≤n} P^{•} & = [⋯ ⟶ P^{n−1} ⟶ Z^{n} ⟶ 0 ⟶ 0 ⟶ ⋯], \end{aligned}

where Z^{n} = Ker(d^{n}). We must show that Z^{n} is projective, using the assumption that H^{n+1}(P^{•}) = 0. Indeed, the sequence

0 ⟶ Z^{n} ⟶ P^{n} ⟶^{d^{n}} P^{n+1} ⟶ 0 (= H^{n+1}(P^{•}))

is exact; as P^{n+1} is projective, the sequence splits, hence P^{n} = Z^{n} ⊕ P^{n+1}. As P^{n} is projective, so is Z^{n}.

3.5.2. Corollary-Definition. If there is a Qis f : P^{•} ⟶ X, where X ∈ A and P^{•} ∈ C^{−}(P_{A}), then Q^{•} = τ_{≤0} P^{•} ∈ C^{≤0}(P_{A}) and the composite morphism g : Q^{•} ↪ P^{•} ⟶^{f} X is also a Qis. In concrete terms, g_{0} : Q^{0} ⟶ X is a morphism in A and the sequence

⋯ ⟶ Q^{−2} ⟶ Q^{−1} ⟶ Q^{0} ⟶^{g_{0}} X ⟶ 0

is exact (we say that Q^{•} is a projective resolution of X).

3.5.3. Proposition. If A has enough projectives, then each X ∈ A has a projective resolution.

Proof. There is an epimorphism f_{0} : P^{0} ⟶ X with P^{0} ∈ P_{A}, which gives rise to an exact sequence 0 ⟶ Ker(f_{0}) ⟶ P^{0} ⟶^{f_{0}} X. Similarly, there is an epimorphism f_{1} : P^{−1} ⟶ Ker(f_{0}) with P^{−1} ∈ P_{A}, which sits in an exact sequence 0 ⟶ Ker(f_{1}) ⟶ P^{−1} ⟶^{f_{1}} Ker(f_{0}) ⟶ 0. Continuing this process, we obtain short exact sequences 0 ⟶ Ker(f_{n}) ⟶ P^{−n} ⟶^{f_{n}} Ker(f_{n−1}) ⟶ 0 with P^{−n} ∈ P_{A}, which give rise (as in 2.1.3(8)) to a long exact sequence ⋯ ⟶ P^{−3} ⟶ P^{−2} ⟶ P^{−1} ⟶ P^{0} ⟶ X ⟶ 0, in which all P^{−n} are projective.

3.5.4. Proposition. If A has enough projectives, then for each X^{•} ∈ C^{−}(A) there exists a Qis P^{•} ⟶ X^{•}, where P^{•} ∈ C^{−}(P_{A}).

Proof. After replacing X^{•} by X^{•}[n] we can assume that X^{•} ∈ C^{≤0}(A). We are going to construct a Qis f : P^{•} ⟶ X^{•}, where P^{•} ∈ C^{≤0}(P_{A}). We use the following sufficient conditions for f to be a Qis:

(A_{q})  Coker(Z^{q}(P^{•}) ⟶ Z^{q}(X^{•})) = 0

(which implies that H^{q}(P^{•}) ⟶ H^{q}(X^{•}) is an epimorphism) and

(B_{q})  Coker(P^{q−1} ⟶^{j_{q−1}} X^{q−1} ×_{Z^{q}(X^{•})} Z^{q}(P^{•})) = 0.

We claim that (B_{q}) implies that H^{q}(P^{•}) ⟶ H^{q}(X^{•}) is a monomorphism. Let us first explain the morphism j_{q−1} (we consider A as a full subcategory of a suitable category of R-modules, so it makes sense to work with elements of X^{n} and P^{n}). The fibre product

X^{q−1} ×_{Z^{q}(X^{•})} Z^{q}(P^{•}) ⟶^{α} Z^{q}(P^{•})
↓^{β}  ↓^{f_{q}}
X^{q−1}  ⟶^{d} Z^{q}(X^{•})

consists of pairs (x, z) ∈ X^{q−1} ⊕Z^{q}(P^{•}) such that dx = f_{q}(z) ∈ Z^{q}(X^{•}) (above, α(x, z) = z and β(x, z) = x). Its universal property implies that the differential d : P^{q−1} ⟶ X^{q−1} factors through a unique morphism j_{q−1} as in (B_{q}) (d = αj_{q−1} and f_{q−1} = βj_{q−1}). If z ∈ Z^{q}(P^{•}) satisfies f_{q}(z) = dx for some x ∈ X^{q−1}, it follows from (B_{q}) that (x, z) = j_{q−1}(p) for some p ∈ P^{q−1}; then z = α(x, z) = αj_{q−1}(p) = dp, which proves the claim. Moreover, (B_{q}) implies (A_{q−1}): if x ∈ Z^{q−1}(X^{•}), then (x, 0) = j_{q−1}(p) for some p ∈ P^{q−1}. As dp = α(x, 0) = 0, we have p ∈ Z^{q−1}(P^{•}) and f_{q−1}(p) = βj_{q−1}(p) = β(x, 0) = x. As a result, it is enough to construct f : P^{•} ⟶ X^{•} satisfying (A_{0}) (by choosing any epimorphism f_{0} : P^{0} ⟶ X^{0} with P^{0} ∈ P_{A}) and (B_{q}) for all q ≤ 0 (having constructed f_{q}, we choose any epimorphism j_{q−1} : P^{q−1} ⟶ X^{q−1} ×_{Z^{q}(X^{•})} Z^{q}(P^{•}) with P^{q−1} ∈ P_{A} and put f_{q−1} = βj_{q−1} : P^{q−1} ⟶ X^{q−1}).

3.5.5. Proposition. Let P^{•} ∈ C^{−}(P_{A}). If Z^{•} ∈ C(A) is acyclic, so is HomA^{•}(P^{•}, Z^{•}). In particular,

Hom_{K(A)}(P^{•}, Z^{•}) = H^{0}(HomA^{•}(P^{•}, Z^{•})) = 0.

Proof. By definition, HomA^{•}(P^{•}, Z^{•}) = Tot^{Π}(K^{•,•}), where K^{i,j} = Hom_{A}(P^{−i}, Z^{j}). Note that ∃i_{0} ∀i < i_{0} ∀j K^{i,j} = 0, since P^{•} ∈ C^{−}(A). Furthermore, each column (K^{i,•}, d_{II}) of K^{•,•} is acyclic, as it is obtained by applying an exact functor Hom_{A}(P^{−i}, •) to an acyclic complex Z^{•}. Proposition then follows from the following Lemma.

3.5.6. Lemma. Let K^{•,•} be a double complex in which each column (K^{i,•}, d_{II}) is acyclic and such that ∃i_{0} ∀i < i_{0} ∀j K^{i,j} = 0. Then the big total complex Tot^{Π}(K^{•,•}) is acyclic (this is false, in general, for the small total complex Tot^{⊕}(K^{•,•})).

Proof. Let x = (x_{ij}, x_{i+1,j−1}, x_{i+2,j−2}, …) ∈ Tot^{Π}(K^{•,•})^{i+j} (x_{ab} ∈ K^{a,b}), dx = 0. As d = d_{I} + d_{II}, we have

d_{II}(x_{ij}) = 0,  d_{I}(x_{ij}) + d_{II}(x_{i+1,j−1}) = 0,  d_{I}(x_{i+1,j−1}) + d_{II}(x_{i+2,j−2}) = 0,  …

As (K^{i,•}, d_{II}) is acyclic, there is y_{i,j−1} ∈ K^{i,j−1} such that x_{ij} = d_{II}(y_{i,j−1}), hence

0 = d_{II}(x_{i+1,j−1}) + d_{I}d_{II}(y_{i,j−1}) = d_{II}(x_{i+1,j−1} − d_{I}(y_{i,j−1})).

As (K^{i−1,•}, d_{II}) is acyclic, there is y_{i+1,j−2} ∈ K^{i+1,j−2} such that x_{i+1,j−1} − d_{I}(y_{i,j−1}) = d_{II}(y_{i+1,j−2}). Continuing this process we obtain an element y = (y_{i,j−1}, y_{i+1,j−2}, y_{i+2,j−3}, …) ∈ Tot^{Π}(K^{•,•})^{i+j−1} satisfying dy = x.

3.5.7. Proposition. Let X^{•}, Y^{•} ∈ C(A) and P^{•}, Q^{•} ∈ C^{−}(P_{A}).

(1) If f : X^{•} ⟶ Y^{•} is a Qis, then the map induced by f

Hom_{K(A)}(P^{•}, X^{•}) ⟶^{f∘} Hom_{K(A)}(P^{•}, Y^{•})

is an isomorphism of abelian groups.

(2) If f : X^{•} ⟶ Y^{•} is a morphism of complexes and u : P^{•} ⟶ X^{•}, v : Q^{•} ⟶ Y^{•} are Qis, then there is a morphism of complexes f′ : P^{•} ⟶ Q^{•}, unique up to homotopy, for which the following diagram commutes up to homotopy:

P^{•}  ⟶^{u} X^{•}
↓^{f′}  ↓^{f}
Q^{•}  ⟶^{v} Y^{•}

(i.e., there is a homotopy vf′ ≃ fu).

(3) If P^{•} is acyclic, then there is a homotopy 0 ≃ id_{P^{•}} (a “contracting homotopy” on the complex P^{•}).

(4) Any Qis α : P^{•} ⟶ Q^{•} is a homotopy equivalence (i.e., there is a morphism of complexes β : Q^{•} ⟶ P^{•} and homotopies αβ ≃ id_{Q^{•}}, βα ≃ id_{P^{•}}).

(5) If α : P^{•} ⟶ Q^{•} is a Qis, so is F (α) : F (P^{•}) ⟶ F (Q^{•}), for any additive functor F : A ⟶ A′ with values in an abelian category A′.

Proof. (1) As f is a Qis, its cone Cone(f) is acyclic, hence so is (see (3.4.6.2))

Cone(HomA^{•}(P^{•}, X^{•}) ⟶^{f∘} HomA^{•}(P^{•}, Y^{•})) ⟶^{∼} HomA^{•}(P^{•}, Cone(X^{•} ⟶^{f} Y^{•})),

by 3.5.5. In other words, the morphism of complexes

HomA^{•}(P^{•}, X^{•}) ⟶^{f∘} HomA^{•}(P^{•}, Y^{•})

is a Qis, hence f induces an isomorphism

Hom_{K(A)}(P^{•}, X^{•}) = H^{0}(HomA^{•}(P^{•}, X^{•})) ⟶^{∼} H^{0}(HomA^{•}(P^{•}, Y^{•})) = Hom_{K(A)}(P^{•}, Y^{•}).

(2) By (1), the map induced by v

Hom_{K(A)}(P^{•}, Q^{•}) ⟶^{v∘} Hom_{K(A)}(P^{•}, Y^{•})

is bijective.

(3) As P^{•} ⟶^{0} 0 is a Qis, we obtain from (1) an isomorphism of abelian groups

Hom_{K(A)}(P^{•}, P^{•}) ⟶^{∼} Hom_{K(A)}(P^{•}, 0) = 0.

(4) By (1), the map induced by α

Hom_{K(A)}(Q^{•}, P^{•}) ⟶^{α∘} Hom_{K(A)}(Q^{•}, Q^{•})

is bijective, which means that there is a morphism of complexes β : Q^{•} ⟶ P^{•}, unique up to homotopy, for which αβ ≃ id_{Q^{•}}. As αβα ≃ id_{Q^{•}} ∘ α = α ∘ id_{P^{•}}, the same bijection implies that βα ≃ id_{P^{•}}.

(5) By (4), α admits a homotopy inverse β; then F (β) is a homotopy inverse of F (α), which implies that F (α) is a Qis.

3.5.8. Corollary. If A has enough projectives, then there is a “resolution functor”

C^{−}(A) ⟶ K^{−}(P_{A}), X^{•} ↦ P^{•}

(factoring through the functor C^{−}(A) ⟶ K^{−}(A)), where P^{•} ∈ C^{−}(P_{A}) is any bounded above complex of projectives equipped with a Qis P^{•} ⟶ X^{•}.

3.5.9. Definition. Assume that A has enough projectives and F : A ⟶ A′ is an additive functor with values in an abelian category A′. Composing the resolution functor 3.5.8 with the functor F = K^{−}(F) : K^{−}(A) ⟶ K^{−}(A′) and with the cohomology functors H^{−n} : K^{−}(A′) ⟶ A′ (n ∈ Z), we obtain the following additive functors (all factoring through C^{−}(A) ⟶ K^{−}(A)):

\begin{matrix}“LF ” : C^{−}(A) ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′),  X^{•} ↦ P^{•} ↦ F (P^{•}) \\ L_{n}F = H^{−n} ∘ “LF ” : C^{−}(A) ⟶ K^{−}(P_{A}) ⟶^{F} K^{−}(A′) ⟶^{H^{−n}} A′ \\ L_{n}F : A ⟶ C^{−}(A) ⟶^{L_{n}F} A′, \end{matrix}  (3.5.9.1)

for any Qis P^{•} ⟶ X^{•} with P^{•} ∈ C^{−}(P_{A}). The functors L_{n}F (resp., L_{n}F) are the classical left derived functors (resp., the left hyper-derived functors) of F (at least when F is right exact; see 3.5.10.?? below).

3.5.10. Proposition (Properties of the left derived functors). Under the assumptions of 3.5.9, the functors “LF ”, L_{n} and L_{n}F have the following proprties.

(1) If X ∈ P_{A}, then L_{0}F (X) = X and ∀n > 0 L_{n}F (X) = 0.

(2) “LF ”(X^{•}[m]) = “LF ”(X^{•})[m].

(3) If f : X^{•} ⟶ Y^{•} is a Qis in C^{−}(A), then “LF ”(f) : “LF ”(X^{•}) ⟶ “LF ”(X^{•}) is an isomorphism in K^{−}(A′); in particular, it is a Qis (hence all maps L_{n}F (f) : L_{n}F (X^{•}) ⟶ L_{n}F (Y^{•}) are isomorphisms).

(4) If H^{m}(X^{•}) = 0 for all m > N, then L_{n}F (X^{•}) = 0 for all n < −N. In particular, L_{n}F = 0 for all n < 0.

(5) If F is right exact, then L_{0}F = F.

(6) If E : 0 ⟶ X^{•} ⟶^{f} Y^{•} ⟶^{g} Z^{•} ⟶ 0 is a short exact sequence in C^{−}(A), then there is a long exact sequence in A′

⋯ ⟶ L_{n}F (X^{•}) ⟶^{L_{n}F (f)} L_{n}F (Y^{•}) ⟶^{L_{n}F (g)} L_{n}F (Z^{•}) ⟶ L_{n−1}F (X^{•}) ⟶^{L_{n−1}F (f)} L_{n−1}F (Y^{•}) ⟶ ⋯,

which is functorial in E. In particular, for each short exact sequence 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 in A there is a long exact sequence in A′

⋯ ⟶ L_{n}F (X) ⟶ L_{n}F (Y) ⟶ L_{n}F (Z) ⟶ L_{n−1}F (X) ⟶ L_{n−1}F (Y) ⟶ ⋯ ⟶ L_{0}F (Z) ⟶ 0.

Proof. (1) We can take P^{•} = X as a projective resolution of X.

(2) If u : P^{•} ⟶ X^{•} (P^{•} ∈ C^{−}(P_{A})) is a Qis, so is u[m] : P^{•}[m] ⟶ X^{•}[m].

(3) If u : P^{•} ⟶ X^{•} is a Qis (P^{•} ∈ C^{−}(P_{A})), so is fu : P^{•} ⟶ Y^{•}, which means that the morphism “LF ”(f) in K^{−}(A′) is equal to (the homotopy class of) id_{F (P^{•})} : F (P^{•}) ⟶ F (P^{•}).

(4) Let u : P^{•} ⟶ X^{•} be a Qis, where P^{•} ∈ C^{−}(P_{A}); then τ_{≤N} P^{•} ↪ P^{•} ⟶^{u} X^{•} is also a Qis and τ_{≤N} P^{•} ∈ C^{≤N}(P_{A}), by 3.5.1. It follows that “LF ”(X^{•}) = F (τ_{≤N} P^{•}) ∈ C^{≤N}(A′), hence H^{−n}(“LF ”(X^{•})) = 0 for all n < −N.

(5) Let X ∈ A; fix a projective resolution P^{•} ⟶ X of X. As F is right exact, it transforms the exact sequence P^{−1} ⟶^{d^{−1}} P^{0} ⟶ X ⟶ 0 into an exact sequence F (P^{−1}) ⟶^{F (d^{−1})} F (P^{0}) ⟶ F (X) ⟶ 0, which implies that (L_{0}F)(X) = H^{0}(F (P^{•})) = Coker(F (d^{−1})) = F (X).

(6) Thanks to (3) and 3.3.15 we can assume that the exact sequence E is of the following special form:

0 ⟶ X^{•} ⟶^{\overline{f}} Cyl(f) ⟶^{π} Cone(f) ⟶ 0.

Fix P^{•}, Q^{•} ∈ C^{−}(P_{A}) and Qis u : P^{•} ⟶ X^{•}, v : Q^{•} ⟶ Y^{•}. By 3.5.7(2) there exists a morphism f′ : P^{•} ⟶ Q^{•} in C(A) and a homotopy h : fu ≃ vf′. The construction from 3.3.16 then gives rise to the following commutative diagram whose rows are split short exact sequences of complexes

0 ⟶ P^{•}  ⟶^{\overline{f}′} Cyl(f′) ⟶ Cone(f′) ⟶ 0
↓^{u}  ↓  ↓
0 ⟶ X^{•}  ⟶^{\overline{f}} Cyl(f) ⟶ Cone(f) ⟶ 0,

and in which all vertical arrows are Qis (u by assumption, the right vertical arrow by the remark after (3.3.9.2), and the middle one by 2.1.10). The required long exact sequence is then the cohomology exact sequence of

0 ⟶ F (P^{•}) ⟶ F (Cyl(f′)) ⟶ F (Cone(f′)) ⟶ 0

(which is exact, as the first row of the diagram is split). 3.5.11. Definition. Assume that A has enough projectives and F : A ⟶ A′ is a right exact additive functor with values in an abelian category A′. We say that X ∈ A is F -acyclic if ∀n > 0 L_{n}F (X) = 0. We denote by A_{F} the full additive subcategory of A consisting of the F -acyclic objects. By 3.5.10(1), P_{A} ⊆ A_{F}.

3.5.12. Proposition. Under the assumptions of 3.5.11, if Y ∈ A and if X^{•} ⟶ Y is a Qis with X^{•} ∈ C^{≤0}(A_{F}) (we say that “X^{•} is an F -acyclic resolution of Y ”), then ∀n ≥ 0 L_{n}F (Y) ⟶^{∼} H^{−n}(F (X^{•})).

Proof. Under construction.

3.5.13. Proposition. Under the assumptions of 3.5.11, if Q^{•} ∈ C^{−}(A_{F}), then there is a Qis “LF ”(Q^{•}) ⟶ F (Q^{•}), hence L_{n}F (Q^{•}) = H^{−n}(F (Q^{•})) for all n ∈ Z.

Proof. Fix a Qis f : P^{•} ⟶ Q^{•} with P^{•} ∈ C^{−}(P_{A}). Fix m ∈ Z such that Cone(f)[m] ∈ C^{≤0}(A_{F}); then Cone(f)[m] is an F -acyclic resolution of 0, hence F (Cone(f)[m]) = Cone(F (f))[m] is also acyclic, by 3.5.12. In particular, the map F (f) : F (P^{•}) ⟶ F (Q^{•}) is a Qis; but F (P^{•}) is canonically isomorphic to “LF ”(Q^{•}) in K^{−}(A′). 3.5.14. In general, the map “LF ”(Q^{•}) ⟶ F (Q^{•}) is not an isomorphism in the homotopy category K^{−}(A′). It becomes an isomorphism only in the derived category

D^{−}(A′) = K^{−}(A′)[Qis^{−1}],

which is obtained from the homotopy category by inverting all Qis.