On Sheaf Theory

Home , Constant sheaf, Local homeomorphism, Resolution (algebra), Sheaf (mathematics), Sheaf of modules, Stalk (sheaf)

Lectures on Sheaf Theory

by C.H. Dowker

Tata Institute of Fundamental Research Bombay 1957 Lectures on Sheaf Theory

by C.H. Dowker

Notes by S.V. Adavi and N. Ramabhadran

Tata Institute of Fundamental Research Bombay 1956 Contents

1 Lecture 1 1

2 Lecture 2 5

3 Lecture 3 9

4 Lecture 4 15

5 Lecture 5 21

6 Lecture 6 27

7 Lecture 7 31

8 Lecture 8 35

9 Lecture 9 41

10 Lecture 10 47

11 Lecture 11 55

12 Lecture 12 59

13 Lecture 13 65

14 Lecture 14 73

iii iv Contents

15 Lecture 15 81

16 Lecture 16 87

17 Lecture 17 93

18 Lecture 18 101

19 Lecture 19 107

20 Lecture 20 113

21 Lecture 21 123

22 Lecture 22 129

23 Lecture 23 135

24 Lecture 24 139

25 Lecture 25 143

26 Lecture 26 147

27 Lecture 27 155

28 Lecture 28 161

29 Lecture 29 167

30 Lecture 30 171

31 Lecture 31 177

32 Lecture 32 183

33 Lecture 33 189 Lecture 1

Sheaves. 1 onto Definition. A sheaf S = (S, τ, X) of abelian groups is a map π : S −−−→ X, where S and X are topological spaces, such that 1. π is a local homeomorphism, 2. for each x ∈ X, π−1(x) is an abelian group, 3. addition is continuous. That π is a local homeomorphism means that for each point p ∈ S , there is an open set G with p ∈ G such that π|G maps G homeomorphically onto some open set π(G). Sheaves were originally introduced by Leray in Comptes Rendus 222(1946)p. 1366 and the modified definition of sheaves now used was given by Lazard, and appeared first in the Cartan Sem. 1950-51 Expose 14. In the definition of a sheaf, X is not assumed to satisfy any separation axioms. S is called the sheaf space, π the projection map, and X the base space. The open sets of S which project homeomorphically onto open sets of X form a base for the open sets of S . Proof. If p is in an open set H, there exists an open G, p ∈ G such that π|G maps G homeomorphically onto an open set π(G). Then H ∩ G is open, p ∈ H ∩G ⊂ H, and η|H ∩G maps H ∩G homeomorphically onto π(H ∩ G) open in π(G), hence open in X.

1 2 Lecture 1

2 π is a continuous open mapping

Proof. Continuity of π follows from the fact that it is a local homeomorphism, and openness follows from the result proved above.

−1 The set S x = π (x) is called the stalk of S at x. S x is an abelian group. If π(p) , π(q), p + q is not deﬁned. S x has the discrete topology.

Proof. This is a consequence of the fact that π is a local homeomorphism.

Let S × S be the cartesian product of the space S with itself and let S + S be the subspace consisting of those pairs (p, q) for which π(p) = π(q). Addition is continuous means that f : S + S → S deﬁned by f (p, q) = p+q is continuous. In other words, if p, q ∈ S and π(p) = π(q), then given an open G containing p + q, there exist open sets H, K with p ∈ H, q ∈ K such that if r ∈ H, s ∈ K and π(r) = π(s), then r + s ∈ G. We may write this as H + K ⊂ G.

Proposition 1. Zero and inverse are continuous.

(i) Writing Ox for the zero element of the group S x, zero is continuous means that f : X → S , where f (x) = Ox, is continuous.

Proof. Let x ∈ X and let G be an open set containing f (x) = Ox. Then there is an open set G1 such that Ox ∈ G1 ⊂ G and π|G1 is a 3 homeomorphism of G1 onto open π(G1). Since Ox + Ox = Ox, and addition is continuous, there exist open sets H, K with Ox ∈ H, Ox ∈ K such that H + K ⊂ G1. Let L = G1 ∩ H ∩ K, then L is open, Ox ∈ L and π|L is a homeomorphism of L onto open π(L). Then x = π(Ox) ∈ π(L) and if y ∈ π(L) there exists q ∈ L with π(q) = y. Then q ∈ H, q ∈ K and hence q + q ∈ G1. But q ∈ G1, and π|G1 is 1−1; hence q = S y∩G1. Therefore, since q+q ∈ S y ∩G1, q+q = q, hence q = Oy. Thus if y ∈ π(L), f (y) = Oy ∈ L. Thus x is in open π(L), with f (π(L)) = L ⊂ G. Hence f is continuous. (Incidentally we have proved that each Ox is contained in an open set which Lecture 1 3

consists of zeros only and which projects homeomorphically onto an open set of X.)

(ii) Writing-p for the inverse of p in the group S π(p) inverse is continuous means that g : S → S where g(p) = −p is continuous.

Proof. Let p ∈ S , and let G be an open set containing-p. Then there exists open L containing Oπ(p) and consisting of zeros only. Since p+(−p) = Oπ(p) and addition is continuous, there exist open H, K, with p ∈ H, −p ∈ K and H + K ⊂ L. Hence if q ∈ H, r ∈ K, π(r) = π(q), then q + r = Oπ(q), i.e. r = −q. We may assume that π|H is a homeomorphism. Let

1 H1 = (π|H) (π(H) ∩ π(K ∩ G)),

then p ∈ H1, and since π is open, continuous, H1 is open. Then if q ∈ H1, there exists r ∈ K∩G with π(r) = π(q); then r = −q = g(q). 4 Thus H1 is open and g(H1) ⊂ G. Hence g is continuous.

Corollary 1. Subtraction is continuous. i.i. f : S + S → S , where f (p, q) = p − q is continuous.

Corollary 2. The set of all zeros is an open set.

Example 1. If X is a topological space, and G is an abelian group fur- nished with the discrete topology, let S = (X × G,π, X) where π(x, g) = x and (x, g1) + (x, g2) = (x, g1 + g2). Each stalk S x is isomorphic to G. Axioms a) and c) are easily veriﬁed. This sheaf is called the constant sheaf associated with G.

Example 2. Form the constant sheaf (A × Z,π, A) where A is the square {(x, y) : 0 ≤ x ≤ 1, 0 ≤ 0 ≤ y ≤ 1}, and Z is the group of integers. Then identify (x, 0) with (1 − x, 1) in A to get a M¨obius band X, and identify (x, 0, n) with (1 − x, 1, −n) in A × Z to get S . The resulting sheaf S = (S,π, X) is the sheaf of “twisted integers” over the M¨obius band. Each S x is isomorphic to the group of integers. 4 Lecture 1

Example 3. Let X be the sphere of complex numbers. Let S x be the additive group of function elements at x, each function element being a power series converging in some neighbourhood of x. Let S = ∪xS x 5 and define π : S → X by π(S x) = x. If p is a function element, a neighbourhood of p in S is defined by analytic continuation. Then S = (S,π, X) is the sheaf of analytic function elements. Each component (maximal connected subset) of S is a Riemann surface without branch points. The sheaf space S is Hansdorff. Lecture 2

Example 4. Let S : {(x, y) : x2 + y2 = 1, x < 1} together with the group 6 of integers Z}. The topology on the former set is the usual induced topology, and the neighbourhoods for an integer n ∈ Z are given by 2 2 2 2 Ga(n) = {n, (x, y) : x + y = 1, a < x < 1}; X : {(x, y) : x + y = 1} and let π : S → X be defined by π(n) = (1, 0), π(x, y) = (x, y). π−1(1, 0) is the group Z, for other pints (x, y) ∈ X, π−1(x, y) = (x, y) is regarded as the zero group. It is easily verified that S = (s,π, X) is a sheaf. Here S is locally euclidean, and has a countable base. The set of all zeros is open (Corollary 2) and compact but not closed; its closure is not compact, S is T1 but not Hausdorff though X is Hausdorff.

Exercise. If X is a To or a T1 space, space, so is S So far we have only defined sheaves of abelian groups. It is now quite clear how we can extend the definition to the case where the stalks are any algebraic systems. A sheaf of rings is a local homeomorphism π : S → X such that each π−1(x) is a ring and addition and multiplication are continuous, i.e f : S + S → S where f (p, q) = p + q, g : S + S → S where g(p, q) = p · q, are continuous. The sheaf of function elements (Example 3) where multiplication 7 of two function elements in the same stalk is defined to be the usual multiplication of power series is a sheaf of rings. In the sheaf of twisted integers (Example 2) each S x is isomorphic to the ring Z, but this sheaf is not a sheaf of rings.

5 6 Lecture 2

A sheaf of rings with unit is a local homeomorphism π : S → X −1 such that each π (x) is a ring with unit element 1x and addition, multiplication and unit are continuous; i,e.,

f : S + S → S, f (p, q) = p + q, g : S + S → S, f (p, q) = p · q,

h : X −→ S,h(x) = 1x are continuous. Example 5. Let A be the ring with elements 0, 1, b, c; where the rules of addition and multiplication are given by

1 + 1 = b + b = c + c = 0; 1 + b = c, 1 + c = b, b + c = 1; b2 = b, c2 = c, bc = cb = 0

[The ring A may be identified with the ring of functions defined on a set of two elements with values in the field Z2.] Let X : {x, k ≤ x ≤ 1}, S : subspace of X × A obtained by omitting the points (k, 1), (k, b); and let π : S → X, be defined by π(x, a) = x. Addition and multiplication in a stalk are defined by

(x, a1) + (x, a2) = (x, a1 + a2),

(x, a1) · (x, a2) = (x, a1 · a2).

8 Then S = (S,π, X) is a sheaf of rings, each S x is a ring with unit, but S is not sheaf of rings with unit. A sheaf of (unitary left) a-modules, where a = (A, τ, X) is a sheaf of rings with unit, is a local homeomorphism π : S → X such that each −1 π (x) is a unitary left Ax-module and addition, and multiplication by elements of Ax (for each x) are continuous; i.e., f : S + S → S, f (p, q) = p + q, g : A + S → S, g(a, p) = ap

are continuous. A + S is the subspace of A × S consisting of all pairs (a, p) for which τ(a) = π(p). Lecture 2 7

[If R is a ring with unit element, we say that M is a unitary, left R-module if it is a left R-module, and if 1 · m = m for each m ∈ M.]

Example 6. Let a denote the sheaf of function elements on the complex sphere X. Let S x consist of all (p, q) of function elements at x. Let S = S x. A neighbourhood of (p, q) is defined by analytic continuation of x∈X pSand q. In each S x addition is defined as (p1, q1) + (p2, q2) = (p1 + q2, q1 + q2); and if a is a function element at x, define multiplication as a(p, q) = (a.p, a.q). Define π : S → X, by π(p, q) = x if p, q are function elements at x. Then (S,π, X) is a sheaf of a-modules.

Any sheaf a of rings with unit can be regarded as a sheaf of a- 9 modules; the product ap for a ∈ Ax, p ∈ Ax being deﬁned as the product ap in Ax. A sheaf of B-modules where B is a ring with unit element is a local homeomorphism π : S → X such that π−1(x) is a unitary left B-module and addition, and multiplication by elements of B are continuous; i.e., f : S + S → S, f (p, q) = p + q,

gb : S → S, gb(p) = b.p for each b ∈ B are continuous. Let S = (S,π, X) be a sheaf of B-modules. This is equivalent to saying that S is a sheaf of B-modules, where B is the constant sheaf (X × B, τ, X). Proof:

B is a ring with unit element. B(X × B, τ, X) is a constant sheaf. S = (S,π, X) is a sheaf of abelian groups. 1) S is a sheaf of B-modules. 1) S is a sheaf of B-modules. This means that b · p is de- This means that (π(p), b) · p is ﬁned such that S x is a uni- deﬁned such that S x is a unitary left B-module. left x × B module. 2) Addition is continuous. 2) Addition is continuous. 3) Multiplication is continuous 3) Multiplication is continuous means that h : B × S → S means that g : (X × B) + S → S h(b, p) = b · p is continuous. g(π(p), b, p) = b.p is continuous. 8 Lecture 2

10 To prove the assertion, it is enough to show that the continuity of h is equivalent to the continuity of g. To do this, deﬁne φ : B × S → (X × B) + S as φ(b, p) = (π(p), b, p), then gφ = h.φ is clearly 1-1. We show that φ is a homeomorphism and the result will follow from this. A base for (X × B) + S is formed by the sets (U × b) + G where G projects homeomorphically onto π(G).

(U × b) + G = (V × b) + G where V = U ∩ π(G) = (V × b) + H where H = (π|G)−1V = (π(H) × b) + H = φ(b × H).

Since the sets b × H form a base for B × S , it follows that φ is a homeomorphism. Thus we may identify sheaves of (X × B, τ, X)-modules with sheaves if B-modules. By abuse of language, we write B for the ring as well as for the constant sheaf (X × B, τ, X).

Example 7. Let C be the ring of complex numbers, S = (S,π, X) be the sheaf of function elements on the complex sphere X and for c ∈ C, p ∈ S deﬁne c · p to be the usual product of a complex number with a 11 power series. Then S becomes a sheaf of C- modules.

Example 8. Let C be the ring of complex numbers, S = (S,π, X) be the sheaf of function elements on the complex sphere X. For c ∈ C and (p, q) ∈ S deﬁne c(p, q) = (c · p, c · q).

Then S becomes a sheaf of C-modules.

Example 9. Let S = (S,π, X) be any sheaf of abelian groups and let Z be the ring of integers. For n ∈ Z and p ∈ S deﬁne n · p = p + ··· + p (n times) if n > 0, n · p = −(−n)p if n < 0, and 0 · p = 0π(p). Thus S may be considered as a sheaf of Z-modules.

Thus sheaves of rings with unit, sheaves of B-modules and sheaves of abelian groups can be considered as special cases of sheaves of a- modules. Lecture 3

Sections.

Deﬁnition . A section of a sheaf (S,π, X) over an open set U ⊂ X is a 12 map f : U → S such that π · f = 1|U where 1|U denotes the identity function on U. (A map is a continuous function).

By abuse of language, the image f (U) is also called a section. For each open set U ⊂ X the function f : U → S , where f (x) = 0x is a section.

Proof. Zero is continuous, and π(0x) = x.

This section will be called the 0-section (zero section).

Proposition 2. If S = (S,π, X) is a sheaf of abelian groups, the set of all sections of S over a non-empty open set U forms an abelian group Γ(U, S ). If S is a sheaf of sings with unit, Γ(U, S ) is ring with unit element. If S is a sheaf of a-modules, Γ(U, S ) is a unitary left Γ(U, a)-module. If S is a sheaf of B-modules, Γ(U, S ) is a unitary left B-module.

Note. The operations are the usual ones for functions. If f, g ∈ Γ(U, S ), a ∈ Γ(U, a), b ∈ B, deﬁne

( f + g)(x) = f (x) + g(x), ( f g)(x) = f (x) · g(x), (a f )(x) = a(x) · f (x), (b f )(x) = b · f (x).

9 10 Lecture 3

The zero element of Γ(U, S ) is the 0-section over U. If S is a sheaf of rings with unit, the unit element of Γ(U, S ) is the unit section 1 : U → S where 1(x) = 1x.

13 Proof. The proposition follows from the fact that addition zero, inverse, unit and multiplication (ring multiplication as well as scalar multiplication) are continuous.

Remark. If a = (X × B, τ, X) is a constant sheaf of rings with units, for each open set U ⊂ X we can identify B with the ring of constant sections fb, b ∈ B where fb(x) = (x, b) over U. Then B ⊂ Γ(U, a) is a subring, and by restricting the ring of scalars, each Γ(U, a)-module becomes a B-module. (B need not be the whole of Γ(U, a)).

Abuse of φ. If S is a sheaf of a-modules, we agree that the unique section over the empty set φ is the 0-section, and the set Γ(φS ) = 0.

Example 6. If S = (S,π, X) is the sheaf of function elements over the complex sphere X, Γ(X, S ) can be identiﬁed with the ring of functions, analytic in U. Then Γ(X, S ) is the ring of functions, analytic every- where, hence is isomorphic to the ring of complex numbers C.

Note. Usually a sheaf S = (S,π, X) may be interpreted as describing some local property of the space X; then Γ(X, S ) gives the corresponding global property. A section f : U → S is an open mapping

Proof. This is proved using the fact that f is continuous and that π is a local homeomorphism.

14 We can now characterise the sections of S . The necessary and suﬃcient condition that a set G ⊂ S is a section f (U) over some open set U ⊂ X is that G is open and π|G is a homeomorphism.

Proof. The suﬃciency is easy to prove. To prove the necessity let f be a section over U, then since f is open f (U) is open. Since f : U → f (U) is 1-1, open, continuous, π| f (U) : f (U) → U is a homeomorphism Lecture 3 11

We have shown that if f is a section over U then f is a homeomorphism of U onto f (U). The sections f (U) form a base for the open sets of S .

Proof. We have already proved that the open sets of S which project homeomorphically onto open sets of X form a base for the open sets of S

The intersection f (U) ∩ g(V) of two sections is a section.

Proof. f (U)∩g(V) is open and projects homeomorphically onto an open set of X since each of f (U) and g(V) has this property.

If f : U → S is a section, the set {x : f (x) = 0x} is open in U.

Proof. {x : f (x) = 0x} = π( f (U) ∩ 0(U)) (0 denotes the 0-section over U), hence is open in U.

Deﬁnition. If f : U → S is a section, the support of f , denoted as supp f , is the set {x : f (x) , 0x}. This set is closed in U. If f is a section over 15 X, supp f is a closedsubset of X

Note . Since the sections of S form a base for the open sets of S , the topology of S = ∪S x can be described by specifying the sections. (See appendix at the end of the lecture).

Let a = (A, τ, X) be a sheaf of rings with unit, and let S = (S,π, X) and R = (R, ρ, X) be sheaves of a-modules (all over the same base space X).

Deﬁnition. A homomorphism h : S → R is a map h : S → R such that ρ·h = π and its restriction h|S x = hx : S x → Rx is an Ax homomorphism for each x ∈ X

This deﬁnition includes as a special case the deﬁnition of homomorphisms of sheaves of B-modules and sheaves of abelian groups. If h : S → R is a homomorphism, the image of each section is a section. 12 Lecture 3

Proof. If f : U → S is a section, then h f : U → R, the image of the section f deﬁned by (h f )(x) = h( f (x)), is continuous and ρ(h f ) = Π · f = 1|U.

(1) h is an open mapping.

(2) h is a local homeomorphism.

Proof. (1) If G ⊂ S is open, then G is a union of sections, hence h(G) is a union of sections of R, hence is open 16 (2) Each p ∈ S is contained in some section f (U). So h f (U) is a section in R and h| f (U) = h f π| f (U), and each of π| f (U) : f (U) → U and h f : U → h f (U) is a homeomorphism.

Appendix. A sheaf may be described by specifying its sections as follows: Sup- pose that we are given a space X and mutually disjoint abelian groups S x, one for each point x ∈ X. Also suppose that we are given a family = {s} of functions with domain open in X and values in ∪xS x, s :P dom(s) → ∪xS x, such that, if x ∈ dom(s), s(x) ∈ S x. Suppose further that

(i) the images for all s ∈ cover ∪S x, P (ii) if s1(x), s2(x) are deﬁned then, for some open U with x ∈⊂ dom(s1) ∩ dom(s2), (s1 + s2)|U ∈ , P (iii) if s(x) = 0x then, for some open U with x ∈ U ⊂ dom(s)s(U) consists entirely of zeros.

If S = ∪S x with {s(U)}, for all s ∈ and open U ⊂ dom(s), as base = for open sets and if π(p) x for p ∈ S Px then (S,π, X) is a sheaf.

Proof. Let p ∈ s(U) ∩ s1(U1) and let x = π(p). By (i) there exists s2 with s2(x) = −p, by (ii) there exists a neighbourhood V of x with (s + s2)|V ∈ and by (iii), since (s + s2)x = p − p = 0x there is a ′ + ′ smaller neighbourhoodP V of x with (s s2)(V ) consisting of zeros. Lecture 3 13

′ + ′ Similarly there is a neighbourhood V1 of x with (s1 s2)(V1) consisting of zeros. = ′ ′ + + = = Let W V ∩ V1, then (s s1 s2)|W s|W s1|W. Then s(W) is in 17 the proposed base and p ∈ s(W) ⊂ s(U) ∩ s1(U1). Therefore the axioms for a base are satisﬁed. Then s : U → S is continuous. For if x ∈ dom(s) and p = s(x), any neighbourhood G of p contains a neighbourhood s1(U1) and again there is an open W with p ∈ s(W) = s1(W) ⊂ s(U1) ⊂ G. Hence s : U → s(U) is a homeomorphism, since s is clearly 1-1 and open. Then π|s(U) is the inverse homeomorphism and its image U is open. Thus π is a local homeomorphism. Addition is continuous. For if p, q ∈ S x, with p+q ∈ s2(U2) suppose p ∈ s(U) and q ∈ s1(U1). By (ii) there is a neighbourhood V of x with (s1 + s2)|V ∈ . Then p+q ∈ s2(U2)∩(s+ s1)(V) and hence for some W, = + with x ∈ W ⊂PU2 ∩ V, s2|W (s s1)|W. Thus, if r ∈ s(W), t ∈ s1(W) and π(r) = π(t), then s(r) + s1(t) = s2(t) ∈ s2(U2).

Lecture 4

If (S ′,π′, X) and (S,π, X) are two sheaves of a-modules with S ′ ⊂ S and 18 if the inclusion map i : S ′ → S is a homomorphism, then S ′ is an open subset of S since i is an open map; further the topology on S ′ is the one ′ = = ′ ′ ′ induced from S . Then π π · i π|S and since i|S x : S x → S x is a ′ homomorphism, S x is a sub - Ax - module of S x. This suggests the following deﬁnition of a subsheaf.

Deﬁnition. (S ′,π|S ′, X) is called a subsheaf of (S,π, X) if S ′ is open in ′ = ′ S and, for each x, S x S ∩ S x is a sub - Ax- module of S x. A subsheaf is a sheaf and the inclusion map i is a homomorphism.

Proof. For each p ∈ S ′ there exists an open set G, p ∈ G ⊂ S , with π|G a homeomorphism. Then G ∩ S ′ is open in S ′ and (π|S ′)|G ∩ S ′ = ′ ′ π|G ∩ S is a homeomorphism. S x is an Ax-module and the operations which are continuous in S are continuous in the subspace S ′. Therefore (S ′,π|S ′, X) is a sheaf. ′ = ′ ′ ′ Since i : S → S is a map, and π · i π|S and i|S x : S x → S x is ′ the inclusion homomorphism of the submodule S x, it follows that i is a homomorphism. The set of all zeros in S is a subsheaf of S .

Proof. The set of zeros is open in S , and 0x is a sub-Ax-module of S x.

This sheaf is called the 0-sheaf (zero sheaf) and is usually identiﬁed 19 with the constant sheaf 0 = (X × 0,π, X).

15 16 Lecture 4

If h : S → R is a homomorphism of sheaves, the set S ′ of p ∈ S ′ such that h(p) = 0π(p) forms a subsheaf S of S called the kernel of h(S ′ = ker h), and the image set S ′′ = h(S ) forms a subsheaf of R called the image of h(S ′′ = imh). Proof. (1) Since h is continuous and 0 (0 denotes the set of zeros in R) ′ = −1 ′ = ′ is open, therefore S h (0) is open. Each S x S ∩ S x is the ′ kernel of h|S x : S x → Rx, hence S x is a sub-Ax-module of S x. ′′ ′′ = ′′ (2) Since h is an open map, S is open. Each S x S x ∩ R is the image ′′ of the homomorphism h|S x : S x → Rx, hence S x is a sub - Ax - module of Rx.

Deﬁnition. A homomorphism h : S → R is called a monomorphism if ker h = 0, an epimorphism if im h = R, and and isomorphism if ker h = 0 and im h = R.

Deﬁnition. A sequence

h j h j+1 ···→ S j−1 −→ S j −−−→ S j+1 →···

of homomorphisms of sheaves is called exact at S j if ker h j+1 = im h j; it is called exact if it is exact at each S j.

If h : S → R is a homomorphism, the sequence

i h 0 → ker h −→ S −→ im h → 0

20 is exact. Here i : ker h → S is the inclusion homomorphism, and h′ : S → im h is deﬁned by h′(p) = h(p). It is a homomorphism. The other two homomorphisms are, of course, uniquely determined.

Proof. The statement is the composite of the three trivial statements:

(i) i : ker h → S is a monomorphism, (ii) ker h′ = ker h, Lecture 4 17

(iii) h′ : S → imh is an epimorphism.

Deﬁnition. A directed set (Ω,<) is a set Ω and a relation <, such that

1) λ < λ (λ ∈ Ω),

2) if λ<µ and µ<ν then λ<ν(λ,µ,ν ∈ Ω), 21

3) if λ, µ ∈ Ω, there exists a ν ∈ Ω such that λ<ν and µ<ν.

That is, < is reﬂexive and transitive and each ﬁnite subset of Ω has an upper bound. (We also write µ > λ for λ<µ).

Example. Let Ω be the family of all compact subsets C of the plane let C < D mean C ⊂ D. Ω is then a directed set.

Deﬁnition . A direct system {Gλ, φµλ} of abelian groups, indexed by a directed set Ω, is a system {Gλ}λ∈Ω of abelian groups and a system {φµλ : Gλ → Gµ}λ<µ of homomorphisms such that

(i) φλλ : Gλ → Gλ is identity,

(ii) if λ<µ<ν, φνµφµλ = φνλ : Gλ → Gν.

Thus if λ<µ<ν and λ< k <ν , then φνµφµλ = φνkφkλ. ′ The deﬁnition of a direct system will be the same even when the Gλ s are any algebraic systems.

Deﬁnition. Two elements a ∈ Gλ and b ∈ Gµ of ∪λ∈ΩGλ are said to be equivalent (aσb) if for some ν, φνλa = φνµb.

This relation is easily veriﬁed to be an equivalence relation. The equivalence class determined by a will be denoted by (a). We will now deﬁne addition of equivalence classes. If (a) and (b) are equivalence classes, a ∈ Gλ, b ∈ Gµ, choose a ν > λ and > µ and 18 Lecture 4

deﬁne (a) + (b) = (φνλa + φνµb).

G Gµ λ TTT jj TTTT jjjj jjTjTjT jjj TTTT G jt jj T* G ν I ν1 II tt II tt I$ tz t Gν2

To show that this does not depend on the choice of ν choose ν1 > λ and >µ, let ν2 >ν and >ν1. Then

+ = + φν2ν(φνλa φνµb) φν2λa φν2µb = + φν2ν1 (φν1λa φν1µb),

+ + hence φνλa φνµb ∼ φν1µb. Clearly the class (φνλa φνλb) is independent of the choice of a and b. If {Gλ, φµλ} is a direct system of abelian groups, the equivalence classes form an abelian group G called the direct limit of the system.

Proof. That G is an abelian group follows easily from the fact that each Gλ is an abelian group.

22 The zero element of Gλ is the class containing all the zeros of all the groups Gλ. Clearly, if each Gλ is a ring, then G is a ring, and similarly for any other algebraic system. The function φλ : Gλ → G, where φλa = (a) is a homomorphism and if λ<µ, φµφµλ = φλ.

Proof.

φλ(a + b) = (a + b) = (a) + (b) = φλa + φλb,

φµ(φµλa) = (φµλa) = (a) = φλa.

Lecture 4 19

Example . Let (N, ≤) be the directed set of natural numbers. For each natural number n let Gn = Z and if n ≤ m, let φmn : Gn → Gm be deﬁned m!a by φ a = . The direct limit is isomorphic to the group of rational mn n! numbers.

Example. Let (N, ≤) be as before. For each natural number n, let Gn be the group of rational numbers modulo 1 and if n ≤ m let φmn : Gn → Gm m!a be deﬁned by φ a = . The direct limit is zero. mn n!

If {Gλ, φµλ} is a direct system of abelian groups and if { fλ : Gλ → H} are homomorphisms into an abelian group H with fµφµλ= fλ , there is a unique homomorphism f : G → H of the limit group G such that f φλ = fλ.

Proof. Since fµφµλ = fλ, all elements of an equivalence class have the same image in H. Then f is uniquely determined by f (φλa) = fλa.

For any two equivalence classes, choose representatives a1, b1 in 23 some Gν. Then

f (φνa1 + φνb1) = f φν(a1 + b1)

= f φνa1 + f φνb1 since f φν = fν is a homomorphism. Thus f is a homomorphism.

Lecture 5

Deﬁnition . Let S be a unitary left A-module and R a unitary left B- 24 module; a homomorphism φ : (A, S ) → (B, R) is a pair (φ′ : A → B, φ′′ : S → R) where

φ′(a + b) = φ′(a) + φ′(b), φ′(ab) = φ′(a)φ′(b), φ′(1) = 1, a, b ∈ A φ′′(s + t) = φ′′(s) + φ′′(t), φ′′(as) = φ′(a)φ′′(s), s, t ∈ S, a ∈ A.

[Remark. For a homomorphism φ : (A, S ) → (B, R), we sometimes write only φ for both φ′ and φ′′.] Direct systems and direct limits are deﬁned for arbitrary algebraic systems. Thus if Ω is a directed set and {Aλ, S λ, φµλ}λ, µ ∈ Ω, where ′ φµλ = (φ ′′ ), is a direct system of unitary modules, the µλ:Aλ→Aµ,φµλ:S λ→S µ direct limit consists of a ring A with unit element, and a unitary left A- module S , and there are homomorphisms φλ : (Aλ, S λ) → (A, S ) such that, if λ < µ, φµφµλ = φλ. The unit element of A is the equivalence class containing all the unit elements of all Aλ, and the zero of A is the class containing all the zeros. Thus, if 1λ = 0λ in some Aλ, 1 = 0 in A and hence for all a ∈ A, a = 1 · a = 0 · a = 0, and for all s ∈ S , s = 0, and the direct limit consists of the pair (0,0). If hλ : (Aλ, S λ) → (B, R) are homomorphisms with hµφmuλ = hλ, there is a unique homomorphism h : (A, S ) → (B, R) such that hφλ = hλ for each λ.

Proof. This is proved exactly as in the last lecture. 25

21 22 Lecture 5

Definition. If Ω is a directed set, a subset Ω′ of Ω is said to be a subdirected set of Ω, if, with the induced order relation, it is a directed set. Definition. A subset Ω′ of a directed set Ω is said to be cofinal in Ω if, for any element λ·∈ Ω, there exists a ν ∈ Ω′ with λ<ν. If Ω′ is cofinal in Ω, Ω′ is a subdirected set.

Proof. This is simple.

′ If = {Aλ, S λ, φµλ}λ,µ∈Ω is a direct system and if Ω is a subdirected Ω ′ = Ω′ set of P, then {Aλ, S λ, φµλ}, λ, µ ∈ is also a direct system. Let ′ ′ ′ ′ ′ Ω′ (A , S ) be its directP limit and φλ : (Aλ, S λ) → (A , S ) for λ ∈ . Since, ′ for λ<µ ∈ Ω φµφµλ = φλ; there is a unique induced homomorphism ′ ′ ′ = i : (A , S ) → (A, S ) with iφµ φµ.

φ φ′ µλ µ ′ ′ (Aλ, S λ) / (Aµ, S µ) / (A , S ) UUUU KK UUUU KK φµ UUUU KKK i φλ UUUU KK UUUUK% * (A, S ) If Ω′ is coﬁnal in Ω, then i : (A′, S ′) → (A, S ) is an isomorphism.

Proof. Each class φλ a of Aλ has a representative in Aλ and if Ω λ∈Ω′ S ′ ′ S′ a, ∈ Aλ and a ∼ o in then a ∼ o in . Thus i : A → A is an λ∈Ω′ isomorphism.S Similarly i′′P: S ′ → S is an anP isomorphism.

26 If {Aλ, S λ, φµλ} and {Bλ, Rλ,θµλ} are direct systems indexed by the same directed set Ω, and if { fλ : (Aλ, S λ) → (Bλ, Rλ) are homomorphisms such that fµφµλ = θµλ fλ : (Aλ, S λ) → (Bµ, Rµ), there is a unique homomorphism f : (A, S ) → (B, R) of the limit modules such that f φλ = θλ fλ.

φµλ φλ (Aλ, S λ) / (Aµ, S µ) (Aλ, S λ) / (A, S ) JJ JJ hλ fλ fµ fλ JJ f JJ J% (Bλ, Rλ) / (Bµ, Rµ) (Bλ, Rλ) / (B, R) θµλ θλ Lecture 5 23

Proof. Let hλ = θλ fλ : (Aλ, S λ) → (B, R). Then hµφµλ = θµ fµφµλ fλ = θµθµλ fλ = θλ fλ = hλ. Therefore there is a unique homomorphism f : (A, S ) → (B, R) with f φλ = hλ, i.e. with f φλ = θλ fλ.

If {Aλ, S λ, φµλ}, {Aλ, Rλ,θµλ}, {Aλ, Qλ, ψµλ} are direct systems of unitary left modules (with direct limits S , R, Q respectively) indexed by the Ω ′ = ′ = ′ same directed set , with φµλ θµλ ψµλ : Aλ → Aµ, and if, for each gλ fλ λ ∈ Ω, S λ −−→ Rλ −→ ϕλ is an exact sequence of homomorphisms of Aλ - modules, and if commutativity holds in

gλ fλ (Aλ, S λ) / (Aλ, Rλ) / (Aλ, Qλ)

θµλ θµλ ψµλ (Aµ, S µ) / (Aµ, Rµ) / (Aµ, Qµ) gµ fµ then the sequence of induced homomorphisms of A-modules

g f S −→ R −→ Q is exact. 27

Proof. Consider the following commutative diagram:

gλ fλ (Aλ, S λ) / (Aλ, Rλ) / (Aλ, Qλ)

φµλ θµλ ψµλ

gµ fµ (Aµ, S µ) / (Aµ, Rµ) / (Aµ, Qµ)

φµ θµ ψµ g f (A, S ) / (A, R) / (A, Q)

(i) im g ⊂ ker f . For if s1 ∈ S , for some λ, φλ s = S 1, s ∈ S λ. Then g(φλ s) ∈ im g, and f (gφλ s) = ψλ( fλgλ s) = ψλ(0) = (0). Hence gφλ s ∈ ker f . 24 Lecture 5

(ii) ker f ⊂ im g. For if r1 ∈ R, then for some λ, θλr = r1, r ∈ Rλ. Let θλr ∈ ker f . Then f (θλ)r = 0 = ψλ fλr, this means that fλr ∼ 0, hence there exists a µ > λ such that

ψµλ( fλr) = 0 = fµ(θµλr), i.e.θµλr ∈ ker fµ

and ker fµ is equal to im gµ by assumption, hence there must exist an s ∈ S µ such that θµλr = gµ s, i.e.

(Q.e.d.) θλr = θµθµλr = θµgµ s = gθµ s ∈ im g.

Deﬁnition. A bihomomorphism f : (A, R, S ) → (B, T), where A, B are commutative rings with unit element, R, S , are unitary A-modules and T is a unitary B-module, is a pair ( f ′ : A → B, f ′′ : R × S → T) such that f ′′ is bilinear. More precisely,

′ ′ ′ ′ f (a1 + a2) = f (a1) + f (a2), f (a1a2) ′ ′ ′ = f (a1) f (a2), f (1) = 1, a1, a2 ∈ A, ′′ ′ ′′ ′ ′′ f (r, a1 s1 + a2 s2) = f (a1) f (r, s1) + f (a2) f (r, s2), r ∈ R, s1, s2 ∈ S, ′′ ′ ′′ ′ ′′ f (a1r1 + a2r2, s) = f (a1) f (r1, s) + f (a2) f (r2, s), r1, r2 ∈ R, s ∈ S.

If A = B and f ′ is the identity, we write f : (R, S ) → T. Given (A, R, S ) there exists a unitary A-module R ⊗A S and a bihomomorphism α : (R, S ) → R ⊗A S where im α generates R ⊗A S such that, for any bihomomorphism f : (A, R, S ) → (B, T) there is a unique homomorphism f¯ : (A, R ⊗A S ) → (B, T) with f¯α = f . The module R ⊗A S together with the bihomomorphism α is called a tensor product of R and S over the ring A; it is uniquely determined upto isomorphism. Hence we can say ’the’ tensor product of R and S .

Proof. For the proof of the existence of the tensor product, see Bour- baki, Algebre multilineaire. We give only the proof of uniqueness. Lecture 5 25

(A, R ⊗A S ) α ll5 O lll lll (A, R, S ) α¯ ′ α¯ RRR RRR α′ RR ( ′ (A, R ⊗A S ) ′ Let R ⊗A S and R ⊗A S be two tensor products of R and S over A. Then by deﬁnition of the tensor product the bihomomorphism α′ ′ ′ induces a homomorphismα ¯ : (A, R ⊗A S ) → (A, R ⊗A S ) such that α¯ ′ · α = α′. Similarly α induces a homomorphismα ¯ such thatαα ¯ ′ = α. ′ ′ ′ Thenα ¯ · α¯ is the identity of R ⊗A S , similarlyα ¯ α¯ is the identity of R ⊗A S . Henceα ¯ is an isomorphism. ′ ′ If we identity R ⊗A S with R ⊗A S under this isomorphism, α will 29 coincide with α

Example . Let A be a commutative ring with unit element, S a unitary A-module, and consider A itself as a unitary A-module. Define α : (A, A, S ) → (A, S ) as α(a, s) = as. Then α is verified to be a bihomomorphism. If f : (A, A, S ) → (B, T) is any bihomomorphism, define the homomorphism f¯ : (A, S ) → (B, T) as f¯′a = f ′a, f¯′′ s = f ′′(1, s). Then

f¯′′α(a, s) ∗ f¯′′(as) = f ′′(1, as) = f ′(a) f ′′(1, s) = f ′′(a, s).

Thus f¯α = f , and S with the bihomomorphism α : (A, S ) → S is the tensor product A ⊗A S . Similarly S ⊗A A = S with α(s, a) = as.

Lecture 6

Deﬁnition. A homomorphism φ : (A, R, S ) → (B, P, Q) consists of ho- 30 momorphisms (A, R) → (B, P) and (A, S ) → (B, Q) (in the sense already deﬁned), where the homomorphism A → B is the same in both cases.

A homomorphism φ : (A, R, S ) → (B, P, Q) induces a map φ : R × S → P × Q. We now consider the following diagram:

φ θ (A, R × S ) / (B, P × Q) / (C, T × U) O OOO βφ α OOO β γ OOO OO (A, R ⊗A S ) / (B, P ⊗B Q) / (C, T ⊗C U) φ¯ θ¯

Here φ, θ are the induced maps and α, β, γ the bihomomorphisms in- cluded in the deﬁnition of tensor products. Further, the homomorphism φ induces a unique homomorphism φ¯ as indicated, such that φα¯ = βφ, and similarly φβ¯ = γθ. From the uniqueness, we have θφ = δbarφ. If φ is the identity then φ¯ also is the identity. The operator of taking the tensor product commutes with the operation of taking the direct limit.

Proof. Let {Aλ, Rλ, S λ, φµλ}λ,µ∈Ω be a direct system, where each Aλ is a commutative ring with unit element, Rλ and S λ are unitary Aλ-modules and φµλ : (Aλ, Rλ, S λ) → (Aµ, Rµ, S µ) are homomorphisms. Then, since φ¯λλ is the identity and φ¯νλ = φνµφµλ = φ¯νλ · φ¯νλ(λ<µ<ν), the system ¯ {Aλ, Rλ ⊗Aλ S λ, φµλ} is a direct system. Let its direct limit be denoted by (A, Q).

27 28 Lecture 6

φµλ φµ (Aλ, Rλ, S λ) / (Aµ, Rµ, S µ) / (A, R, S ) J JJ f JJJ JJJ J% αλ αµ β (B, T) 9 tt ttt tt f¯ ttt (Aλ, Rλ ⊗Aλ S λ) / (Aµ, Rµ ⊗Aµ S µ) / (A, Q) φ¯µλ φ¯µ 31 We deﬁne a bihomomorphism β : (A, R, S ) → (A, Q) as

β(r, s) = φ¯λαλ(rλ, sλ),

where rλ and sλ are respresentatives of r ∈ R, s ∈ S , for the same λ. Since

φ¯µαµ(φµλrλ, φµλ sλ) = φ¯µφ¯µλαλ(rλ, sλ) = φ¯λαλ(rλ, sλ), β : (R, S ) → Q is independent of the choice of representatives. For a suitable choice of representatives, + ′ = ¯ + ′ β(r, bs cs ) φλαλ(rλ, bλ sλ cλ sλ) = ¯ + ′ φλ((bλ · αλ(rλ, sλ) cλ · αλ(rλ, sλ)) = ¯ ¯ + ¯ ′ φλ(bλ) · φλαλ(rλ, sλ) φλ(cλ)φλαλ(rλ, sλ) = b · β(r, s) + c · β(r, s′)

and similarly β(br + cr′, s) = b · β(r, s) + c · β(r′, s)

Thus β : (R, S ) → Q is veriﬁed to be a bihomomorphism. Clearly im

β generates Q, for, each q ∈ Q has a representative qλ in some Rλ ⊗Aλ S λ = k and, since im αλ generates Rλ ⊗Aλ S λ, qλ i=1 ai. αλ(ri, si), ai ∈ Aλ, ri ∈ Rλ, si ∈ S λ. Then P k k q = φλ(ai)φ¯λαλ(ri, si) = φλ(ai)β(φλri, φλ si) Xi=1 Xi=1 Lecture 6 29

32 which proves that im β generates Q. We now show that Q together with the bihomomorphism β : (R, S ) → Q is the tensor product R ⊗A S . To do this, let f : (A, R, S ) → (B, T) be any bihomomorphism, then for each λ, f φλ : (Aλ, Rλ, S λ) → (B, T) is also a bihomomorphism, hence it induces a unique homomorphism f¯λ : ¯ ¯ = ¯ = (Aλ, Rλ ⊗Aλ S λ) → (B, T) where for each λ<µ, fµφµλ · αλ fµαµφµλ = = ¯ ¯ ¯ = f φµφµλ f φλ fλαλ. Hence, since im αλ generates Rλ ⊗Aλ S λ, fµφµλ f¯λ. Therefore there is a unique homomorphism f¯ : (A, Q) → (B, T) with f¯φ¯λ = f¯λ. Then

f¯β(r, s) = f¯φ¯λαλ(rλ, sλ) = f¯λαλ(rλ, sλ) = f φλ(rλ, sλ) = f (r, s); and the proof of the statement is complete. Presheaves. Let Ω be the set of all open sets of X, with the order relation ⊃, i.e. U ⊃ V is equivalent to saying that U < V. then U ⊃ U and if U ⊃ V and V ⊃ W then U ⊃ W and given U, V there exists W = U ∩ V with U ⊃ W, V ⊃ W; hence Ω is a directed set. Definition. A presheaf of modules over a base space X is a direct system {AU, S U, φVU } indexed by Ω, such that (Aφ, S φ) = (0, 0), where φ is the empty set. For a presheaf over X, the index set Ω is always the family of all open sets of X. The definition of a presheaf includes, as a special case, the definition 33 of a presheaf of B-modules, presheaf of rings with unit element and a presheaf of abelian groups.

Example 10. Let X be the complex sphere, AU the ring of all functions analytic in U if U is a non-empty open set and Aφ = O; and, if f ∈ AU and U ⊃ V, let φVU f = f |V, i.e. φVU is the restriction homomorphism. Then {AU, φVU } is a presheaf of rings with unit element.

Presheaf of sections. Let a be a sheaf of rings with unit and S a sheaf of a-modules. For each open U, the ring Γ(U, S ) is a unitary Γ(U, a)- module. If V ⊂ U, let

φVU : (Γ(U, a), Γ(U, S )) → (Γ(V, a), Γ(V, S )) 30 Lecture 6

denote the restriction homomorphism. By convention Γ(φ, a) = o, Γ (φ, S ) = o where φ denotes the empty set. Thus {Γ(U, a), Γ(U, S ), φVU } is a presheaf denoted by (a¯, S¯), and is called the presheaf of sections of (a, S ). For each x ∈ X let Ωx denote the family of all open subsets of X containing x. Then Ωx is a subdirected set of Ω. If {AU, S U, φVU } is a presheaf, let (Ax, S x) denote the direct limit of the subsystem {AU, S U , φVU} indexed by Ωx, and let φxU : (AU , S U ) → (Ax, S x) be the homomorphism which sends each element into its equivalence class. If a ∈ AU , its image φxU a = ax is called the germ of a at x; similarly 34 for s ∈ S U . We will denote bya ¯ : U → Ax the function for which x a¯(x) = ax, and similarly fors ¯ : U → S xS. For each W ⊂ U we write x aW = a¯(W) = {ax : x ∈ W}, and similarlyS for sW . [For instance, in Example 10, if f is analytic in U, x ∈ U the germ fx is the class of those functions each of which coincides with f in some neighbourhood of x.] = = Let A x Ax, S x S x. Deﬁne τ : A → X, π : S → X by = = τ(Ax) x, π(S x) x. ThenSτa¯ : U → U, πs¯ : U → U are the identity maps on U. We can take {aU }U, a ∈ AU as a base for open sets in A. For, {aU } covers A and if p ∈ aU ∩ bV , x = τ(p), we have p = ax = bx with a ∈ AU, b ∈ AV and x ∈ U ∩ V. Then for some W with x ∈ W ⊂ U ∩ V, φWU a = φWV b = c say. Since

φxU a = φxW φWU a = φxW c = φxW φWV b = φxV b for each x ∈ W,

we have cW = aW = bW . Then p ∈ cW = aW = bW ⊂ aU ∩ bV . Similarly the sets {sU} form a base for open sets in S . Lecture 7

With the notations introduced in the last lecture, we prove 35

Proposition 3. If {AU, S U, φVU } is a presheaf of modules over the space X, then a = (A, τ, X) is a sheaf of rings with unit and S = (S,π, X) is a sheaf of a-modules.

Proof. If a ∈ AU,a ¯ : U → A is continuous. For, if ax ∈ bV with b ∈ AV, there exists c ∈ AW with x ∈ W ⊂ U ∩ V such thata ¯(W) = aW = cW = bW ⊂ bV . Alsoa ¯ : U → A is an open mapping since, for open V ⊂ U, aV = (φVU a)V is open by deﬁnition.

Hencea ¯ : U → aU being 1-1 is a homeomorphism and the inverse τ|aU : aU → U is a homeomorphism of aU onto the open set U. Similarlys ¯ : U → S is continuous and π|sU : sU → U is a homeomorphism. Thus τ and π are local homeomorphisms. −1 −1 For each x ∈ X, τ (x) = Ax is a ring with unit element, and π (x) = S x is a unitary left Ax-module.

Addition is continuous, for if ax, bx ∈ Ax (with a ∈ AU, b ∈ AU1 , + x ∈ U ∩ U1) and ax bx ∈ cU2 with c ∈ AU2 , then for some W with + = x ∈ W ⊂ U ∩ U2 ∩ U2, φWU a φWU1 b φWU2 c. Thus ax ∈ aW , bx ∈ bW and for any p ∈ aW , q ∈ bW with τ(p) = τ(q) = y say, we have + = + = p q ay by cy ∈ cU2 . The unit is continuous, for if 1x ∈ Ax is the unit element of Ax and 1x ∈ bU , then for some V with x ∈ V ⊂ U, φVUb is the unit element of AV. Then bV ⊂ bU and consists of the unit elements 1y for y ∈ V. Similarly the other operations of a and S are continuous. 36

31 32 Lecture 7

We remark that aU , sU ··· are sections and that such sections form a base for open sets of A, S .

Remark . The “sheaves” introduced originally by Leray were actually presheaves with the indexing set Ω consisting of the family of all closed sets instead of the family of all open sets.

Example 11. Let X be the circle |z| = 1; for each open set U of X let S U be the abelian group of all integer valued functions in U and let φVU be the restriction homomorphism. This system is a presheaf and the induced sheaf has Example 4 as a subsheaf.

Example 12. Let X be the real line, and S U the R module (R denotes the ring of real numbers) of all real indefinitely differentiable functions in U and let φVU denote the restriction homomorphism. This system is a presheaf and the sheaf space S of the induced sheaf is not Hausdorff.

Let (a, S ) be a sheaf, (a¯, S¯) its presheaf of sections and (a′, S ′) the sheaf determined by (a¯, S¯). We show that (a′, S ′) and (a, S ) are canonically isomorphic.

(a¯,¯) < F z F z F z F z h " (a, ) o (a′, ′)

If x ∈ U and f ∈ Γ(U, a), let hxU f = f (x) ∈ Ax. Similarly, if s ∈ Γ(U, S ), let hxU s = s(x) ∈ S x. Then

hxU : (Γ(U, a), Γ(U, S )) → (Ax, S x)

37 is a homomorphism and, if x ∈ V ⊂ U, hxV φVU = hxU . Then there is an ′ ′ = induced homomorphism hx : (···x , S x) → (Ax, S x) with hxφxU hxU . In fact hx is an isomorphism. For if p ∈ Ax, then there is some section f : U → A with f (x) = p, then

p = f (x) = hxU f = hxφxU f ∈ imhx, Lecture 7 33

′ ′ ′ = Γ ′ and if p ∈ Ax with hx p O, choose a representative f ∈ (U, a) for p . ′ Then f (x) = hxU f = hx p = O. Hence, for some V, with x ∈ V ⊂ U, f |V = o. Therefore

′ p = φxU f = φxV φVU f = φxV OV = (O). ′ ′ Thus hx : Ax → Ax is an isomorphism and similarly hx : S x → S x is an isomorphism. ′ ′ ′ ′ = Γ Let h : (A , S ) → (A, S ) be given by h|(Ax, S x) hx. If f ∈ (U, a) ′ and fU is the induced section in a , given by f¯(x) = φxU f , then

h f¯(x) = hx f¯(x) = hxφxU f = hxU f = f (x) and thus h( f¯(U)) = f (U). The same holds if s ∈ Γ(U, S ). Thus h is an isomorphism of stalks for each x and, since it maps section fU onto sections f (U), h is a local homeomorphism and hence is continuous. Thus h : (a′.S ) → (a, S ) is a sheaf isomorphism. We identify (a′, S ′) with (a, S ) under this isomorphism. 38 If a ∈ A, h−1a is the class of all sections f : U → A where f (U) contains a, and similarly for h−1 s. ′ = ′ ′ ′ = Deﬁnition. If {AU, S U , φVU} and {AU, S U , φVU} are preshea- = ′ ves over X, aP homomorphism f : P→ is a system { fU} of homo- ′ ′ ′ = morphisms fU : (AU , S U ) → (AU, SPU ) suchP that fV φVU φVU fU, that is, the following diagram is commutative.

′ ′ ′ φVU ′ ′ (AU , S U) / (AV , S V )

fU fV (AU , S U) / (AV , S V ). φVU

Let (a′, S ′), (a, S ) be the sheaves determined by ′, . Then ′ the homomorphism f : → induces a sheaf homomorphismP P f : ′ ′ (a , S ) → (a, S ). P P ′ ′ Proof. For each x, { fU}x∈U induces a homomorphism fx : (Ax, S x) → ′ = (Ax, S x), with fxφxU φxU fU and these homomorphism fx deﬁne a function f : (A′, S ′) → (A, S ). 34 Lecture 7

′ ′ ′ = If for a ∈ AU, fUa a, then = = ′ = ′ ′ = ′ ax φxU a φxU fUa fxφxU a fx(ax). ′ = Thus f (aU ) aU and f is a local homeomorphism, hence is continuous. Hence f is a sheaf homomorphism, and this completes the proof. 39 Let be a presheaf which determines the sheaf (a, S ), (a¯, S¯) the ′ ′ presheafP of sections and (a , S ) the sheaf determined by it. The functions fU : (AU , S U ) → Γ(U, a), Γ(U, S ) (where fU a is the section a¯ : U → A determined by a, and similarly for fU s), determine a homomorphism, f = { fU} : → (a¯, S¯). In general, the homomorphism f is neither an epimorphismP nor a monomorphism, hence obviously not a isomorphism. The induced homomorphism f : (a, S ) → (a′, S ′) is the identifying isomorphism h−1.

Proof. Let a ∈ A and suppose that a = φxU b, b ∈ AU. f (a) is the class at x containing fU b which is a section with ( fU b)(x) = φxU b = a. Thus f (a) is the class h−1 a of all sections g : U → A with g(x) = a.

f / (¯a, s¯) u: u P u u u f (a, s) / (a′, s′). h−1 Lecture 8

g f Proposition 4. If ′ −→ −→ ′′ is an exact sequence of homomor- 40 phisms of presheaves,P i.e.P if eachP sequence g ′ U fU ′′ (AU , S U ) −−→ (AU , S U) −−→ (AU, S U )

g f is exact, then the induced sequence S ′ −→ S −→ S ′′ of sheaves is also exact.

′ ′′ Proof. The sequences (Ax, S x) → (Ax, S x) → (Ax, S x ) are exact by a property of the direct limit.

Induced homomorphism of presheaves of sections. If S ′, S are sheaves of a-modules and g : S ′ → S is a homomorphism, there is a homomorphismg ¯ : S¯ ′ → S¯ of the presheaves of sections with ′ g¯U : Γ(U, S ) → Γ(U, S ) deﬁned byg ¯U ( f ) = g f . This homomorphism takes all sections through s′ ∈ S ′ into sections through gs′. Thus, with the usual identiﬁcation,g ¯ inducts the sheaf homomorphism g : s′ → s. Quotient Sheaves.

Proposition 5. If S is a sheaf of a-modules and S ′ is a subsheaf of S , there is a unique sheaf S ′′, whose stalks are the quotient modules ′′ = ′ ′′ = ′′ S x S x/S x, such that j : S → S where j|S x jx : S x → S x is the natural homomorphism, is a sheaf homomorphism. S ′′ is the quotient sheaf S ′′ = S /S ′. ′′ = ′ Proof. If S x S x/S x is to have a topology such that j : S → 41 ′′ ′′ S is a sheaf homomorphism,S S must be covered by sections which

35 36 Lecture 8

are image j f (U) of sections of S , and this uniquely determines the topology of S ′′. This topology has the property that a set V of S ′′ is open if and only if j−1V is open. Thus S ′′ if it exists, is unique.

′ ′ Γ(U, S ) is a sub-Γ(U, a)-module of Γ(U, S ) and let φVU|Γ(U, S ) = ′ ′′ = Γ Γ ′ Γ ′′ φVU. Let S U (U, S )/ (U, S ) and let jU : (U, S ) → S U be ′′ ′′ ′′ the natural homomorphism. Let φVU : S U → S V denote the homomor- ′′ = ′′ ′′ phism induced by φVU. Then {AU , S U, φVU} is a presheaf and the f j sequence 0 → (a¯, S¯′) −→ (a¯,PS¯) −→ ′′ → 0 is exact, where 0 is the presheaf {AU, 0U ,...}. Then the inducedP sequence of sheaves

i i 0 → S ′ −→ S −→ S ′′ → 0,

where S ′′ is the sheaf determined by ′′, is exact. That is, for each x, the sequence P ′ ix jx ′′ 0 → S x −→ S x −→ S x → 0 ′ ′′ is exact. Thus jx induces an isomorphism S x/S x → S x , and if we ′′ ′ ′ identify S x with S x/S x, jx : S x → S x/S x is the natural homomorphism. Thus a sheaf S ′′ having the required properties exists.

Deﬁnition . A homomorphism f : (a, S ) → (B, R) consists of maps 42 f ′ : A → B and and f ′′ : S → R, commuting with the projections, such ′ = ′ ′′ = ′′ that the restrictions fx f |Ax → Bx and fx f |S x : S x → Rx give a = ′ ′′ homomorphism fx ( fx , fx ) : (Ax, S x) → (Bx, Rx). If f : (a, S ) → (B, R) is a sheaf homomorphism and if S ′, R′ are subsheaves of S , R respectively such that f (S ′) ⊂ R′, there are induced homomorphisms

′ ′ → ′ ′′ ′ → ′ f : (a, S ) (B, R ), f : (a, S /S ) (B, R/R) with f i = i f ′, f ′′ j = j f , where i denotes the inclusion homomorphism and j the natural homomorphism of a sheaf onto a quotient sheaf.

Proof. The result is clear for stalks, and the fact that f ′, f ′′ are homomorphisms follows from the fact they are continuous. Lecture 8 37

Example 13. Let X be the circle |z| = 1, let S be the constant sheaf (X × Z,π, X) of integers over X and let S ′ be the subsheaf obtained by omitting the points (1, n) for n , o. Then S /S ′ is isomorphic to the sheaf of Example 4.

Example 14. Let X be the complex plane. Let S U be the additive abelian ′′ group of functions analytic in U, let S U be the multiplicative abelian ′′ group of non-vanishing analytic functions in U and let jU : S U → S U 2πi f be the homomorphism deﬁned by jU f = e . The system { jU} gives a homomorphism of presheaves and there is an induced exact sequence of sheaves 0 → Z → S → S ′′ → 0 where Z is the constant sheaf of integers. An element of Γ(X, S ) is a 43 function analytic in the whole plane.

More generally, in this example X can be replaced by a complex analytic manifold. Tensor products of sheaves. Let a, B be sheaves of commutative rings with unit element let R, S be sheaves of a-modules and let J be a sheaf of B-modules. Deﬁnition . A bihomomorphism f : (a, R, S ) → (B, J) consists of maps f ′ : A → B, f ′′ : R+S → T, which commute with the projections, such that, for each x ∈ X, the restriction fx : (Ax, Rx, S x) → (Bx, Tx) is a bihomomorphism.

If r ∈ Γ(U, R), s ∈ Γ(U, S ) there is a section t : U → T deﬁned by t(x) = f ′′(r(x), s(x)). That t is a map follows from the fact that it is the composite of the two maps

r,s f ′′ U = U + U −−→ R + S −−→ T; + = ′′ where U U is the set of points (x, x), x ∈ U. We write t fU (r, s). Then fU : (AU, RU, S U ) → (BU, TU ) where AU = Γ(U, a), etc., is a bihomomorphism, as follows from the property at each x, e.g. ′′ = ′′ = ′′ fU (ar, s)(x) f ((ar)(x), s(x)) f (a(x)r(x), s(x) 38 Lecture 8

= ′ ′′ = ′ ′′ f (a(x)). f (r(x), s(x)) ( fU (a))(x). fU (r, s)(x) = ′ ′′ ( fU (a). fU (r, s))(x),

and similarly the other properties can be proved. Clearly fU commutes 44 with the restriction of functions; fVφVU = θVU fU .

fU (AU, RU, S U ) / (BU, TU )

φVU θVU (AV , RV, S V ) / (BV, TV ) fV

φxV θxV (Ax, Rx, S x) / (Bx, Tx) fx

Also fxφxU (r, s) = f (r(x), s(x)) = fU (r, s)(x) = θxU fu(r, s); i.e., fxφxU = θxU fU . Thus the bihomomorphism f is determined by the system of bihomomorphisms { fU}. Proposition 6. If a is a sheaf of commutative rings with unit, and R, S are sheaves of a-modules, there exists a sheaf Q of a-modules and a bihomomorphism α : (R, S ) → Q with im αx generating Qx for each x, such that if f : (a, R, S ) → (B, J ) is any bihomomorphism there is a (unique) homomorphism f¯ : (a, Q) → (B, J ) with f¯.α = f. The sheaf Q together with the bihomomorphism α is called the tensor product R⊗a S and is unique upto isomorphism. Each Qx together with

αx : (Rx, S x) → Qx is the tensor product Rx ⊗Ax S x. The sections q(U) = k Γ Γ where q(x) i=1 αx(ri(x), si(x)) with ri ∈ (U, a), si ∈ (U, S ) and Ω= = o < k < ∞, formP a base for the open sets of x Qx x Rx ⊗Ax S x. S S 45 Proof. Let {AU, RU, S U , φVU} be the presheaf (a¯, R¯, S¯) i.e., AU = Γ(U, a), etc. For each φVU, φVU : (AU , RU, S U ) → (AV , RV, S V ) there is ¯ an induced homomorphism φVU : (AU , RU ⊗AU S U ) → (AV, RV ⊗AV S V ) ¯ and the system AU, RU ⊗AU S U , φVU is a presheaf determining some sheaf (a, Q). Since tensor products and direct limits commute, for each x, there is a unique induced bihomomorphism αx : (Rx, S x) → Qx with αxφxU = Lecture 8 39

¯ φxU αU and Qx together with αx is the tensor product Rx ⊗Ax S x. An = k arbitrary element q i=1 αU (ri, si) ∈ RU ⊗AU S U determines a section Ω qU : U → where P

k k qU (x) = φ¯ xU q = φ¯ xU αU(ri, si) = αxφxU (ri, si) Xi=1 Xi=1 k = αx(ri(x), si(x)) Xi=1 and such sections from a base for Q. If f : (a, R, S ) → (B, J ) is any bihomomorphism, there is an induced homomorphism { fU} : (a¯, R¯, S¯) → (B¯, J¯) of presheaves. ¯ Then, if fU : (AU , RU ⊗AU S U ) → (BU, TU ) is the homomorphism induced by fU , ¯ ¯ ¯ fU : AU, RU ⊗AU S U, φVU → B, J is a homomorphism of presheaves which induces a homomorphism f¯ : (a, Q) → (B, J ) of sheaves. Then

φVU φxV (AU, RU, S U ) / (AV, RV , S V ) / (Ax, Rx, S x) r PP H r PP fV HH fx rr α α P α H rr U V PPP x HH rr PPP HH rr φVU φxV P H rr ⊗ ⊗ H L (AU, RU AU S U ) / (AV , RV AV S V ) / (Ax, Qx) LL ww LLL nnn ww L f¯ f¯ nn f¯ ww LL U V nnn x ww LLL nn ww % θVU nv n θxV w{ (BU, TU) / (BV, TV ) / (Bx, Tx)

f¯xαxφxU (r, s) = θxU f¯U αU(r, s) = θxU fU(r, s) = fxφxU (r, s); ¯ = ¯ = = thus fxαx fx and hence f ·α f . Since imαx generates Rx⊗Ax S x Qx, f¯x is uniquely determined by fx; hence f¯ is unique. That Q and α are unique upto isomorphism is proved in the usual manner.

Corollary. If φ : (a, R, S) → (B, J, U) is a homomorphism, there is a unique induced homomorphism φ¯ : (a, R ⊗a S ) → (B, J ⊗B U) with 40 Lecture 8

φα¯ = βφ φ (a, R, ) / (B, J, U)

φ β φ¯ (a, R ⊗a S / (B, J ⊗β U), Lecture 9

1 Cohomology groups of a space with coeﬃcients in a presheaf 47 Deﬁnition. A covering (an indexed covering) {Ui}i∈I of a space X is a system of open sets whose union is X.

Definition . If = {S U , ρVU} is a presheaf of A-module where A is a fixed ring withP unit element, a q-cochain f (q = 0, 1,...) of a covering U = {Ui}i∈I with values in is an alternating function of q + 1 indices with P ∈ ∩···∩ f (io, i1,..., iq) S Uio Uiq or more briefly f (σ) ∈ S Uσ where σ is the simplex io,..., iq. In partic- = = ular f (i0, i1,..., iq) 0 if Ui0 ∩···∩ Uiq φ. ( A function f is called alternating if

(i) f (i0, i1,..., iq) = 0 if any two of i0,..., iq are the same,

(ii) f ( j0, j1,..., jq) = ± f (i0, i1,..., iq) according as the permutation j0,..., jq of i0,..., iq is even or odd).

We will often write ρ(V, U) for ρVU . The q-cochains of U with values in form an A-module Cq(U , ). q For q < 0, we deﬁne C (U , ) = 0. P P P 41 42 Lecture 9

+ Deﬁnition . The coboundary δq 1 f (or simply δ f ) of f ∈ Cq(U , ) is the function on (q + 1)-simplexes deﬁned by P q+1 q+1 j (δ f )(σ) = (−1) ρ(Uσ, U∂ jσ) f (∂ jσ), Xj=0

48 where ∂ jσ = i0,..., iˆj,..., iq+1 = i0,..., i j−1, i j+1,..., iq+1 is the j-th face of σ = io,..., iq+1. + If f ∈ Cq(U , ), then δ f ∈ Cq 1(U , ). P P Proof. It is suﬃcient to verify that f is an alternating function, e.g., = δ f (i1, io,..., iq+1) ρ(Uσ, U∂1σ) f (∂1σ) − ρ(Uσ, U∂oσ) f (∂oσ) q+1 j + (−1) ρ(Uσ, U∂ jσ) f (i1, io,..., iˆj,..., iq+1) Xj=2 = + −ρ(Uσ, U∂o jσ) f (∂oσ) ρ(Uσ, U∂1σ) f (∂1σ) q+1 j − (−1) ρ(Uσ, U∂ jσ) f (io, i1,..., iˆj,..., iq+1) Xj=2

= −δ f (io, i1,..., iq+1)

and, if io = i1, = δ f (io, i1,..., iq+1) ρ(Uσ, U∂oσ) f (∂oσ) − ρ(Uσ, U∂1σ) f (∂1σ∂) = 0,

where σ = io,..., iq+1.

It follows, since ρ(Uσ, U∂ jσ) is a homomorphism, that

+ + δq 1 : Cq(U , ) → Cq 1(U , ) X X is a homomorphism. One veriﬁes by computation that δq+1δq f = 0 for f ∈ Cq−1(U , ), using the fact that for j ≤ k P ∂k∂ jσ = δk(io,..., iˆj,..., ik+1,..., iq+1) = io,..., iˆj,..., ikˆ+1,..., iq+1 1. COHOMOLOGY GROUPS OF..... 43

= ∂ j∂k+1σ.

49 (The computation is carried out at the end of the lecture). Thus in δq ⊂ ker δq+1 in the sequence

δ′ 0 → Co(U , ) −→ C′(U , ) →··· X X + δq δq 1 → Cq−1(U , ) −→ Cq(U , ) −−−→ X X + The quotient module Hq(U , ) = ker δq 1/ im δq is called the q-th cohomology module of U with coePfficients in the presheaf . q q+1 The elements of the module Z (U , ) = ker δ areP called q- q q cocycles and the elements of the module PB (U , ) = im δ are called o o o q-coboundaries. Since B (U , ) = 0, we have HP(U , ) ≈ Z (U , ). P P P Definition . A covering W = {V j} j∈J is said to be a refinement of the covering U = {Ui}i∈I if for each j ∈ J there is some i ∈ I with V j ⊂ Ui.

If W is a reﬁnement of U , choose a function τ : J → I with V j ⊂ Uτ( j). Then there is a homomorphism

+ τ : Cq(U , ) → Cq(W , ) X X deﬁned by + τ f (σ) = ρ(Vσ, Uτ(σ)) f (τσ)

where σ = j0,..., jq; and τσ = τ( j0),...,τ( jq). τ+ commutes with δ since

q+1 q+1 + k + δ τ f (σ) = (−1) ρ(Vσ, V∂kσ)τ f (∂kσ) Xk=0 q+1 = k (−1) ρ(Vσ, V∂kσ)ρ(V∂kσ, Uτ(∂kσ)) f (τ(∂kσ)), Xk=0 q+1 = k (−1) ρ(Vσ, Uτ(∂kσ)) f (τ(∂kσ), Xk=0 + q+1 τ δ f (σ) = ρ(Vσ, Uτσ)δ f (τσ) 44 Lecture 9

q+1 = k (−1) ρ(Vσ, Uτσ)ρ(Uτσ, U∂kτσ) f (∂kτσ), Xk=0 q+1 = k (−1) ρ(Vσ, U∂kτσ) f (∂kτσ) Xk=0

50 and τ∂kσ = ∂kτσ. Hence there is an induced homomorphism,

q q τW U : H (U , ) → H (W , ). X X q q The homomorphism τW U : H (U , ) → H (W , ) is independent of the choice of τ. P P Proof. Let τ : J → I, τ′ : J → I be two such choices. Let the set J be linearly ordered and deﬁne the function

kq−1 : Cq(U , ) → Cq−1(W , ) X X by q−1 q−1 = h (k f )(σ) (−1) ρ(Vσ, Uτhσ) f (τhσ) Xh=0

for σ = j0,..., jq−1 with j0 < j1 < ··· < jq−1, where

′ ′ τhσ = τ( j0),...,τ( jh), τ ( jh),...,τ ( jq−1),

and let kq−1 f be alternating. Then kq−1 is a homomorphism, since ρ(V , U ) : S → S − is a homomorphism. σ τhσ Uτh σ V σ

51 Using the facts that, for σ = j0, j1,..., jq,

τh∂i = ∂iτh+1 (0 ≤ i ≤ h ≤ q − 1),

τh∂i = ∂i+1τh (0 ≤ h < i ≤ q),

∂hτh−1 = ∂hτh (1 ≤ h ≤ q), ′ ∂0τ0 = τ , ∂q+1τq = τ,

one ﬁnds that + + + δqkq−1 f + kqδq 1 f = τ′ f − τ f. 1. COHOMOLOGY GROUPS OF..... 45

(The computation is given at the end of the lecture.) This holds for q = 0 with the obvious meaning of k−1 : C0(U , ) → q ′+ + 0. Thus if r ∈ H (U , ) is represented by a cocycle f , τ f − τ Pf is a coboundary and τW U rPis uniquely determined. If the covering W is a refinement of W and W is a refinement of U , = then τW W τW U τW U and τU U is the identity. = Proof. If W Wk k∈K is a refinement of W , choose τ1 : K → J so that Wk ⊂ Vτ1k. Then W k ⊂ Vτ1k ⊂ Uττ1k and τ2 : K → I can be chosen to be ττ1. Then + + = + (τ1 τ f )(σ) ρ(Wσ, Vτ1σ)(τ f )(τ1σ) = ρ(Wσ, Vτ1σ)ρ(Vτ1σ, Uττ1σ) f (ττ1σ) = ρ(Wσ, Uτ2σ) f (τ2σ) = + (τ2 f )(σ). + + = + Thus τ1 τ τ2 and so for the induced homomorphisms, 52 = q q τW W τWU τWU : H (U , ) → H (W , ). X X Similarly, for the refinement U of U , τ : I → I can be chosen to be q q the identity, hence τU U : H (U , ) → H (U , ) is the identity. 53 P P

(1) δq+1δq = 0. (2)δqkq−1 + kqδq+1 = τ′+ − τ+. q+1 q+1 q = j q (1) (δ δ f )(σ) (−1) ρ(Uσ, U∂ jσ)(δ f )(∂ jσ) Xj=0 q+1 q = j k (−1) (−1) ρ(Uσ, U∂ jσ)ρ(U∂ jσ, U∂k∂ jσ) f (∂k∂ jσ) Xj=0 Xk=0 k q = j+k + (−1) ρ(Uσ, U∂ j∂k+1σ) f (∂ j∂k+1σ) Xj=0 Xk=0 q+1 q j+k (−1) ρ(Uσ, U∂k∂ jσ) f (∂k∂ jσ) j=Xk+1 Xk=0 46 Lecture 9

= 0 q q q−1 = i q−1 (2) (δ k f )(σ) (−1) ρ(Vσ, V∂iσ)(k f )(∂iσ) Xi=0 q q−1 = i+h (−1) ρ(Vσ, Uτh∂iσ) f (τh∂iσ) Xi=0 Xh=0 h q−1 = i+h (−1) ρ(Vσ, U∂iτh+1σ) f (∂iτh+1σ) Xi=0 Xh=0 q q−1 + i+h (−1) ρ(Vσ, U∂i+1τhσ) f (∂i+1τhσ) i=Xh+1 Xh=0 h−1 q = i+h−1 (−1) ρ(Vσ, U∂iτhσ) f (∂iτhσ) Xi=0 Xh=0 q+1 q−1 + i+h−1 q q+1 (−1) ρ(Vσ, U∂iτhσ) f (∂iτhσ)(k δ f )(σ) i=Xh+2 Xh=0 q = h q+1 (−1) ρ(Vσ, Uτhσ)(δ f )(τhσ) Xh=0 q q+1 = i+h q q−1 + q q+1 (−1) ρ(Vσ, U∂iτhσ) f (∂iτhσ), (δ k f k δ f )(σ) Xh=0 Xi=0 q q = ρ(Vσ, U∂hτhσ) f (∂hτhσ) − ρ(Vσ, U∂h+1τhσ) f (∂h+1τhσ) Xh=0 Xh=0 q q+1 = ρ(Vσ, U∂hτhσ) f (∂hτhσ) − ρ(Vσ, U∂hτh−1σ) f (∂hτh−1σ) Xh=0 Xh=1 = ρ(Vσ, U∂0τ0σ) f (∂0τ0σ) − ρ(Vσ, U∂q+1τqσ) f (∂q+1τqσ) ′ = ρ(Vσ, Uτ′σ) f (τ σ) − ρ(Vσ, Uτσ) f (τσ) = (τ′+ f )(σ) − (τ+ f )(σ).

54 Lecture 10

′ Let h = {hU } : Σ → be a homomorphism of presheaves, i.e., each 55 ′ ′ = hU : S U → S U is a homomorphismP and, if V ⊂ U, hV ρVU ρVUhU . We deﬁne, for each q ≥ 0, the mapping

+ h : Cq(U , Σ′) → Cq(U , ) X + = + by (h f )(σ) hUσ f (σ). Then, h is a homomorphism since each hUσ is a homomorphism. h+ commutes with δ

Proof.

+ q+1 = q+1 (h δ f )(σ) hUσ (δ f )(σ) q+1 = j ′ hUσ (−1) ρ (Uσ, U∂ jσ) f (∂ jσ), Xj=0

q+1 q+1 + = j + and (δ h f )(σ) (−1) ρ(Uσ, U∂ jσ)(h f )(∂ jσ) Xj=0 q+1 = j (−1) ρ(Uσ, U∂ jσ)hU∂ jσ f (∂ jσ) Xj=0 q+1 = j ′ hUσ (−1) ρ (Uσ, U∂ jσ) f (∂ jσ) Xj=0

47 48 Lecture 10

Q. e. d. + q Σ′ q + Hence h induces a homomorphism hU : H (U , ) → H (U , ). h + commutes with τ P Proof.

+ + = + (h τ f )(σ) hVσ (τ f )(σ) = ′ hVσ ρ (Vσ, Uτσ) f (τσ) = ρ(Vσ, Uτσ)hUτσ f (τσ) + + = (τ h f )(σ).

= q Σ′ q 56 Hence hW τW U τW U hU : H (U , ) → H (W , ), i.e., the following diagram is commutative. P

hU Hq(U , ′) / Hq(U , )

τW U P τWP U Hq(W , ′) / Hq(W , ). hW P P h g If ′ −→ −→ ′′ is a sequence of homomorphisms of presheaves, then ghP inducesP a homomorphismP ′ ′′ + (gh) : Cq(U , ) → Cq(U , ) X X such that (gh)+ = g+h+ and, if h is the identity, h+ is the identity.

Proof.

+ = ((gh) f )(σ) (gh)Uσ f (σ) = gUσ hUσ f (σ) = (g+h+ f )(σ).

+ If h is the identity, i.e., if each hU is the identity, then (h f )(σ) = = + hUσ f (σ) f (σ); so h is the identity, q.e.d. Lecture 10 49

If one has the commutative diagram of homomorphisms of presheaves, i.e., gh = h1g1, then

h / ′

g1P Pg

′′ ′′ 1 / h1 P P + + = + = + = + + g h (gh) (h1g1) h1 g1 , = and hence gU hU h1U g1U . ′ h g ′′ If U = {Ui}i∈I is a covering and the sequence −→ −→ of 57 homomorphisms of presheaves is exact, then the sequenceP P P

′ + ′′ h+ g Cq(U , ) −−→ Cq(U , ) −−→ Cq(U , ) X X X is also exact. + + Proof. (i) If f ∈ Cq(U , ) is an element of im h , clearly f ∈ ker g , + + hence im h ⊂ ker g P. (ii) Linearly order the index set I, and let f ∈ Cq(U , ) be an ele- + = ment of ker g . Then f (σ) ∈ ker gUσ imhUσ for eachP q-simplex σ, hence there is an element r in the module corresponding to Σ′ = the open set Uσ, of the presheaf , such that hUσ (r) f (σ). If σ = (i0,..., iq) with i0 < ··· < iq, define the function t on σ by ′ t(σ) = r. If σ = ( j0,..., jq) is a permutation of σ = (i0,..., iq) define t on σ′ by t(σ′) = ±t(σ) according as σ′ is an even or odd permutation of σ. If σ is a q-simplex in which two indices are repeated, define t(σ) to be zero. It then follows that t ∈ Cq(U , Σ′) and it is easily verified that h+(t) = f , hence ker g+ ⊂ imh+.

i j If the sequence 0 → ′ −→ −→ ′′ → 0 of homomorphisms of presheaves is exact, thereP is an inducedP P homomorphism ′′ ′ q q+1 δU : H (U , ) → H (U , ). X X 50 Lecture 10

Proof. Since the homomorphisms i+, j+ commute with the homomorphism δ, there is commutativity in the following diagram: (∗)

δ δ δ + i+ j 0 / Cq(U , ′) / Cq(U , ) / Cq(U , ′′) / 0

δ P δ P δ P + + + i + j + 0 / Cq 1(U , ′) / Cq 1(U , ) / Cq 1(U , ′′) / 0 δ P δ P δ P 58 Since the sequence 0 → ′ → → ′′ → 0 is exact, each row of the diagram is an exactP sequenceP of homomorphisms.P We will + construct a homomorphism θ : Zq(U , ′′) → Hq 1(U , ′) which is q ′′ zero on B (U , ), and hence θ willP induce a homomorphismP from q ′′ q+1 ′ H (U , ) → HP (U , ). q ′′ q + To doP this, let r ∈ Z (PU , Σ ), and choose s ∈ C (U , ) with j s = + + + r. Since δ j s = j δs = δr = 0, δs ∈ ker j and by exactness,P there is + + + + a unique t ∈ Cq 1(U , Σ′) with i t = δs. Then i δt = δi t = δδs = 0, + hence δt = 0. Let τ ∈ Hq 1(U , Σ′) be the element represented by t. To show that τ is unique. let s1, t1 be the result of a second such choice, + + q ′ then j (s − s1) = r − r = 0 and s − s1 = i u for a unique u ∈ C (U , Σ ). Then since i+ is a monomorphism and + + + i (t − t1) = δ(s − s1) = δi u = i δu,

hence t − t1 = δu. Thus t and t1 represent the same element τ ∈ + Hq 1(U , Σ′). q ′′ + Let τ = θ(r). If r = ar1 + br2 ∈ Z (U , Σ ), suppose that r1 = j s1, + + + δs1 = i t1 and that r2 = j s2, δs2 = i t2, and let τ1 = θ(r1), τ2 = θ(r2) + + 59 be the elements represented by t1 and t2. Then since j , δ and i are homomorphisms, + + r = j (as1 + bs2), δ(as1 + bs2) = i (at1 + bt2) Lecture 10 51

and, since at1 + bt2 represents aτ1 + bτ2, we have

θ(r) = aθ(r1) + bθ(r2).

+ Thus θ : Zq(U , Σ′′) → Hq 1(U , Σ′) is a homomorphism. q ′′ q−1 + If r ∈ B (U , Σ ), let r = δv. For some w ∈ C (U , )v = j w. + + q+1 ′ Then j (δw) = δ j w = r and there exists a unique t ∈ CP (U , Σ ) with i+t = δ(δw) = 0, hence t = 0; i.e., θ(r) = 0. Thus θ induces a homomorphism

q Σ′′ q+1 Σ′ δU : H (U , ) → H (U , ).

Q.e.d. τW ,U commutes with δU , i.e., the following diagram is commutative. δU + Hq(U , ′′) / Hq 1(U , ′)

τW U P τWP U Hq(W , ′′) / Hq+1( , ′) δW W P P Proof. τ+ commutes with j+, δ, i+.

i j If 0 → ′ −→ −→ ′′ → 0 is exact, then the sequence P P P ′ ′ iU 0 → H0 U , →···→ Hq U , −−→      X  X     ′′ ′   jU  δU + Hq U , −−−→ Hq U , −−−→ Hq 1 U ,     X X X         is exact.

Proof. The exactness of this sequence is the result of six properties of 60 the form ker ⊂ im and im ⊂ ker. Each can be easily veriﬁed in (∗). (See Eilenberg-Steenrod, Foundations of Algebraic Topology, p. 128). 52 Lecture 10

′ i j ′′ ′ i1 j1 ′′ If 0 → −→ −→ → 0 and 0 → 1 −→ 1 −→ 1 → 0 ′ ′′ ′ ′′ are exact sequence,P P andP if h : ( , , ) →P ( 1,P1, 1 )Pis a homomorphism commuting with i,P j, i1P, andP j1, thenP hUPcommutesP with δU . Proof. The homomorphism h+ commutes with the homomorphisms j+, δ and i+, q.e.d. With the same assumptions as in the above statement, we then have the following commutative diagram, in which each row is exact.

iU jU δU 0 / H0(U , ′) / ... / Hq(U , ) / Hq(U , ) / Hq(U , ′′) / Hq+1(U , ′) / ... P P P P P ′ ′′ ′ hU hU hU hU hU

i1U j1U δ1U 0 U ′ ... q U ′ q U q U ′′ q+1 U ′ ... 0 / H ( , 1) / / H ( , 1) / H ( , 1) / H ( , 1 ) / H ( , 1) / P P P P P Deﬁnition. A proper covering of X is a set of open sets whose union is X.

A proper covering U = {U} of X may be regarded as an indexed covering UU U∈U if each open set of the covering is indexed by itself. Every covering Ui i∈I has a refinement which is a proper covering, e.g., = the set of all open sets U such that U Ui for some i ∈ I. 61 Let Ω be the set of all proper coverings of X and let U < W mean that W is a refinement of U . Then Ω is a directed set, for 1) U < U 2) if U < W and W < W then trivially U < W and 3) given U , W there exists W with U < W , W < W , e.g, W may be chosen to consists of all open sets W with W = ∪ ∩ V for some U ∈ U , V ∈ W . (There is no set of all indexed coverings). q The system H (U , ), τW U U ,W ∈Ω is then a direct system. Its q direct limit H (X, ) is calledP the q-th cohomology module (over A) of q q X with coefficientsP in . Let τU : H (U , ) → H (X, ) denote the usual homomorphismP in to the direct limit. P P If h : ′ → is a homomorphism of presheaves, there is a induced homomorphismP P ′ ∗ q q ∗ = h : H (X, ) → H (X, )with h τU τU hU . X X Lecture 10 53

= Proof. This follows from the fact that hU τW U τW U hU . Q.e.d.

i j If O → ′ −→ −→ ′′ → O is an exact sequence of presheaves, there is an induced exact sequence P P P

0 / H0(X, ′) / ··· / Hq(X, ′) / Hq(X, ) / Hq(X, ′′) / Hq+1(X, ′) / ··· P P P P P Proof. This is a consequence of the fact that the direct limits of exact sequences is again an exact sequence.

′ ′′ ′ ′′ If h : ( , , ) → ( 1, 1, 1 ) is a homomorphism of exact 62 sequence ofP presheavesP P P P P

i j 0 / ′ / / ′′ / 0

hP′ hP h′′P

′ i1 j1 ′′ 0 / 1 / 1 / 1 / 0 P P P ∗ and h commutes with i, j, i1, j1 then the following diagram, where h is the homomorphism induced from h, is a commutative diagram.

∗ i∗ j δ∗ ... / Hq(X, ′) / Hq(x, ) / Hq(x, ′′) / Hq(x, ′) / ... P P P P h∗ h∗ h∗ h∗ ∗ ∗ ∗ i1 j1 δ1 ... q ′ q q ′′ q+1 ′ ... / H (X, 1) / H (X, 1) / H (X, 1 ) / H (X, 1) / P P P P Proof. The result is a consequences of the fact that hU commutes with iU , jU and δU .

Lecture 11

If the coeﬃcient presheaf is the presheaf of sections of some sheaf S 63 of A-modules, we write Cq(U , S ) instead of Cq(U , S¯) etc. Then, if = = Uσ Uio ∩···∩ Uiq is called the support of the simplex σ io,..., iq, a q-cochain f ∈ Cq(U , S ) is an alternating function which assigns to each q-simplex σ a section over the support of σ. o If U = {Ui}i ∈ I is any covering of X, H (U , S ) is isomorphism to Γ(X, S ).

0 Proof. A 0-cochain belonging to C (U , S ) is a system ( fi)i∈I, each fi being a section of S over Ui. In order that this cochain be a cocycle, it is necessary and suﬃcient that fi − f j = O over Ui ∩ U j; in other words, that there exist a section f ∈ Γ(X, S ) which coincides with fi Γ on Ui for each i ∈ I. Thus there is an isomorphism φU : (X, S ) → Z0(U , S ) → H0(U , S ).

Proposition 7. H0(X, S ) can be identiﬁed with Γ(X, S ). = Proof. Since τW U φU φW , there is an induced isomorphism φ : Γ 0 = (X, S ) → H (X, S ) with τU φU φ. A homomorphism h : S → S1 of sheaves induces a homomorphism {hU } of the presheaves of section ∗ and hence induced homomorphism hU ′ , h with commutativity in

φU τU Γ(X, S ) / H0(U , S ) / H0(X, S )

∗ hX hU h

φU τU Γ 0 0 (X, S1) / H (U , S1) / H (X, S1)

55 56 Lecture 11

64 Thus we can identify Γ(X, S ) with Ho(X, S ) under φ if we also ∗ o o identify hx : Γ(X, S ) → Γ(X, S1) with h : H (X, S ) → H (X, S1).

Definition. A system Ai i∈I of subset of a space X is called finite if I is finite, countable if I is countable. The system is said to be locally finite if each point x ∈ X has a neighbourhood V such that V ∩ Ai = φ expect for a finite number of i. (This finite number may also be zero).

We notice that a locally finite system Ai i∈I is always point finite. (A system Ai i∈I of subsets of X is said to be point finite if each point x ∈ X belongs to Ai for only a finite number of i). If Ai i∈I is locally finite, so is B j j∈J if J ⊂ I and each B j ⊂ A j. If ¯ ¯ Ai i∈Iis locally finite, so is Ai i∈I, where Ai denotes the closure of Ai, and Ai = A¯i. In particular, if each Ai is closed, so is Ai i∈I i∈I i∈I S S S Definition . The order of a system {Ai}i∈I of subsets of X is −1 if Ai is the empty set for each i ∈ I. Otherwise the order is the largest integer n + ′ such that for n 1 values of i ∈ I, the Ai s have a non-empty intersection, and it is infinity if there exists no such largest integer.

Definition. The dimension of X, denoted as dim X, is the least integer n 65 such that every finite covering of X has a refinement of order ≤ n, and the dimension is infinity if there is no such integer.

Deﬁnition. A space X is called normal, if for each pair E, F of closed sets of X with E ∩ F = φ, there are open sets G, H with E ⊂ G, F ⊂ H and G ∩ H = φ.

Deﬁnition. A covering U = {Ui}i∈I of the space X is called shrinkable if there is a reﬁnement W = {Vi}i∈I of U with V¯ i ⊂ Ui for each i ∈ I.

X is normal if and only if every locally ﬁnite covering of X is shrinkable.

Proof. See S. Lefschetz, Algebraic Topology, p.26.

If X is normal, dim X ≤ n if and only if every locally ﬁnite covering of X has a reﬁnement of order ≤ n. Lecture 11 57

Proof. See C.H.Dowker, Amer.Jour of Math. (1947), p.211.

Definition. A space X is called paracompact if every covering of X has a refinement which is locally finite. If X is paracompact and normal, dim X ≤ n if and only if every covering of X has a refinement of order ≤ n.

Proof. (i) Since X is paracompact, every covering of X has a locally finite refinement and since X is normal and dim X ≤ n, using the above result, every locally finite covering has a refinement of order 66 ≤ n, thus every covering has a refinement of order ≤ n.

(ii) Since every covering of X has a refinement of order ≤ n, in particular, every locally finite of X has a refinement of order ≤ n, hence, since X is normal, using the above result, we obtained dim ≤ n.

Remark. Since a paracompact Hausdorﬀ space is normal, (see J. Dieu- donne, Jour. de Math. 23, (1944), p.(66), this result holds, in particular, when X is a paracompact Hausdorﬀ space.

If a covering U of X has a reﬁnement W of order ≤ n, then it has a proper reﬁnement W of order ≤ n.

Proof. If W = {V j} j∈J has order ≤ n, let W be the proper covering formed by all open sets W such that, for some j ∈ j, W = V j. Then W has order ≤ n and is a reﬁnement of U . If X is paracompact and normal and dim X ≤ n, then Hq(X, ) = 0 for q > n and an arbitrary presheaf . P P Proof. Replace the directed set Ω of all proper coverings of X by the co- ﬁnal sub-directed set Ω′ of all proper covering of order ≤ n. If U ∈ Ω′, ∈ q ∈ = = q > n and f C (U , ), then f (U0, U1,..., Uq) S Uo∩···∩Uq S φ 0 + for any q 1 distinctP open sets of U . If the open sets U0, U1,..., Uq are not all distinct, then f (U0, U1,..., Uq) = 0 since f is alternating. Hence Cq(U , ) = 0 and hence also Hq(U , ) = 0. Therefore q H (X, ) = 0, q > nP. P P 58 Lecture 11

67 If is a presheaf which determines the zero sheaf, then Ho(X, ) = 0. P P Proof. For any element η ∈ H0(X, ) choose a representative 0 0 f ∈ z (U , )SH (U , ), where U is someP proper covering of X, so = = that τU f Pη. For eachP x ∈ X choose an open set U τ(x) such that x ∈ τ(x) ∈ U . Since determines the 0-sheaf, one can choose = an open set Vx such that x P∈ Vx ⊂ τ(x) and ρ(Vx, τ(x)) f (τ(x)) 0. = + = Then W Vx x∈X is a reﬁnement of U , and, for each x, (τ f )(x) = + = ρ(Vx, τ(x)) f (τ(x )) 0, hence τ f 0. If W is a proper reﬁnement + = of W , choose τ1 : W → X so that each W ⊂ Vτ1(W) and (ττ1) f + + = = = = = τ1 τ f 0. Thus τW U f 0 and hence η τU f τW τW U f 0. Hence Ho(X, ) = 0. P This result is not true in general for the higher dimensional cohomology groups. However, if the space X is assumed to be paracompact and normal, we will prove the result to be true for the higher dimensional cohomology groups. Lecture 12

Proposition 8. If X is paracompact and normal and if is a presheaf 68 q which determines the zero sheaf, the H (X, ) = 0 for allP q ≥ 0. q P Proof. Let f ∈ C (U , ) where U = {Ui}i∈I is any locally finite cov- = ering. Since X is normal,P we can shrink U to W , W {Wi}i∈I with W¯ i ⊂ Ui. For each x ∈ X choose a neighbourhood Vx of x such that the following conditions are satisfied: a) If x ∈ Ui, Vx ⊂ Ui, b) If x ∈ Wi, Vx ⊂ Wi, c) if x < W¯ i, Vx ∩ Wi = φ, ∈ ∩···∩ = = d) if x Uio Uiq Uσ, ρVxUσ f (σ) 0 Conditions a) and b) can be satisfied, for the coverings U and W being locally finite, each x is contained only in a finite number of sets of the coverings. To see that condition c) can be satisfied, consider all W¯ i for which x < W¯ i. The union of these sets is the closed since W is locally finite, and x is in the open complement of this union. Next, by condition a), Vx ⊂ Uσ, and since determines the 0-sheaf, we can choose Vx small enough so that d) isP satisfied. We can thus always choose Vx small enough so that the above conditions are fulfilled.

If the Vx are chosen as above, the covering Vx x∈X is a reﬁnement 69 of U . Choose the function τ : X → I so that x ∈ Wτ(x), then by b), Vx ⊂ Wτ(x) ⊂ Uτ(x). Then + + τ f (σ) = τ f (xo,..., xq) = ρ(Vσ, Uσ) f (τ(xo),...,τ(xq)).

59 60 Lecture 12

= + = , If Vσ φ, τ f (σ) 0. If Vσ φ then Vxo meets each Vx j, hence ¯ ¯ meets each Wτ(x j) and hence by c), xo ∈ Wτ(x j). Then since xo ∈ Wτ(x j) ⊂

Uτ(xi), by a) Vxo ⊂ Uτ(xi) for each j and hence Vxo ⊂ Uτ(σ). Hence

τ+ f (σ) = ρ f (τσ) = ρ ρ U f (τσ) VσUτ(σ) VσVx0 Vx0 τ(σ) = 0 by d).

Thus τ+ f (σ) = 0 for all σ, hence τ+ f = 0. q If U1 is a proper covering and f ∈ C (U1, ), there is a locally finite refinement U of U1 (since X is paracompact).P Then there is a refinement W of U (found as above), a proper refinement W1 of W (existence of W1 is trivial) and a function τ1 : W1 → U1 with V ⊂ τ1(V) for + = q V ∈ W1 such that τ1 f 0. Hence every element of H (U1, ) (U1proper) is equivalent to zero, i.e., the direct limit Hq(X, ) consistsS only of zero.P P Example 15. Let X be the space with four points a, b, c, d and let a base for the open sets be the sets (a, c, d), (b, c, d), (c), (d). Let be the = = = presheaf for which S U z, the group of integers, if U (c, d); andP S U 70 0 otherwise. The homomorphisms ρVU are the obvious ones. Then 1 determines the 0-sheaf, but H (X, ) = Z. The space X paracompactP but not normal. P i j If 0 → S ′ −→ S −→ S ′′ is an exact sequence of sheaves, then

iU jU 0 → Γ(U, S ′) −−→ Γ(U, S ) −−→ Γ(U, S ′′)

is exact and hence 0 → S¯ → S¯ → S¯′′ is exact

Proof. We will show that for ker jU ⊂ imiU (the rest is trivial). Let f ∈ ker jU . Then, x ∈ U, j f (x) = ( jU f )(x) = 0x and hence by exactness, = ′ ′ ′ ′ ′ ′ f (x) ip for some p ∈ S x. Thus f (U) ⊂ i(S ). But i : S → i(S ) is homeomorphism. Then g : U → S ′, where g(x) = 1−1 f (x), is a section ′ of S over U, and f = iU g. Lecture 12 61

One cannot in general complete the sequence

0 → Γ(U, S ′) → Γ(U, S ) → Γ(U, S ′′) by a zero on the right as the following example shows.

Example 16. Let X be the segment {x : 0 ≤ x ≤ 1}. Let G be the 4- group with elements 0, a, b, c. Let S be the subsheaf of the constant sheaf G = (X ×G,π, X) formed by omitting the point (0, a), (0, c), (1, b), (1, c). Let S ′ be the subsheaf of S formed by omitting all the points ′′ ′′ (x, a), (x, b). Let S = (X × Z2π, X) and let j : S → S be the 71 homomorphism induced by j : G → Z2 where j(a) = j(b) = 1, j(c) = j(0) = 0. The the sequence i j 0 → S ′ −→ S −→ S ′′ → 0 is exact, but the sequence 0 → Γ(X, S ′) → Γ(X, S ) → Γ(X, S ′′) → O, i.e.,

0 −→ O −→ O −→ Z2 −→ O is not exact.

i j If 0 → ′ −→ −→ ′′ is an exact sequence of presheaves, there is ′′ ′′ an image presheafP P 0 P⊂ and a quotient presheaf Q such that the sequences P P ′ ′′ i j O → −→ −→o → 0, X X X0 ′′ ′′ ¯i ¯j O → −→ −→ Q → 0 X0 X are exact. These sequences are ‘natural’ in the sense that if h is a homomorphism of exact sequences, commuting with i and j:

i j 0 / ′ / / ′′

hP hP Ph ′ i j ′′ 0 / 1 / 1 / 1 P P P 62 Lecture 12

then there are induced homomorphisms h∗ of the exact cohomology sequences

∗ ∗ ∗ j δ ... q ′ i q 0 q ′′ 0 q+1 ′ ... / H (X, ) / H (X, ) / H (X, 0 ) / H (X, ) / P P P P h∗ h∗ h∗ h∗ ∗ ∗ ∗ j0 δ0 ... q ′ i q q ′′ q+1 ′ ... / H (X, 1) / H (X, 1) / H (X, 10) / H (X, 1) / P P P P 72 and

¯∗ ¯j∗ ¯∗ ... q ′′ i q ′′ q δ q+1 ′′ ... / H (X, 0 ) / H (X, ) / H (X, Q) / H (X, 0 ) / P P P h∗ h∗ h∗ h∗

∗ ¯i∗ ¯j δ¯∗ ... q ′′ q ′′ q q+1 ′′ ... / H (X, 10) / H (X, 1 ) / H (X, Q1) / H (X, 10) / P P P ∗ ∗ ∗ ¯∗ ¯∗ ¯∗ commuting with i , j0, δ0 and i , j , δ respectively.

′′ = ′′ ′′ ′′ = ′′ Proof. If {S U, ρVU }, let S oU im jU . Then since j : → is a ′′ ′′ = homomorphism,P ρVU maps im jU into im jV . Hence, writingPS oU Pim jU = ′′ ′′ ′′ and QU S U /S oU , there are induced homomorphisms ρoVU andρ ¯VU with comutativity in

¯ ¯ joU ′′ iU ′′ jU S U / S oU / S U / QU

′′ ′′ ρVU ρoVU ρVU ρ¯VU ¯ ¯ joV ′′ iV ′′ jV S V / S oV / S V / QV . ′′ = ′′ ′′ = Clearly the systems o {S oU, ρoVU } and Q {QU, ρ¯VU } are presheaves and the sequencesP

Σ′ i Σ j Σ′′ O → −→ −→ o → O

and ¯ ¯ Σ′′ i Σ′′ j O → o −→ −→ Q → O Lecture 12 63 are exact. Since h commutes with i, j,

′ i j ′′ jU ′′ 0 / / / S U / S U

P ′ P P ′′ ′′ h h h hU hU ′ i1 j1 ′′ j1U ′′ 0 / 1 / 1 / 1 S 1U / S 1U ′′ ′′ =P P ′′ =P hU maps S oU im jU into S 1oU im j1U . Hence there are induced 73 ′′ ¯ homomorphisms hoU , hU with comutativity in ¯ joU ′′ ¯iU ′′ jU S U / S oU / S U / QU

′′ ′′ hU hoU hU h¯U ¯ joU ′′ ¯iU ′′ jU S 1U / S 1oU / S 1U / Q1u .

Since h is a homomorphism of presheaves, hU commutes with ρVU ′′ ′′ ¯ and hU with ρVU. Hence, since joU and jU are epimorphisms and joU, ¯ ′′ ′′ ¯ jU commute with ρ and h, hoU and commutes with ρoVU and hU with ρ¯VU, i.e., the diagrams given below are commutative:

ρ′′ ρ¯ ′′ oVU ′′ VU S oU / S oV QU / QV =^ @ <^ A = Ñ < ¯ ¯ Ò ==joU joV ÑÑ << jU jV ÒÒ == ÑÑ << ÒÒ = ρ Ñ < ρ′′ Ò VU Ñ ′′ VU ′′ Ò S U / S V S U / S V

h′′ h h h′′ ¯ ′′ ′′ ¯ oU U V oV hU hU hV hV ρ ′′ VU ρ S S ′′ VU ′′ 1U / 1V S 1U / S 1V Ñ == < jo ÑÑ == jo ¯j ÒÒ << ¯j ÑÑ == ÒÒ << Ñ ′′ = Ò < ÑÐ ρ Ò ρ¯ < S ′′ oVU S ′′ ÒÐ VU 1oU / 1oV Q1U / Q1V ′′ ′′ ¯ Thus h, ho , h , h are homomorphisms of presheaves commuting with jo, ¯i, ¯j. ¯ jo ′′ ¯i ′′ j / o / / Q

P ′′P P ′′ h ho h h¯ ¯ jo ′′ ¯i ′′ j 1 / 1o / 1 / Q1 . P P P 64 Lecture 12

74 Thus {h} is a homomorphism of the exact sequences

′ j jo ′′ 0 / / / o / 0

P′ P ′′P h h ho ′ i jo ′′ 0 / 1 / 1 / 1o / 0 P P P commuting with i and jo, and a homomorphism of the exact sequences

¯ ′′ ¯i ′′ j 0 / o / / Q / 0

′′P ′′P ho h h¯ ′′ ¯i ′′ ¯j 0 / 1o / 1 / Q1 / 0 P P commuting with ¯i and ¯j. Therefore the induced homomorphisms of ∗ ∗ ∗ ¯∗ ¯∗ the exact cohomology sequences commute with i , jo, δo and i , j , δ¯∗ respectively. Lecture 13

Example 17. Let X consist of the natural numbers together with two 75 special points p and q. Each natural number forms an open set. A neighbourhood of p (resp. q) consists of p (resp; q) together with all but a finite number of the natural numbers. Let S U = Z if U consists of all but a finite number of the natural numbers and if V ⊂ U is another such set, let ρVU : Z → Z be the identity. If U is an open set not containing all but a finite number of the natural numbers or if it contains either p = or q, let S U O and let ρVU , ρUW be the zero homomorphisms. Then = 1 = {S U , ρVU } is a presheaf determining the 0-sheaf, but H (X, Z) Z. TheP space X is T1 and paracompact but not normal.

R Example 18. Let R be a set with cardinal number N1 let S = 2 be the set of all subsets of R and let T = 2S be the set of all subsets of S . If r ∈ R, let r′ ∈ T be the largest subset of S , which is such that, each of its elements considered as a subset of R contains the elements r. Let ′ ′ R ⊂ T consists of all r, for all r ∈ R and let T1 = T − R .

Let X be a space consisting of (1) all elements r ∈ R and (2) all triples (t, r1, r2) with t ∈ T1, r1, r2 ∈ R and r1 , r2. Each point (t, r1, r2) is to form an open set. Neighborhoods of points r of the ﬁrst kind are sets N(r; s1,..., sk), where o ≤ k < ∞ and s1,..., sk ∈ S , consisting of r together with all points (t, r1, r2) with r ∈ (r1, r2) and, for each 76 i = 1,..., k, either r ∈ si ∈ t or r < si < t. [cf. Bing’s Example G, Canadian Jour, of Math. 3 (1951) p.184]. For sets U ⊂ X of cardinal number ≥ 2 and consisting of points of the second kind, let S U = Z and, if V ⊂ U is another such set, let

65 66 Lecture 13

ρVU : Z → Z be the identity. If U is an open set containing any point of the ﬁrst kind or consisting of at most one point, let S U = 0. Then = ′ , S U, ρVU is a presheaf determining the 0-sheaf, but H (X, ) 0 P(although dim X = 0). X is a completely normal, Hausdorﬀ spaceP but is not paracompact. (A space X is said to be completely normal if each subspace of X is normal). i j If 0 → S ′ −→ S −→ S ′′ → 0 is an exact sequence of sheaves, let j ′′ ′ i ′′ S¯ o, Q,¯ be the image and quotient presheaves in 0 → S¯ −→ S −→ S for which the sequences

¯ i ¯ jo ¯′′ 0 → S −→ S −→ So → 0, ¯ ¯ ¯′′ i ¯′′ j ¯ 0 → S0 −→ S −→ Q → 0, are exact. Then Q¯ determines the zero sheaf.

Proof. An arbitrary element of a stalk Qk of the induced sheaf has the ¯ Γ ′′ formρ ¯ xU jU f where x ∈ U and f ∈ (U, S ). Since j maps S onto S , there is an open set V, x ∈ V ⊂ U, for which f |V im jV . Then ′′ = | ∈ = ¯ ¯ ′′ = ρVU f f V im jV im iV and by exactness jV ρVU f 0. Hence ¯ = ¯ = ¯ ′′ = ρ¯xU jU f ρ¯xV ρ¯VU jU f ρ¯ xV jVρVU f 0.

77 Therefore the sheaf determined by Q¯ is the 0-sheaf.

′′ ′′ Note . In example 16, if Γo(U, S ) = im jU, we have Γo(U, S ) = ′′ ′′ ′′ Γ(U, S ) for all U expect X, but Γo(X, S ) = 0, Γo(X, S ) = Z2. Thus Q¯ X = Z2, Q¯ U = 0 for all smaller U, and thus Q¯ determine the 0-sheaf. i j Proposition 9. If X is paracompact and normal and if 0 → S ′ −→ S −→ S ′′ → 0 is an exact sequence of sheaves, there is an exact cohomology sequence

i∗ 0 → Ho(X, S ′) →···→ Hq(X, S ′) −→ ∗ j δ∗ + Hq(X, S ) −→ Hq(X, S ′′) −→ Hq 1(X, S ′) → . Lecture 13 67

′ ′′ ′ ′′ If h : (S , S , S ) → (S 1, S1, S 1) is a homomorphism of exact sequences, commuting with i and j,

i j 0 / S ′ / S / S ′′ / 0

h h h ′ i j ′′ 0 / S1 / S1 / S1 / 0 the induced homomorphisms S ∗ of the cohomology sequences commute with i∗, j∗ and δ∗, i.e. the following diagram is commutative:

∗ i∗ j δ∗ ... / Hq(x, S ′) / Hq(X, S ) / Hq(x, S ′′) / Hq+1(X, S ′) / ...

h∗ h∗ h∗ h∗

∗ j∗ ∗ ... q ′ i q q ′′ δ q+1 ′ ... / H (X, S1 ) / H (X, S 1) / H (X, S1 ) / H (X, S1 ) /

¯′′ ¯ Proof. As before, if So , Q denote the image and quotient presheaves in the exact sequence of presheaves of sections

i j 0 → S¯ ′ −→ S¯ −→ S¯′′, we obtain the exact sequence of presheaves 78

j ¯ ′ i ¯ 0 ¯′′ 0 → S −→ S −→ S0 → 0, and ¯i ¯j 0 → S¯′′ −→ S¯′′ −→ Q¯ → 0,

From these exact sequences of presheaves we obtain the following 68 Lecture 13

exact cohomology sequences:

Hq−1(X, Q¯) = 0

δ¯∗

∗ j∗ δ∗ ... q ′ i q o q ¯′′ o q+1 ′ ... / H (X, S ) / H (X, S ) / H (X, So ) / H (X, S ) / LL 8 LL qqq LL ¯i∗ qq j∗ LLL qqδ∗ L% qqq Hq(X, S ′′)

¯j∗ Hq(X, Q¯) = 0

Since Q¯ determine the 0-sheaf, by Proposition 8, Hq(X, Q¯) = 0 for all q ≥ 0 and hence, by exactness, ¯i∗ is an isomorphism. Hence if ∗ = ∗ ¯∗ −1 q ′′ q+1 ′ δ δo(i ) : H (X, S ) → H (X, S ), the cohomology sequence

0 → Ho(X, S ′) → Ho(X, S ) → Ho(X, S ′′) →··· + ···→ Hq(X, S ′) → Hq(X, S ) → Hq(X, S ′′) → Hq 1(X, S ′) →···

is exact. 79 Next, since the homomorphism h commutes with i and j, the induced homomorphism h of presheaves also commutes with i and j:

i j 0 / S¯′ / S¯ / S¯ ′′

h h h ¯′ i ¯ ¯′′ 0 / S1 / S1 / S1 .

Hence, the induced homomorphism h∗ of the cohomology modules ∗ ∗ ¯∗ ∗ commutes with i , jo, i and δo. Thus in the exact cohomology sequences, h∗ commutes with i∗, j∗ and δ∗. Lecture 13 69

Note. If X is not paracompact and normal, in general, ¯i∗ is not an isomorphism, to be precise, the cohomology sequence is not deﬁned. One does, however, have the exact sequence

0 → Ho(X, S ′) → Ho(X, S ) → Ho(X, S ′′) → H1(X, S ′) → H1(X, S ) → H1(X, S ′′) as one sees from the exact sequences 70 Lecture 13 ) ) ) ¯ Q ′′ ′′ o , X S S ( , , o ∗ X X i ¯ H ( ( & 1 1 L = L H H L / 0 L L ∗ o j ∗ L j L L L ) L S , X ( 1 H / ∗ i ) ′ S , X ( 1 8 q H q / q q q ∗ o q δ q . q 0 q ) ) q ′′ ′′ q o = ) S S ¯ , , Q 0 ∗ , X X i ¯ ( ( X & o o ( M o M H H M / M H M ∗ o M j ∗ j M M M ) M M S , X ( o H / ∗ i ) ′ S , X ( o H / 0 Lecture 13 71

The following examples show that Proposition 9 is not true, in gen- 80 eral,unless the space is both paracompact and normal.

Example 19. Let X consist of the unit segment I with the usual topology and of two points p and q. A neighbourhood of p (resp. q) consists of p (resp. q) together with the whole of I. Let S ′, S , S ′′ be the sheaves S ′, S , S ′′ of Example 16 over I together with zeros at p and q. A neighbourhood of 0p (resp. 0q) consists of the zeros over a neighbourhood of p (resp. q). Then there is an exact sequence

i j 0 → S ′ −→ S −→ S ′′ → 0 where i, j correspond to those in Example 16. Since H1(X, S ) = 2 ′ 1 ′′ H (X, S ) = 0 and H (X, S ) = Z2, there is no exact cohomology sequence. The space X is paracompact but not normal.

Example 20. Let X consists of a sequence of copies In of the unit segment together with two special points p and q. A neighbourhood of p (resp q) consists of p (resp. q) together with all but a ﬁnite number of the segments In. Let G be the 4-group and let S be the subsheaf of (X × G,π, X) consisting of zero at p and q and on each In a copy of ′′ the sheaf S of Example 16. Let S be the subsheaf of (X × Z2,π, X) formed by omitting the points (p, 1), (q, 1) and let the homomorphism ′′ j : S → S be induces by j : G → Z2 as deﬁned in Example 16. Then there is exact sequence

i j 0 → S ′ −→ S −→ S ′′ → 0 but H1(X, S ) = H2(X, S ′) = 0 while H1(X, S ′′) , 0. Thus there is 81 no exact cohomology sequence. The space X is paracompact and T1 but not normal.

Example 21. Let R, S , T1 be as in Example 18. Let X be the space consisting of (1) the elements r ∈ R and (2) segments Intr1r2 where n is a natural number, t ∈ T1, r1 and r2 are in R, and r1 , r2. Neighbourhoods of the points r are sets N(r; n, s1,., sk) where n is a natural number and s1,..., sk ∈ S , consisting of r together with all segments Imtr1r2 with 72 Lecture 13

m > n, r ∈ (r1, r2) and, for each i = 1,..., k, either r ∈ si ∈ t or r < si < t.

Let G be the 4-group and let S be the subsheaf of (X × G,π, X) consisting of zero at each r and a copy of the sheaf S of Example 16 on ′′ each Intr1r2 . Let S be the subsheaf of (X ×Z2,π, X) formed by omitting (r, 1) for all r, and let the homomorphism j : S → S ′′ be induced by j : G → Z2 as mentioned before. Then there is an exact sequence

0 → S ′ → S → S ′′ → 0 but H1(X, S ) = H2(X, S ′) = 0 while H1(X, S ′′) , 0. Thus again, there is no exact cohomology sequence. X is a perfectly normal Haus- dorﬀ space but is not paracompact. (A space X is said to be perfectly normal if, for each closed set C of X there is a continuous real valued function deﬁned on X and vanishing on C but not at point x ∈ X − C. Perfectly normal spaces are completely normal.) Lecture 14

Deﬁnition. A resolution of a sheaf G of A-modules is an exact sequence 82 of sheaves A-modules

e d1 dq 0 → G −→ S o −−→ S 1 →···→ S q−1 −−→ S q →··· such that Hq(X, S q) = 0, p ≧ 1, q ≧ 0. There are than induced homomorphisms

+ do d1 dq dq 1 + 0 −→ Γ(X, S o) −→· · · → Γ(X, S q−1) −→ Γ(X, S q) −−−→ Γ(X, S q 1) →···

+ + for which im dq ⊂ ker dq 1, i.e., dq 1dq = 0. The A-modules Γ(X, S k) (k ≧ 0) together with the homomorphisms dq form a formal cochain complex denoted by Γ(X, S ). Let the q − th cohomology module of the + complex Γ(X, S ) be denoted by HqΓ(X, S ) = ker dq 1/ im dq.

Example 22. Let X be the unit segment x : o ≦ x ≦ 1 , and let G be the subsheaf of the constant sheaf Z2 formed by omitting the points (0,1) and (1,1). A resolution

e d1 (1) 0 → G −→ S o −−→ S 1 → 0 of G is obtained by identifying G with the sheaf S ′ of Example 16 and taking S , S ′′, i, j for S o, S 1, e, d1. (That H p(X, S q) = 0, p ≧ 1, q ≧ 0 can be veriﬁed.) The induced sequence

0 → Γ(X, S o) → Γ(X, S ′) → 0

73 74 Lecture 14

is 83 0 → 0 → Z2 → 0.

Another resolution

e d1 d2 (2) 0 → G −→ S o −−→ S 1 −−→ S 2 → 0

o 1 2 1 of G is obtained by taking e, S as before, S = Z2+Z2, S = Z2, d = k j where k(x, 1) = (x, (1, 0)), and d2 with d2(x, (1, 1)) = d2(x, (0, 1)) = (x, 1). Another resolution

e d1 (3) 0 → G −→ S o −−→ S 1 → 0

o of G is obtained by taking S to be subsheaf of Z2 formed by omitted (0,1), with e : G → S o as the inclusion homomorphism and with d1 as the natural homomorphism onto the quotient sheaf S 1 = S o/G . Yet another resolution

e d1 (4) 0 → G −→ S o −−→ S 1 → 0

o 1 o of G is obtained by taking S = Z2 and S = S /G . 1 p In each case H Γ(X, S ) = Z2, H Γ(X, S ) = 0, p > 1.

Example 23. Let X be the sphere x2 + y2 + z2 = 1, and let G be the constant sheaf Z2. Let R denote the constant sheaf Z2 + Z2 with i : G → 84 R deﬁned by i(1) = (1, 1). Let R′ ⊂ R consist of all zeros together with ((x, y, z), (0, 1)) for z < 0; let S o = R/R′ and let j : R → S o be the natural homomorphism. Let e = ji : G → S o.

o o Let Io be the quotient sheaf S /e(G ) and let h : S → Io be the natural homomorphism. The stalks of Io are Z2 on the equator and 0 elsewhere. Let I be the quotient sheaf of R consisting of Z2 + Z2 on the equator and 0 elsewhere. Identify Io with the subsheaf of I consisting of all zeros and all ((x, y, 0), (1, 1)), and let k : Io → I be the inclusion homomorphism. Let I′ be for y > 0 and ((x, y, 0), (1, 0)) for y < 0. Let Lecture 14 75

S ′ = I/I′ and let l : I → S 1 be the natural homomorphism. Let 1 = o 1 2 = 1 o 2 d 1kh : S → S . Let S S /d1 S and let d be the natural homomorphism. Then from the diagram:

0 0

R′ I′

k R Io / I > = i }} || }} j || }} || h }} e || d′ d2 G / S o / S ′ / S 2 we see that e d1 d2 0 → G −→ S o −−→ S 1 −−→ S 2 → 0 is a resolution of G and the induced sequence

0 → Γ(X, S o) → Γ(X, S 1) → Γ(X, S 2) → 0 is 85 0 → Z2 + Z2 → Z2 + Z2 → Z2 + Z2 → 0 and o 2 1 H Γ(X, S ) = H Γ(X, S ) = Z2, H Γ(X, S ) = 0

Proposition 10. If X is paracompact normal and if

e dq 0 → G −→ S o →···→ S q−1 −−→ S q →··· is a resolution of G , there is a uniquely determined isomorphism η : HqΓ(X, S ) → Hq(X, G ). If

1 q e o d 1 q−1 d q 0 → G1 −→ S1 −−→ S1 →···→ S1 −−→ S1 →··· 76 Lecture 14

is a resolution of another sheaf G1 and if

o 1 o 1 h : (G , S , S ,...) → (G1, S1 , S1 ,...) is a homomorphism commuting with e, d1, d2,..., then the induced homomorphism H∗ commutes with η.

η HqΓ(X, S ) / Hq(X, G )

h∗ h∗ qΓ q H (X, S1) η / H (X, G1) .

∗ q q 86 Proof. The homomorphism h : H (X, G ) → H (X, G1) is the usual induced homomorphism. Now, since h commutes with dq, q ≧ 1, h also commutes with the homomorphisms

dq : Γ(X, S q−1) → Γ(X, S q) (q ≧ 1),

and hence there is an induced homomorphism

∗ q q h : H Γ(X, S ) → H Γ(X, S1).

+ Let zq = im dq = ker dq 1 ⊂ S q; then there are exact sequences

1 e do 0 → G −→ S o −−→ z1 → 0 (1) +  q q 1 i do +  0 → zq −→ S q −−−→ zg 1 → 0 (q ≧ 1)  q q  where i is the inclusion homomorphism and do is the homomorphism q q q = q q q induced by d : i do d . Since h commutes with d, h maps z in z1, and commutes with i, do.

1 e do 0 / G / S o / z1 / 0,

h h h 1 e o do 1 0 / G1 / S1 / z1 / 0, Lecture 14 77

q+1 iq do + 0 / zq / S q / zq 1 / 0,

h h h + q dq 1 zq i q o q+1 0 / 1 / S1 / z1 / 0.

Hence the induced homomorphism h∗ of the corresponding exact ∗ ∗ cohomology sequences also commutes also commutes with e , do, δ ∗ ∗ ∗ and i , do, δ respectively.

e d1 Case 1. q = 0. Since 0 → G −→ S o −−→ S 1 is exact, so is the se- 87 e d1 quence 0 → Γ(X, G ) −→ Γ(X, S o) −−→ Γ(X, S 1). Then Ho(X, S ) = ker d1 = ime, but e is a monomorphism and Γ(X, G ) = Ho(X, G ), hence e : Ho(X, G ) → HoΓ(X, S ) is an isomorphism commuting with h∗. Let η = e−1. Case 2. q > 0. The exact cohomology sequence corresponding to that exact sequences (1) for q − 1 (where zo = G ) is

q−1 q ∗ i do δ 0 → Γ(X, zq−1) −−→ Γ(X, S q−1) −→ Γ(X, zq) −→ H1(X, zq−1) → 0 →··· since H1(X, S q−1) = 0. Thus δ∗ induces an isomorphism

∗ q q 1 q−1 δ : Γ(X, z )/imdo → H (X, z ) (q ≧ 1).

q q+1 q+1 q Since im i = ker do = ker d , the monomorphism i induces an isomorphism

q∗ q q q q i : Γ(X, z )/ im do → im i / im d + = ker dq 1/ im dq = HqΓ(X, S ).

Thus we have an isomorphism

∗ δ∗(iq )−1 : HqΓ(X, S ) → H1(X, zq−1) (q ≧ 1) 78 Lecture 14

commuting with h∗, since h∗ commutes with δ∗ and (i∗)−1.

q ∗ do δ Γ(X, S q−1) / Γ(X, zq) / H1(X, zq−1) / 0 N NNN NN iq dq NNN NN' Γ(X, S q) N NN q+1 q+1 NNNd do NN NNN + ' + 0 / Γ(X, zq 1) / Γ(X, S q 1). iq+1 88 Also, the exact cohomology sequences corresponding to (1) contain

q−p+1 ∗ ∗ q−p ∗ (do ) + δ (i ) 0 −−−−−−−→ H p−1(X, zq−p 1) −→ H p(X, zq−p) −−−−−→ 0

for 1 < p < q and

∗ 1 ∗ ∗ do δ e 0 −−→ Hq−1(X, z1) −→ Hq(X, G ) −→ 0 for p = q.

Thus we have isomorphisms (q ≧ 1),

∗ δ∗(iq )−1 δ∗ HqΓ(X, S ) −−−−−−→ H1(X, zq−1) →···→ Hq−1(X, z1) −→ Hq(X, G )

commuting with h∗. Let η be the composite of these isomorphisms, η : HqΓ(X, S ) → Hq(X, G ).

Theorem 1 (Uniqueness theorem). If X is paracompact normal, and if

e d1 dq 0 → G −→ S o −−→ S 1 →···→ S q−1 −−→ S q →··· , 1 q e o d 1 q−1 d q 0 → G −→ S1 −−→ S1 → → S1 −−→ S1 →··· ,

89 are two resolutions of the same sheaf G of A-modules, there is a canon- Lecture 14 79 ical isomorphism

q q φ : H Γ(X, S ) → H Γ(X, S1).

o 1 2 o 1 2 Moreover, if h : (S , S , S ,...) → (S1 , S1 , S1 ,...) is a homomorphism commuting with e, d1, d2,...,

d1 d2 dq S o / 1 / ... / q−1 / S q / ... > S S e || || || || 0 / G h h h h @@ @@ e @@ @@ 1 2 q o d 1 d ... q−1 d q ... S1 / S1 / / S1 / S1 / then the induced homomorphism

∗ q q h : H Γ(X, S ) → H Γ(X, S1) is the isomorphism φ.

Proof. We have the canonical isomorphisms η, η1,

Q η q η1 q H Γ(X, S ) −→ H (X, G ) ←−− H Γ(X, S1);

= −1 let φ η1 η. There is commutativity in the diagram:

η HqΓ(X, S ) / Hq(X, G )

h∗ h∗ q η1 q H Γ(X, S1) / H (X, G ), where the homomorphism h∗ on the right is the identity. Hence the ∗ −1 = homomorphism h on the left is equal to η1 η φ.

Lecture 15

We now given example to show that the uniqueness theorem fails in 90 more general spaces.

Example 24. Let X consist of the unit segment I = {x : o ≦ x ≦ 1} together with two points p, q. A neighbourhood of p (resp. q) consists of p (resp. q) together with all of I. Let G be the subsheaf of the constant sheaf Z2 formed by omitting the points (p, 1), (q, 1), (0, 1), (1, 1). Let o S be the subsheaf of the constant sheaf Z2 + Z2 formed by omitting (p, a), (q, a) for all a , 0 and (0, (1, 0)), (0, (1, 1)), (1, (0, 1)), (1, (1, 1)). 1 2 Let S be the subsheaf of Z2 formed by omitting (q, 1). Let S have the stalk Z2 at p and 0 elsewhere; a neighbourhood of (p, 1) consists of (p, 1) together with all the zeros over I. Then there is a resolution

e d1 d2 d3 0 → G −→ S o −−→ S 1 −−→ S 2 −−→ 0 and the corresponding sequence

d1 d2 d3 0 → Γ(X, S o) −−→ Γ(X, S 1) −−→ Γ(X, S 2) −−→ 0 is 0 → 0 → 0 → Z2 → 0; 2 so H Γ(X, S ) = Z2.

There is also a resolution 91

e d1 0 → G −→ Ro −−→ 0

81 82 Lecture 15

where e is an isomorphism; then H pΓ(X, R) = 0 for all p. There is even a homomorphism h, commuting with e, d1,...,

d1 o / 0 / 0 / 0 =R e {{ {{ {{ {{ 0 / h G B BB BB e BB B d1 S o / S 1 / S 2 / 0.

[The space X is not normal.]

Deﬁnition . A sheaf S of A-modules is called ﬁne if for every closed set E in X and open set G in X with E ⊂ G, there is a homomorphism h : S → S such that

i) h(s) = s if π(s) ∈ E,

ii) h(s) = 0π(s) if π(s) < G¯

example of a ﬁne sheaf.

Example 25. For each open subset U of X, let S U be the A-module

of all functions f : U → A. If V ⊂ U, deﬁne ρVU to be restriction homomorphism. Let S be the sheaf of germs of functions determined = by the presheaf {S U, ρVU }. If E ⊂ G with E closed and G open, let hU : S U → S U beP deﬁned by

(hU f )(x) = f (x) χG(x)

92 where x ∈ U and χG is the characteristic function of G.

(χG(x) = 1 ∈ A if x ∈ G, χG(x) = 0 ∈ A if x < G).

Then {hU } : → is a homomorphism. If h : S → S is the induced homomorphism,P Ph(s) = s if π(s) ∈ E and h(s) = 0 if π(s) < G¯, hence the sheaf is ﬁne. Lecture 15 83

Exercise. If M is any non-zero A-module and the space X is normal, the constant sheaf (X χM,π, X) is ﬁne if and only if dim X ≤ 0.

Note. The set of all endomorphisms h : S → S forms an A-algebra, in general, non commutative, where h1·h2 is the composite endomorphism. The identity 1 : S → S is the unit element of the algebra.

If S = (S,π, X) is a sheaf and X1 is a subset of X, let X1 and S 1 = −1 π (X1) have the induced topology. Then (S 1,π|S 1, X1) is a sheaf called the restriction of S to X1. If X is normal, the restriction of a ﬁne sheaf S to any closed set C is ﬁne.

Proof. Let E be any closed subset of C and G any open subset of C with E ⊂ G. Extend G to an open set H of X, G = H ∩ C. Then, since X is normal, E closed in X, H open in X with E ⊂ H, there is an open subset V of X with E ⊂ V ⊂ V¯ ⊂ H. Since S is ﬁne, there is a homomorphism h : S → S with

h(s) = s if π(s) ∈ E,

= 0π(s) if π(s) ∈ X − V¯ .

93 Then if S1 is the restriction of S to C, h|S 1 : S 1 → S 1 is a homomorphism h1 : S1 → S1 and we have

h1(s) = h(s) = s if π(s) ∈ E

= 0π(s) if π(s) ∈ C − G¯ ⊂ C − G ⊂ X − H ⊂ X − V¯ .

Proposition 11. If X is normal, U = {Ui}iεIa locally finite covering of X, and if the restriction of S to each U¯ i is fine (in particular, if S is fine), there is a system {li}i∈I of homomorphisms li : S → S such that

i) for each i ∈ I there is a closed set Ei ⊂ Ui such that 1i(S x) = 0x if x < Ei, 84 Lecture 15

ii) 11 = 1. i∈I P (1 denotes the identity endomorphism S → S ). Proof. Using the normality of X, we shrink the locally finite covering U = {Ui}i∈I to the covering U {Vi}i∈I with V¯ i ⊂ Ui and we further shrink the locally finite covering U to the covering U = {Wi}i∈I with W¯ i ⊂ Vi. Since the restriction Si of S to U¯ i is fine, there is a homomorphism gi : Si → Si with

gi(s) = s if π(s) ∈ W¯ i,

= oπ(s) if π(s) ∈ U¯ i − V¯ i.

94 Let the homomorphism hi : S → S be deﬁned by

hi(s) = gi(s) if π(s) ∈ U¯ i,

= 0π(s) if π(s) ∈ X − Ui.

(This deﬁnition is consistent, since gi(s) = 0π(s) on U¯ i − Ui). This hi : S → S is continuous and is a homomorphism with

hi(s) = s if π(s) ∈ W¯ i,

= 0π(s) if π(s) ∈ X − V¯ i.

Let the set I of indices be well ordered and deﬁne the homomorphisms li : S → S by

1i  (1 − h j) hi, Yj

Each point x ∈ X has a neighbourhood Nx meeting Ui for only a ﬁnite number of i, say i1, i2,..., iq with i1 < i2 < ··· < iq. If π(s) ∈ Nx, = = = li(s) (1 − hi1) ··· (1 − hik−1 )hik (s), i ik, k 1,..., q, = 0π(s)for all other i. Lecture 15 85

Clearly li(S x) ⊂ S x and li|S x : S x → S x is a homomorphism. The −1 function li is continuous on each π (Nx) and coincides on the overlaps of two such neighbourhoods, hence li : S → S is continuous. Thus li : S → S is a homomorphism, and

hi(S x) = 0x, x < V¯ i, hence li(S x) = 0x, x < V¯ i. Take Ei = V¯ i ⊂ Ui. Let π(s) ∈ Nx; then for some ik, 1 ≦ k ≦ q, = π(s) ∈ Wik and hence hik (s) s. Hence 95 = (1 − hi1 ) ··· (1 − hiq )(s) 0.

Therefore

= + + + 1i(s) hi1 (s) (1 − h11 )hi2(s) ··· (1 − hi1 ) ··· (1 − hiq−1 )hiq (s) Xi∈I = s − (1 − hi1 ) ··· (1 − hiq )(s) = s.

Note. The homomorphisms li are usually not uniquely determined and they cannot therefore be expected to commute with other given homomorphisms.

Lecture 16

Let {li}i∈I be a system of endomorphisms of a ﬁne sheaf S correspond- 96 ing to a locally ﬁnite covering {Ui}i∈I of a normal space X. Each li gives a homomorphism li(U) : Γ(U, S ) → Γ(U, S ) for each open U and li(U) has a meaning and is the identity endomorphism of Γ(US ). i∈I AlsoP li determines a homomorphism

l¯i(U) : Γ(Ui ∩ U, S ) → Γ(U, S ) deﬁned by

(l¯i(U)g)(x) = li(g(x)) if x ∈ Ui ∩ U,

= 0 if x ∈ (X − Ui) ∩ U. One veriﬁes that the following diagrams are commutative.

li(U) l¯i(U) Γ(U, S ) / Γ(U, S ) Γ(Ui ∩ U, S ) / Γ(U, S ) n7 l¯i(U) n S nn S ρ ρ(Ui∩U, ) nnn ρ(Ui∩U, ) ρ(Ui∩V,Ui∩U) VU nnn nn l¯i(V) Γ(Ui ∩ U, S ) / Γ(Ui ∩ U, S ) Γ(Ui ∩ V, S ) / Γ(V, S ) li(Ui∩U)

If X is normal, S is ﬁne and U = {Ui}i∈I is a locally ﬁnite covering of X, then Hq(U , S ) → 0 for q ≧ 1.

Proof. Let kq−1 : Cq(U , S ) → Cq−1(U , S ) for q ≧ 1 be the homomorphism deﬁned by

q−1 (k f )(σ) = l¯i(Uσ)( f (iσ)), Xi∈I

87 88 Lecture 16

where iσ = i, io,..., iq−1 if σ = io,..., iq−1. (This infinite sum of sec- 97 tions is finite neglecting zeros, in some neighbourhood of each point.) Using the fact that ∂o(iσ) = σ and ∂ j(iσ) = i∂ j−1(σ) for j > 0, one verifies that + δqkq−1 f + kqδq 1 f = f. (The computation is given at the end of the lecture.) Hence each cocycle f is a coboundary q.e.d.

Proposition 12. For a ﬁne sheaf S over a paracompact, normal space X,Hq(X, S ) = 0 for q ≧ 1

Proof. Hq(X, S ) = 0 q ≧ 1 for each locally ﬁnite covering U of the space X. Since the space X is paracompact, this means that Hq(X, S ) = 0, q ≧ 1.

Corollary . If X is paracompact and normal, any exact sequence of sheaves

e d1 dq 0 → G −→ S o −−→· · · → S q−1 −−→ S q → ...,

where each S q(q ≧ 0) is ﬁne, is a resolution of G .

Definition. A sheaf S is called locally fine, if for each open U and each x ∈ U, there is an open V with x ∈ V ⊂ U such that the restriction of S to V¯ is fine.

If X normal, a ﬁne sheaf S is locally ﬁne.

Proof. The restriction of S to an arbitrary closed set is ﬁne.

98 Proposition 13. If X is paracompact normal, a locally ﬁne sheaf S is ﬁne.

Proof. Let E ⊂ G1, with E closed and G1 open. If G2 = X − E then {Gi}i=1,2 is a covering of X. Since S is locally fine, for each x ∈ Gi, there is an open Vx with x ∈ Vx ⊂ Gi such that the restriction of S to V¯ x is fine. Since X is paracompact, there is a locally finite refinement U {U j} j∈J of {Vx}x∈X, hence U is also a refinement of {Gi}. If U j ⊂ Vx, Lecture 16 89

then U¯ j ⊂ V¯ x, and V¯ x, being closed in X, is normal. Since the restriction of S to V¯ x is ﬁne, the restriction of S to U¯ j is also ﬁne. Now by proposition 11, there exist endomorphisms l j such that l j = 1 and l j j∈J P is zero outside a closed set E j ⊂ U j. Choose the function τ : J → (1, 2) so that U j ⊂ Gτ( j) and let

li = l j, i = 1, 2. τX( j)=i

Then I1 + I2 = l and

li(s) = 0 if π(s) ∈ X − U j ⊃ X − Gi. τ[( j)=i

Hence l1(s) = 0 if π(s) ∈ X − G1 and l1(s) = s if π(s)εX − G2 = E.

l1 thus gives the required function, and this completes the proof.

Corollary . If X is paracompact and normal, any exact sequence of sheaves e d1 dq 0 → G −→ S o −−→ ... S q−1 −−→ S q ··· , where each S q(q ≧ 0) is locally ﬁne, is a resolution of G . 99

The following examples shows that, in more general spaces, ﬁneness need not coincide with local ﬁneness.

Example 26. Let X have points a, b,..., h with base for open sets consisting of ( f ), (g), (h), (d, f, h), (e, g, h), (c, f, g), (b, e, g, h), (a, d, e, f, g, h). Let S be the subsheaf of the constant sheaf Z2 formed by omitting (c, 1), ( f, 1). Then S is fine but not locally fine. In fact, V = (c, f, g) is the least open set containing c and the restriction of S to V¯ = X − (h) is not fine. (X is not normal.) 90 Lecture 16

Example 27. Let T be the space of ordinal numbers ≦ ω1 with the usual topology induced by the order. Let A be the space or ordinal numbers ≦ ωo and let X be the subspace of TXA formed by omitting the point (ω1,ωo). Let S be the constant sheaf Z2 over X. Then S is locally ﬁne, for every point has a closed neighbourhood which is normal and zero dimensional. But S is not ﬁne. If B is the set of even numbers, then B ⊂ A. Let E = ω1 × B and G = T × B. Then E ⊂ G ⊂ X with E closed and G open. There is no endomorphism of S which is the identity on E and is zero outside G¯. (X in neither paracompact nor normal.)

100 Example 28. The space M of Quart. Jour. Math. 6 (1955), p. 101 is normal and locally zero dimensional but not zero dimensional. Therefore 101 the constant sheaf Z2 is locally ﬁne but not ﬁne. (M is not paracompact).

+ δqkq−1 f + kqδq 1 f = f. q q q−1 = j q−1 δ k f (σ) (−1) ρ(Uσ, U∂ j σ)(k f )(∂ jσ) Xj=0 q = j ¯ (−1) ρ(Uσ, U∂ j σ) Ii(U∂ j σ) f (i∂ jσ) Xj=o Xi q = j ¯ (−1) l1(Uσ)ρ(Uiσ, U j∂ j σ) f (i∂ jσ). Xj=0 Xi q q+1 q+1 k δ = l¯1(Uσ)(δ f )(iσ) Xi q+1 = ¯ j li(Uσ) (−1) ρ(Uiσ, U∂ j iσ) f (∂ jiσ) Xi Xj=0

= l¯i(Uσ)ρ(Uiσ, Uσ) f (σ) Xi q+1 j + l¯i(Uσ) (−1) ρ(U jσ, Ui∂ j−1∂) f (i∂ j−1σ) Xi Xj=1 Lecture 16 91

q = ¯ j+1 li(Uσ) f (σ) li(Uσ) (−1) ρ(Uiσ , Ui∂ jσ) f (i∂ jσ). Xi Xi Xj=0 q q−1 q q+1 δ k f (σ) + k δ = σili(Uσ) f (σ) = f (σ)

Lecture 17

In this and the next lecture, we shall give a proof of de Rham’s theorem. 102 Let X be an indefinitely differentiable (C∞) manifold of dimension n, which is countable at infinity (i.e) a countable union of compact sets); we assume that X is a Hausdorff space. Then X is paracompact and normal. (Dieudonne, Jour. de Math. 23 (1944)). The set E p(U) of all C∞ (alternating) differential p-forms on an open set U forms a vector space over the field of real numbers. Exterior differentiation gives a homomorphism dp,

dp : E p−1(U) → E p(U) with dp+1dp = 0. In particular, there is a sequence

do d1 dp dn dn+1 0 −−→ E o(X) −−→ E 1(X) →··· −−→ E p(X) →··· −−→ E n(X) −−−→ 0 with im dp ⊂ ker dp+1. Let

+ H p(E (X)) = ker dp 1/ im dp.

This vector space of the closed p-forms modulo the derived p-forms is called the p-th de Rham cohomology vector space of the manifold X. If V ⊂ U, the inclusion map i : V → U induces a homomorphism

= −1 p p ρVU i : E (U) → E (V)

p = p which commutes with d. Thus the system E {E (U), ρVU } is a 103

93 94 Lecture 17

presheaf which determines a sheaf Ωp, called the sheaf of germs of p-forms, and dp : E p−1 → E p is a homomorphism of presheaves which induces a homomorphism

dp : Ωp−1 −→ Ωp.

There is a constant presheaf {R, ρVU } where R is the ﬁled of real

numbers and ρVU : R → R is the identity. This presheaf determines the constant sheaf R. There is a homomorphism e : R → E o(U), (E o(U) is the space of C∞ functions on U) where e(r) is the function on U with

the constant value r, and further e commutes with ρVU . Thus, there is an induced sheaf homomorphism e : R → Ωo. Hence, we have a sequence of homomorphisms of sheaves

e d1 dp 0 → R −→ Ωo −−→· · · → Ωp−1 −−→ Ωp → ...,

with dp+1dp = 0. There is a homomorphism

p Ω¯p { fU } : {E (U), ρVU }→ ,

(Ω¯ p denotes the presheaf of sections of Ωp), where the image of an el- 104 ement of E p(U) is the section over U which it determines in Ωp. Then d commutes with fU and, in particular, with fx. Thus we have the commutative diagram:

dp E p−1(X) / E p(X)

fX fX dp Γ(X, Ωp−1) / Γ(X, Ωp).

If a p-form ω ∈ E p(X) is not zero, there is some point x ∈ X at which it does not vanish, and hence ρxXω , 0. Thus fXω , 0, i.e., fX is a monomorphism. (In the same way, fU is a monomorphism for each p open set U.) fX is also onto, hence an isomorphism. For, if g ∈ Γ(X, Ω ), Lecture 17 95 then since Ωp is the sheaf of germs of p-forms, for each x ∈ X there is a neighbourhood Ux of x and a p-form ωx deﬁned on Ux such that section g and the section determined by ωx coincide on Ux. Then {Ux} forms a covering for X, and since the section determined by ωx and ωy coincide on Ux ∩ Uy, using the fact that fU is a monomorphism for each open set U, we see that the forms ωx and ωy themselves coincide on Ux ∩ Uy; hence they deﬁne a global from ω, such that fX(ω) = g. Thus fX gives an isomorphism of the sequences:

d1 dp 0 / E o(X) / ... / E p−1(X) / E p(X) / ...

fX fX fX d1 dp 0 / Γ(X, Ωo) / ... / Γ(X, Ωp−1) / Γ(X, Ωp) / ...

Hence there is an induced isomorphism of the cohomology vector 105 spaces: ∗ q qΓ Ω fX : H (E (X)) → H (X, ) (q ≧ 0). Poincare’s lemma. The sequence

d1 dp 0 → R → Ωo −−→· · · → Ωp−1 −−→ Ωp →··· is exact.

Proof. We have to prove that for each point a ∈ X, the sequence

d1 p e Ωo Ωp−1 d Ωp 0 → Ra −→ a −→··· → a −−→ a ··· Ωo is exact, where the subring Ra of a, consisting of the germs of constant functions at a, is identified with the filed R of real number. Choose a coordinate neighbourhood W of a with coordinates (x1,..., xn) and p suppose that a = (0,..., 0). Then Ωa is the direct limit of the system p E (U), ρVU aεU′ where U belongs to the cofinal set of those spherical 2 + + 2 2 neighbourhoods x1 ··· xn < r which are contained in W. For each such U, let

h : E o(U) → R 96 Lecture 17

and kp−1 : E p(U) → E p−1(U) (p ≧ 1) be the homomorphisms deﬁned by

h( f ) = f (0,..., 0),

and

p−1 k ( f (x1,..., xn)dxi ··· dxip ) 1 P = p−1 j−1 ˆ ( f (tx1,..., txn)t dt) · (−1) xi j dxi1 ... dxi j ... dxip Z0 Xj=1

106 respectively. (The formula on the right is an alternating function of p−1 o p i1,..., ip; h and k are then extended by linearity to E (U) and E (U) respectively.) One now veriﬁes that

eh + k0d1 = 1, + dpkp−1 + kpdp 1 = 1 (p ≧ 1),

where 1 denotes the identity map. (The computation is carried out at the end of the lecture.) Thus f ∈ ker d′ implies that f ∈ im e and ω ∈ ker dp+1 implies that ω ∈ im dp. Hence ker d1 = im e and ker dp+1 = imdp, since already im e ⊂ ker d1 and im dp ⊂ ker dp+1. Hence the sequence

1 p e o d p−1 d p 0 → RU −→ E (U) −−→ ... E (U) −−→ E (U) → ... is exact, and since exactness is preserved under direct limits, therefore the limit sequence

1 p e Ωo d Ωp−1 d Ωp 0 → Ra −→ a −−→· · · → a −−→ a → ... is exact, q.e.d. The sheaf Ωp is ﬁne.

Proof. Since the space X is paracompact and normal, by Proposition 13 (Lecture 16), it is enough to prove that the sheaf Ωp is locally ﬁne. Let U be an open set of X, and let a ∈ U. Lecture 17 97

107 We may assume that U¯ is compact and that it is contained in some coordinate neighbourhood N of a. Let V be an open subset with a ∈ V and V¯ ⊂ U. We will now prove that the restriction Ωp of Ωp to V¯ is fine. V¯ Let E ⊂ G with E closed and G open in V¯ . Extend G to an open set H ⊂ U, so that G = V¯ ∩ H. Then U¯ is covered by a finite number of spherical neighbourhoods S i contained in N, such that S¯ i either does not meet E or is contained in H. For each i, choose an indefinitely differentiable function fi which is positive inside S i and vanishes out side S i. We construct one such n 2 2 function as follows: For the spherical neighbourhood (x j −bi j) < ri , j=1 let P

gi(r) = 0 (r ≧ ri), ri 1 r = i ≦ ≦ , exp ri dt ( r ri) Zr ((t − 2 )(t − ri)) 2 ri 1 r = exp dt (0 ≦ r ≦ i ) − ri − Zri/2 ((t 2 )(t ri)) 2

and deﬁne fi by

n 2 fi(x1,..., xn) = gi( (x j − bi j) . tuv Xj=1

Let ϕ1(x) = fi(x), summed for all i for which S¯ i meets E and let = + ϕ2(x) fi(x), summedP for all the remaining i. Then ϕ1 ϕ2 is positive in U and,P if θ(x) = ϕ1(x)/(ϕ1(x) + ϕ2(x)), θ is indefinitely differentiable in U, is zero outside H and is constant, 108 equal to 1, in a neighbourhood of E. Let h : E p(W) → E p(W), for open W ⊂ U be defined by h(ω) = Ω θ · . Then h is a homomorphism commuting with ρYW , Y open in U, Y ⊂ W. Hence h induces a homomorphism h : Ωp(U) → Ωp(U) for which

h(ωb) = ωb if b ∈ E, 98 Lecture 17

h(ωb) = 0b if b ∈ V¯ − G¯ ⊂ U − H.

(ωb denotes the germ determined by ω at b ∈ U.) Proposition 14. There is an isomorphism

∗ p P η fX : H (E (X)) → H (X, R). Proof. By the corollary to Proposition 12, the exact sequence

e d1 dp 0 → R −→ Ωo −−→· · · → Ωp−1 −−→ Ωp → ...

is a resolution of the constant sheaf R and hence, by Proposition 10, there is an isomorphism

η : H pΓ(x, Ω) → H p(X, R);

but we already have (as proved in the earlier part of this lecture) an isomorphism ∗ p pΓ Ω fX : H (E (X)) → H (X, ).

+ 109 (1)eh + k0d1 = 1. (2)dpkp−1 + kpdp 1 = 1 (p ≧ 1).

0 (1) If f (x) = f (x1,..., xn) ∈ E (U),

n 1 eh f (x) = f (0,..., 0) and d f (x) = Di f (x)dxi, (∗), Xi=1

1 1 d hence kod1 f (x) = n D f (tx)dt · x = f (tx)dt = f (x) − i=1 0 i i 0 dt f (0), thus eh f (x) +Pkod1Rf (x) = f (x). R = (2) If ω f (x1,..., xn)dxi1 ··· dxip ,

1 p p p−1 = p p−1 j−1 d k d ( f (tx)t dt · (−1) xi j dxi1 ··· dxˆi j ··· dxip Z0 Xj=1 Lecture 17 99

n 1 p = p j−1 ( ( Di f (tx)t dt)) · (−1) xi j dxi1 ··· dxˆi j ... dxip Xi=1 Z0 Xj=1 1 + p−1 ( f (tx)t dt) · pdxi1 ··· dxip . Z0 n p p+1 = p k d ω k ( Di f (x)dxidxi1 ··· dxip ) Xi=1 n i p ( Di f (tx)t · dt) xidxi1 ··· dxip Xi=1 Z0 p j−1 − (−1) xi j dxidxi1 ··· dxˆi j ··· dxip . Xj=1

(∗) Di denotes partial derivation with respect to the i - th variable con- cerned. + Thus dpkp−1ω + kpdp 1ω 110

1 = p−1 ( f (tx)p · t dt)dxi1 ... dxip Z0 n 1 + p Di f (tx)t · dt · xidxi1 ... dxip Xi=1 Z0 1 1 n = p p f (tx)t − Di f (tx) · xit dt dxi1 ... dxip 0 Z0 Xi=1 n 1 + p Di f (tx) · t dt · xidxi1 ... dxip Xi=1 Z0 = f (x)dxi1 ... dxip = ω.

Lecture 18

Let sp be a ﬁxed p-simplex in Euclidean p-space Rp, with vertices 111 p a0, a1,..., ap, i.e. s is the convex set spanned by points a0,..., ap which are in general position. We may assume that ao is the origin p p−1 and a1,..., ap are unit points of a coordinate axes in R , and that s is p the face opposite ap in s .

Deﬁnition . A diﬀerentiable singular p-simplex in a C∞ manifold X is aC∞− map t : sp → X. The image, im t, is called the support of the p−1 singular simplex t. The j-th face ∂ jt is the composite map td j : s → X p−1 p where d j : s → s is the linear map which maps ao,..., ap−1 into ao,..., aˆ j,..., ap.

The support of ∂ jt is contained in the support of t.

Definition . A differentiable singular p-cochain in an open set U ⊂ X is a real valued function of differentiable p-simplexes with supports in U; f (t) ∈ R if suppt ⊂ U.

p ﬀ The set S U of of all di erentiable P− cochains in U forms a real P → p vector space. There is a restriction homomorphism ρVU : S U S V for p p−1 p V ⊂ U and a coboundary homomorphism d : S U → S U deﬁned by

p p j (d f )(t) = (−1) f (∂ jt). Xj=0

p o p+1 The homomorphisms ρVU and d commute, and imd ⊂ ker d .

101 102 Lecture 18

In particular, there is a sequence

p o d1 1 p−1 d p 0 → S X −−→ S X →···→ S X −−→ S X → ... + + 112 with dp 1dp = 0, i.e., im dp ⊂ ker dp 1. Let p p+1 p H (S X) = ker d /imd . This vector space is called the p−th cohomology vector space of X based on differentiable singular cochains. Since a singular 0-simplex may be identified with the point which is o its support, S U can be identified with the vector space of all functions f : U → R. The vector space E o(U) of C∞− functions on U is a o subspace of S U , and the space R of constant functions on U is a subspace o o o of E (U), i.e., R ⊂ E (U) ⊂ S U . p p The presheaf S U, ρVU determines a sheaf S and since ρVU com- p 0 mutes with d and with the inclusion homomorphism e : R → S U , there are induced homomorphisms

p e d1 d 0 → R −→ S 0 −−→· · · → S p−1 −−→ S p → ... Here the constant sheaf R is identiﬁed with the sheaf of germs of constant functions. There is a homomorphism

p → ¯p gU : S U , ρVU S , p where the image of an element of S U is the section which it determines. Then d commutes with gU and, in particular, with gX. Then we have the commutative diagram:

p S o ... p−1 d S p ... 0 / X / / S X / X /

gX gX gX

 dp 0 / Γ(X, S o) / ... / Γ(X, S p−1) / Γ(X, S p) / ... 113 ∗ p Γ The induced homomorphism gX : H (S X) → (X, S )is an isomorphism for each p. Lecture 18 103

Proof. For each covering of X, let S p denote the vector space of U U all real valued functions of differentiable p−simplexes, which are defined for each simplex t with supp t contained in some open set of U . dpmapsS p−1 into S p and, if is a refinement of , there is the ob- U U W U vious restriction homomorphism g : S p → S p commuting with W U U W p p d . Then S , gW U is a direct system and it can be proved, using a U method similar to the one used in the proof of Proposition 8 (Lecture 12), that its direct limit is Γ(X, S p). The induced homomorphisms g∗ : H p(S ) → H p(S ) are W U U U W isomorphisms. (See Cartan Seminar, 1948-49, Exposé8, §3.). Hence g∗ : H p(S ) → H pΓ(X, ) is an isomorphism and, in particular, tak- U U S ing U as the covering by one open set X, ∗ p pΓ gX : H (S X) → H (X, S ) is an isomorphism, q.e.d. The sequence

e dp 0 → R −→ S →···→ S p−1 −−→ S p → ... is exact.

Proof. It is suﬃcient to show that 114

p e 0 p−1 d p 0 → R −→ S U →···→ S U −−→ S U → ... is exact for a coﬁnal system of neighbourhoods U of each point a ∈ X. For this system, take the spherical neighbourhoods U contained in a coordinate neighbourhood N of a. The result is proved using the conical homotopy operator. (See Cartan Seminar, 1948-49, Expose 7, §6 ; his formula should be replaced by

y(λ0,...... λp+1) = φ(λ0)x (λ1/(1 − λ0),...) λ0 , 1,

= 0 λ0 = 1; where the indeﬁnitely diﬀerentiable function φ is chosen so that 0 ≦ φ(λ)0) ≦ 1 for 0 ≦ λ0 ≦ 1, φ(0) = 1 and φ(λ0) = 0 for λo in some neighbourhood of 1.) 104 Lecture 18

The sheaf S p is ﬁne. p p Proof. Let E ⊂ G with E closed and G open. Deﬁne hU : S U → S U by

(hU f )(t) = f (t) if supp t ⊂ G = 0 otherwise.

Then hU is a homomorphism commuting with ρVU and induces a p p p p homomorphism h : S → S such that hx : S x → S x is the identity if x ∈ G, and is zero if x ∈ X − G¯. Thus S p is ﬁne, q.e.d.

p p 115 Now let hU : E (U) → S U be the homomorphism deﬁned by

−1 (hU ω)(t) = t ω, Z sp

−1 where t ω is the inverse image of the form ω by t. Clearly hU com- Ωp p mutes with ρVU , hence induces a homomorphism h : → S with commutativity in ρ p xU p E (U) / Ωx

hU hx ρ p xU p S U / S x . Hence there is an induced homomorphism h : Γ(X, Ωp) → Γ(X, S p) with commutativity in

fX E p(X) / Γ(X, Ωp)

hX h g p X Γ p S X / (X, S ).

p hU commutes with d . Proof.

p −1 p (hU d ω)t = t (d ω) Z sp Lecture 18 105

= dp(t−1ω) ( sincedp commutes with t−1) Z sp p = (−1) j t−1ω( by Stokes’ theorem for sp), Z Xj=0 p d j s p j = (−1) (hU ω)∂ jt Xj=0 p = (d hU ω)t.

∗ p p Thus hX induces a homomorphism hX : H (E (x)) → H (S X). Also 116 the homomorphisms h : Ωp → S P and hence the induced homomorphisms h : Γ(X, Ωp) → Γ(X, S P) commute with dp, and thus there are induced homomorphisms h∗ : H pΓ(X, Ω) → H pΓ(X, S ). There is commutativity in

f ∗ H p(E (X)) X / H pΓ(X, Ω)

∗ ∗ hX h g∗ p X p H (S X) / H Γ(X, S ). The homomorphism h∗ is an isomorphism.

Proof. We have the two resolutions

dp Ω0 / ... / Ωp−1 / Ωp / ... > e }} }} }} }} 0 / R h h h AA AA AA e AA dp S 0 / ... / S p−1 / S p / ... of R. The homomorphism h commutes with dp, and commutativity in the triangle follows from the fact that R ⊂ Ωo → S o and e, h and e are inclusion homomorphisms. Hence h∗ is the isomorphism of the uniqueness theorem, q.e.d. 106 Lecture 18

Theorem 2 (de Rham). The homomorphism

∗ p p hX : H (E (X)) → H (S X)

is an isomorphism.

Proof. The following diagram is commutative :

f ∗ H p(E (X)) X / H pΓ(X, Ω)

∗ ∗ hX h g∗ p X p H (S X) / H Γ(X, S ). 117 ∗ ∗ ∗ Since fX, gX, and h are isomorphisms, and the above diagram is ∗ = ∗−1 ∗ ∗ ∗ commutative, we have hX gX h fX. Therefore, hX is an isomorphism. Lecture 19

Deﬁnition. A double complex K is a system of A− modules (A is a com- 118 mutative ring with unit element K p,q , indexed by pairs (p, q) of integers, together with homomorphisms d1 and d2 with p,q p−1,q p,q p,q p,q−1 p,q d1 : K → K , d2 : K → K , p+1,q p,q = p,q+1 p,q = d1 · d1 0, d2 d2 0, p+1,q p+1,q−1 + p+1,q p,q = d2 d1 d1 d2 0,

(i.e., d1 and d2 are diﬀerential operators of bi degree (1, 0) and (0, 1) respectively, which anticommute. Usually we omit the superscripts attached to d1 and d2.) We have then the anticommutative diagram:

 d2 d2 ... / K p−1, q−1 / K p−1, q / K p−1, q+1 / ...

d1 d1 d1

 d2 d2 ... / K p,q−1 / K p,q / K p,p+1 / ...

(Each row and column of a double complex forms a (single) complex with the homomorphisms d2 and d1 respectively.) Deﬁnition. A subcomplex L of K is a system of submodules Lp,q ⊂ K p,q p−1,q p,q p,q−1 p,q stable under d1 and d2; thus d1(L ) ⊂ L and d2(L ) ⊂ L . 119

107 108 Lecture 19

If L is a subcomplex of a double complex K, then clearly K/L = p,q p,q K /L with the homomorphisms induced by d1 and d2 is again a double complex. p,q p,q p+1,q p,q Let Z1 (K) be the kernel of d1 : K → K and let B1 (K) be p−1,q p,q 2 = p,q p,q p,q the image of d1 : K K . Since d1 0, B1 ⊂ Z1 ⊂ K . P−1,Q = p,q p,q−1 Now d1(Z )1 0 ∈ Z1 and, since d1d2(Z1 ) = p,q−1 = p,q−1 p,q = p,q −d2d1(Z1 ) 0, d2(Z1 ) ⊂ Z1 . Thus Z1(K) Z1 is a subcomplex of K. P−1,q = p,q p,q−1 = p−1,q−1 = Also d1(B1 ) 0 ∈ B1 and d2(B1 ) d2d1(K ) p−1,q−1 p,q = p,q −d1d2(K ) ⊂ B1 . Thus B1(K) B1 is a subcomplex of = = p,q = p,q p,q Z1(K). Let H1(K) Z1(K)/B1(K) with H1(K) H1 (K) Z1 /B1 . In the double complex H1(K), the homomorphism induced by d1 is the trivial (zero) homomorphism.

... p−1,q−1 d2 p−1,q d2 p−1,q+1 ... / H1 / H1 / H1 /

d d + ... p,q−1 2 H p,q 2 p,q 1 ... / H1 / 1 / H1 /

p,q = p,q = Similarly if Z2 ker d2 and B2 im d2 there is a double complex = p,q = p,q p,q H2(K) with H2(K) H2 (K) Z2 |B2 . In H2(K), the homomorphism induced by d2 is the trivial homomorphism.

 p−1,q−1 p−1,q p−1,q+1 H2 H2 H

d1 d1 d1 + p,q−1 H p,q p,q 1 H2 2 H2

 Lecture 19 109

120 In particular, there is a double complex H2(H1(K)), which we write = p,q p,q = p,q p,q p,q = as H12(K) H12 (K) , where H12 Z12 B12 and Z12 ker d2 : p,q p,q+1 = p,q−1 p,q H1 → H1 ; B12 im d2 : H1 → H1 . In the double complex H12(K), the induced homomorphisms d1 and d2 are the trivial homomorphisms. Similarly there is a double complex

= p,q = H21(K) H21 (K) H1(H2(K)). p,q = q p Notations. In terms of the more usual notation, H12 (K) HII(HI (K)) p,q = p q and H21 (K)) HI (K)(HII(K)). p,q To the double complex K = K , d1, d2 we can now associate the (single) complex Kn, d Kn being the direct sum Kn = K p,q (each p+q=n n P n−1 K is an A− module) with the diﬀerential operator d = d1 +d2 : K → n 2 = 2 + + + 2 = K . (d is a homomorphism and d d1 d1d2 d2d1 d2 0).

dn dn+1 ···→ Kn−1 −−→ Kn −−−→ Kn+1 →··· Thus im dn ⊂ ker dn+1 and there are cohomology modules Hn(K) = ker dn+1/imdn.

Deﬁnition. A homomorphism f : K → L (of bidegree (r, s)) of double 121 complexes is a system of homomorphisms f : K p,q → Lp+r,q+s. Deﬁnition. A map f : K → L of double complexes is a homomorphism of bidegree (0, 0), which commutes with d1 and d2. Clearly a map f : K → L induces homomorphisms

+ p,q p,q ∗ p,q p,q f : H1 (K) → H1 (L), f : H12 (K) → H12 (L) ∗ p,q p,q and f : H21 (K) → H21 (L) Also, f determines obvious homomorphisms f : Kn → Ln which ∗ commute with d = d1 + d2 and there are induced homomorphisms f : Hn(K) → Hn(L) 110 Lecture 19

Deﬁnition. A sequence

hr hr+1 ···→ Kr−1 −→ Kr −−−→ Kr+1 →···

of homomorphisms of bidegree (0, 0) of double complexes is called exact if, each pair (p, q), the sequence

p,q p,q p,q ···→ Kr−1 → Kr → Kr−1 →··· is exact.

Given an exact sequence of maps of double complexes

i j 0 → K′ −→ K −→ K′′ → 0,

there is an exact cohomology sequence

∗ i∗ j d∗ + ···→ Hn(K′) −→ Hn(K) −→ Hn(K′′) −−→ Hn 1(K′) →···

122 Proof. The sequences

i j 0 → K′n −→ Kn −→ K′′n → 0

are clearly exact for each each n, and d commutes with i and j. Then, using the standard arguments of Lecture 10, we obtain the result.

Deﬁnition. Two maps of double complexes, f : K → L and g : K → L p+1,q are called homotopic ( f ◦g) if there exist homomorphisms h1 : K → p,q p,q+1 p,q L and h2 : K → L (i.e., h1 and h2 are homomorphisms K → L of bidegree (−1, 0) and (0, −1) respectively) such that

d1h1 + h1d1 + d2h2 + h2d2 = g − f,

d1h2 = −h2d1, d2h1 = −h1d2.

(Homotopy of maps is obviously an equivalence relation.) Homotopic maps f :→ L and g : K → L induce the same homomorphisms Lecture 19 111

∗ = ∗ p,q p,q (i) f g : H12 (K) → H12 (L), ∗ = ∗ p,q p,q (ii) f g : H21 (K) → H21 (L), (iii) f ∗ = g∗ : Hn(K) → Hn(L).

Proof. Let (h1, h2) be the pair of homomorphisms K → L which express the homotopy between f and g. Since h2, like d2, anticommutes with d1, there are induced homomorphisms + p,q+1 p,q h2 : H1 (K) → H1 (L). Further, since 123

d1h1 + h1d1 = g − f − d2h2 − h2d2, h1 expresses the homotopy of g and f + d2h2 + h2d2 from a column complex of K to the corresponding column complex of L.

Hence + + + + + = + p,q p,q f d2h2 h2 d2 g : H1 (K) → H1 (L), + + + = + + i.e., d2h2 h2 d2 g − f . + + + Thus h2 expresses the homotopy of g and f from a row complex p,q p,q of H1 (K) to the corresponding row complex of H1 (L) . Hence ∗ = ∗ p,q p,q f g : H12 (K) → H12 (L). This proves (i). The proof of (ii) is carried out in a similar manner, using the other anti-commutativity d2h1 = −h1d2. n+1 n To prove (iii), let h = h1 + h2 : K → L . Then dh + hd = (d1 + d2)(h1 + h2) + (h1 + h2)(d1 + d2)

= (d1h1 + h1d1 + d2h2 + h2d2) + (d1h2 + h2d1) + (d2h1 + h1d2) = g − f.

Thus h is a homotopy of the complexes Kn and Ln , and we obtain ∗ ∗ n n f = g : H (K) → H (L). 112 Lecture 19

Note . In the double complexes which occur in the usual applications, one has commutativity d1d2 = d2d1, d1h2 = h2d1 and d2h1 = h1d2 124 rather than anti-commutativity. The commutative case can be trans- formed into the anti-commutative case and vice versa by replacing d2 P p,q−1 p,q P p,q−1 p,q by (−1) d2 : K → K and h2 by (−1) h2 : K → K . These substitutions do not change ker d2, im d2, etc., and so the cohomology p,q p,q modules H1 , H12 , etc., remain unchanged. But if K is a commutative double complex, Kn, d, Hn(K) are understood to refer to the associated anticommutative double complex. Lecture 20

Deﬁnition. If Ω is a directed set, a direct system of double complexes 125

Kλ, φµλ is a system of double complexes Kλ and maps λ,µ∈Ω

φµλ : Kλ → Kµ (λ<µ), such that

(i) φλλ is the identity,

(ii) φνµφµλ is homotopic to φνλ for λ<µ<ν.

If Kλ, φµλ is a direct system of double complexes, there are uniquely determined direct limits: p,q = p,q ∗ (i) H12 (K) direct limit {H12 (Kλ), φµλ}, p,q = p,q ∗ (ii) H21 (K) direct limit{H21 (Kλ), φµλ}, n = n ∗ (iii) H (K) direct limit{H (Kλ), φµλ}.

p,q ∗ Proof. (i) The system H12 (Kλ), φµλ is a direct system, as ∗ (a) φλλ is the identity, since φλλ is the identity, ∗ ∗ = ∗ = ∗ (b) φνµφµλ (φνµφµλ) φνλ, since homotopic maps induce the same homomorphism on the cohomology groups.

113 114 Lecture 20

The proofs of (ii) and (iii) are carried out in a similar manner.

Deﬁnition. We say that a system {Kλ} of double complexes is bounded p,q = 126 on the right if there is some integer m, such that Kλ 0 whenever q > m (for all p and λ). The system is said to be bounded on the left p,q = (resp. above, below) if there is an integer m such that Kλ 0 whenever q < m (resp. p < m, p > m).

Proposition 15. If Kλ, φµλ is a direct system of double complexes p,q = which is bounded above or on the right and if H12 (K) 0 for all p and q, then Hn(K) = 0 for all n.

n n Proof. Let α ∈ H (K), let αλ be its representative in some H (Kλ) and n = p,q + p−1,q+1 + let aλ ∈ Z (Kλ) represent the class αλ. Let aλ aλ aλ ..., + = m,n−m n−m,m where p q n and the sum terminates with aλ (resp. aλ ). n Since aλ ∈ Z (Kλ), daλ = (d1 + d2)aλ = 0, i.e., = p,q + p,q + p−1,q+1 + daλ d1aλ (d2aλ d1aλ ) ··· , p,q = p,q+ p−1,q+1 = and the sum being direct, we have d1aλ 0 and d2aλ d1aλ 0. p,q p,q p,q p,q+1 p,q Thus aλ ∈ Z1 (Kλ) and d2aλ ∈ B1 (Kλ). Therefore, aλ represents p,q p,q = an element of Z12 (Kλ). Since H12 0, there is some µ > λ such p,q = p,q p,q that aµ φµλaλ represents an element of B12 (Kµ). Thus, for some p,q−1 p,q p,q p,q = + b ∈ Z1 (Kµ), aµ − d2b ∈ B1 (Kµ), and hence aµ d2b d1c for p−1,q some c ∈ Kµ .

Let aµ = φµλaλ and let

eµ = aµ − d(b + c)

= aµ − d1b − d2b − d1c − d2c p,q = aµ − aµ − d2c ( since d1b = 0) p−1,q+1 p−2,q+1 = (aµ − d2c) + aµ + ··· p−1,q+1 p−2,q+2 = eµ + aµ + ···

p−1,q+1 p−1,q+1 127 where eµ = aµ −d2c. Then since eµ and aµ represent the same Lecture 20 115

n n class αµ ∈ H (Kµ), they represent the same element α ∈ H (K). The reason for choosing this representative eµ is the fact that its (p, q)− th component is zero. Continuing thus, after a ﬁnite number k of steps for ′ a suitable ρ, α is represented by eρ = −d2c , consisting of a single term p−k,q+k in Kρ . Continuing the construction still further, since the system of double complexes is bounded above or to the right, after a ﬁnite number ′′ of steps, we obtain a representative eν = −d2c = 0, i.e., α is represented n by 0 ∈ Z (Kν) for a suitable ν. Hence α = 0.

Proposition 15-a. If Kλ, φµλ is a direct system of double complexes p,q = which is bounded below or on the left, and if H21 (K) 0 for all p and q then Hn(K) = 0 for all n.

Proof. This is carried out exactly as in Proposition 15, except that we eliminate the component of aλ with highest second degree q instead of the one with highest ﬁrst degree, and −d1c plays the role of −d2c.

Proposition 16. If Kλ, φµλ is a direct system of double complexes p,q = which is bounded above or on the right and if H12 (K) 0 except (at = q o,q most) for p 0, then there exist isomorphisms θ : H (K) → H12 (K) for all q.

If another direct system Klambda, φµλ is bounded above or on the p,q ′ = = ′ right with H12 (K ) 0 except for p 0 and if hλ : Kλ → Kλ are maps ′ = with hµφµλ φµλhλ, then there are induced homomorphisms 128

∗ q ′ q ∗ o,q ′ o,q h : H (K ) → H (K) and h : H12 (K ) → H12 (K) commuting with the isomorphisms θ.

θ q ′ / 0,q ′ H (K ) H12 (K )

h∗ h∗ q θ 0,q H (K) / H12 (K). 116 Lecture 20

Proof. Let Lλ be the subcomplex

Lλ :

 ... −1,q−1 −1,q ... / Kλ / Kλ /

 ... / 0,q−1 / 0,q / ... Z1 Z1

 0 0

p,q = p,q p,q = o,q = of Kλ, with Lλ Kλ for p < 0; Lλ 0 for p > 0 and Lλ o,q Z1 (Kλ). Since Lλ is stable under d1 and d2, it is a subcomplex of Kλ. Let Mλ be the subcomplex

Mλ :

 ... −1,q−1 −1,q ... / Kλ / Kλ /

 ... 0,q−1 0,q ... / B1 / B1 /

 0 0

p,q = p,q = p,q p,q = p,q = 129 of Lλ, with Mλ Lλ Kλ for p < 0; Mλ Lλ 0 for p > 0 o,q = o,q and Mλ Bλ (Kλ). Since Mλ is stable d1 and d2, it is a subcomplex of Lλ.

′ ′ Since hλ : Kλ → Kλ commutes with d1 and d2, we have hλ : Lλ → ′ ′ ′ Lλ and hλ : Mλ → Mλ. Thus there are induced maps hλ : Kλ/Lλ → ′ ′ Kλ/Lλ and hλ : Lλ/Mλ → Lλ/Mλ which commute with i and j in the Lecture 20 117 exact sequences

′ i ′ j ′ ′ 0 / Lλ / Kλ / Kλ/Lλ / 0

hλ hλ hλ i j 0 / Lλ / Kλ / Kλ/Lλ / 0, and ′ i ′ j ′ ′ 0 / Mλ / Lλ / Lλ/Mλ / 0

hλ hλ hλ i j 0 / Mλ / Mλ / Lλ/Mλ / 0. ∗ ∗ ∗ ∗ Hence hλ commutes with d , i and j in the exact cohomology sequences:

∗ ∗ j∗ ... n−1 ′ ′ d n ′ i n ′ n ′ ′ / H (Kλ/Lλ) / H (Lλ) / H (Kλ) / H (Kλ/Lλ) /

∗ ∗ ∗ ∗ hλ hλ hλ hλ ∗ d∗ i∗ j ... n−1 n n n / H (Kλ/Lλ) / H (Lλ) / H (Kλ) / H (Kλ/Lλ) and

∗ j∗ ∗ ... n ′ i n ′ n ′ ′ d n+1 ′ ... / H (Mλ) / H (Lλ) / H (Lλ/Mλ) / H (Mλ) /

∗ ∗ ∗ ∗ hλ hλ hλ hλ ∗ i∗ j d∗ ... n n n n+1 ... / H (Mλ) / H (Lλ) / H (Lλ/Mλ) / H (Mλ) / In the direct limit, we have the following commutative diagram 130 where each row is exact.

∗ d∗ ∗ j n−1 ′ ′ n ′ i n ′ n ′ ′ (A1) ··· / H (K /L ) / H (L ) / H (K ) / H (K /L ) / ...

h∗ h∗ h∗ h∗

∗ d∗ ∗ j n−1 n i n n (A2) ··· / H (K/L) / H (L) / H (K) / H (K/L) / ... and

∗ ∗ j d∗ n ′ i n ′ n ′ ′ n+1 ′ (B1) ··· / H (M ) / H (L ) / H (L /M ) / H (M ) / ...

h∗ h∗ h∗ h∗

∗ ∗ j d∗ n i n n n+1 (B2) ··· / H (M) / H (L) / H (L/M) / H (M) / ... 118 Lecture 20

The quotient double complex Kλ/Lλ is

Kλ/Lλ : 0 0

 ... 0,q−1 0,q−1 0,q 0,q ... / Kλ /Z1 / K /Z1 /

 ... 1,q−1 1,q ... / Kλ / Kλ /

 

o,q o,q 1,q and since the sequence 0 → Kλ /Z1 → Kλ is exact, we have H1(Kλ/Lλ):

0 0

 ... / 1,q−1 / 1,q / ... H1 (Kλ) H1 (Kλ)

 ... / 2,q−1 / 2,q / ... H1 (Kλ) H1 (Kλ)

 

131 p,q = ≦ p,q Thus H12 (Kλ/Lλ) 0 for p 0, and is equal to H12 (Kλ) for p > 0, p,q = p,q = ≦ hence H12 (K/L) direct limit H12 (Kλ/Lλ) 0 for p 0, and by hypothesis is also zero for p > 0, hence is zero for all pairs (p, q). Since Kλ/Lλ is bounded above or to the right, by Proposition 15, we n have H (K/L) = 0 for all n. Thus, in the sequence (A2), we see that i∗ : Hn(L) → Hn(K) is an isomorphism. Lecture 20 119

−1,q o,q Again, since the sequence Kλ → B1 → 0 is exact, we have for

H1(Mλ) :

 ... / −1,q−1 / −1,q / ... H1 (Kλ) H1 (Kλ)

 0 0

p,q = p,q Thus, H12 (Mλ) 0 for p > 0 and is equal to H12 (Kλ) for p < 0, p,q = p,q hence H12 (M) direct limit H12 (Mλ) is equal to zero for p ≥ 0, and by hypothesis, is also zero for p < 0, hence is zero for all pairs (p, q). As before, the conditions of Proposition 15 being satisﬁed, we 132 n have H (M) = 0 for all n. Thus, in the sequence (B2), we see that j∗ : Hn(L) → Hn(L/M) is an isomorphism.

The quotient double complex Lλ/Mλ is given by

Lλ/Mλ : 0 0

 ... / 0,q−1 / 0,q / ... H1 (Kλ) H1 (Kλ)

 0 0

q = o,q = o,q−1 o,q Thus (Lλ/Mλ) H1 (Kλ) and d d2 : H1 (Kλ) → H1 (Kλ). q = o,q q ′ ′ = o,q ′ Hence H (Lλ/Mλ) H12 (Kλ); similarly H (Lλ/Mλ) H1 (Kλ). Fur- q = o,q q ′ ′ = thermore, in the limit we have H (L/M) H12 (K) and H (L /M ) o,q ′ H1 (K ). From the sequences (A1), (A2), (B1), (B2), we have the commutative 120 Lecture 20

diagram :

∗ i∗ j q ′ o q ′ / q ′ ′ = 0,q ′ H (K ) H (L ) H (L /M ) H12 (K )

h∗ h∗ h∗ h∗ ∗ i∗ j Hq(K) o Hq(L) / Hq(L/M) = 0,q H12 (K). Then there is an isomorphism q o,q θ : H (K) → H12 (K), = ∗ ∗ −1 q q q = o,q where θ j (i ) : H (K) → H (L) → H (L/M) H12 (K), θ 133 being an isomorphism since we have proved that each of i∗ and j∗ is an isomorphism. Further, form (I), we have obviously commutativity in the following diagram : q ′ θ 0,q ′ H (K ) / H12 (K )

h∗ h∗ q θ 0,q H (K) / H12 (K).

Proposition 16-a. If Kλ, φµλ is a direct system of double complexes p,q = which is bounded below or on the left and if H21 (K) 0 except for = p p,o q 0, then there exist isomorphisms θ : H (K) → H21 (K) for all p. ′ ′ If another direct system Kλ, φµλ is bounded below or on the left p,q ′ = = ′ with H21 (K ) 0 except for q 0, and if hλ : Kλ → Kλ are maps with = hµφµλ φµλhλ, then there are induced homomorphisms ∗ p ′ p ∗ p,o ′ p,o h : H (K ) → H (K) and h : H21 (K ) → H21 (K) which commute with θ. Remark . In particular, all the propositions proved in this lecture are true for a double complex K = K p,q satisfying the conditions stated in the propositions. We have only to replace the φµλ by the identity map K → K. Lecture 20 121

Example 29. Let 134

K p,q = Z (ring of integers) if q ≧ 0 and p = −q or − q − 1, = 0 otherwise.

−q−1,q −q,q For q ≧ 0, let d1 : K → K and

d2 : K−q−1,q → K−q−1,q+1 be the identity isomorphisms of Z onto itself. The other homomorphisms are all the trivial homomorphisms. Then K = K p,q is a double complex with

p,q = p,q = = H21 (K) H2 (K) Z if (p, q) (0, 0), = 0 otherwise ; p,q = p,q = H12 (K) H1 (K) 0 for all (p, q), and

Hn(K) = Z if n = 0, = o otherwise .

This double complex is bounded below and on the left, but is un- bounded above and on the right.

Lecture 21

Introduction of the family Φ 135 Let Φ be a family of paracompact normal closed subsets of a topological space X such that

(1) if F ∈ Φ, then every closed subset of F is in Φ,

(2) if F1, F2 ∈ Φ, then F1 ∪ F2 ∈ Φ, (3) if F ∈ Φ, there is an open U with F ⊂ U and U¯ ∈ Φ.

For example, if X is paracompact and normal, Φ can be taken to the family of all closed subsets of X, and, if X is locally compact and Hausdorﬀ, then Φ can be taken to be the family of all compact sets of X. Sections with supports in the family Φ. If S is a sheaf of A− modules, the set of all sections f ∈ Γ(X, S ) such that supp f = x : f (x) , 0x is in Φ, forms an A− module (if supp f1 ∈ Φ and supp f2 ∈ Φ then supp ( f1 ± f2) ⊂ (supp f1 ∪ supp f2) is in Φ), a submodule of Γ(X, S ), which we denote by ΓΦ(X, S ). Any homomorphism h : S 1 → S of two sheaves of A− modules induces a homomorphism

h : ΓΦ¯ (X, S ′) → ΓΦ¯ (X, S ), since a homomorphism of sheaves decreases supports (i.e., supp h f ⊂ supp f ).

123 124 Lecture 21

Deﬁnition . A Φ- covering of X is a locally ﬁnite proper covering U 136 such that, if X < Φ, there is a special open set U∗ ∈ U with U¯ ∈ U∈U −(U∗) Φ. S

Remark. If X < Φ, U∗ is unique, and is not the empty set, for otherwise, in each case, X would belong to Φ. If X ∈ Φ, a Φ- covering is just a locally ﬁnite proper covering of X.

The Φ- coverings of X form a subdirected set Ω∗ of the directed set of all locally ﬁnite proper coverings of the space X.

Proof. (i) If X ∈ Φ, Ω∗ is the directed set of all locally ﬁnite proper coverings of X.

(ii) If X < Φ and U , W are any two Φ- coverings of X, let W = W : W = U∩V for some U ∈ U and some V ∈ W with W∗ = U∗ ∩V∗. Then W is a locally ﬁnite proper covering of X and

W¯ = ( U¯ ) ∪ ( V¯ )

W∈W[−(W∗) U∈U[−(U∗) V∈W[−(V∗)

is in Φ, since each set contained in brackets is in Φ. Thus W is a Φ-covering which is a common reﬁnement of U and W .

Remark. If U and W are two Φ coverings with special sets U∗ and V∗ then U∗ ∩ V∗ is not empty, and if W is reﬁnement of U , then V∗ ⊂ U∗ and V∗ is not contained in any other U ∈ U . In particular, if W is equivalent to U , then V∗ = U∗.

137 Cohomology groups with supports in the family Φ. If U is a Φ- covering and a presheaf of A- modules, we deﬁne P p p CΦ U , = C U , for p > 0, p X o X and CΦ U , ⊂ C U , for p = 0, X X Lecture 21 125

p o whereCΦ (U , ) is the submodule of C (U , ) consisting of those zero cochains whichP assign to U∗ the zero of S U∗ .P Then we have a mapping

p p−1 p δ : CΦ U , → CΦ U , . X X Let 138 p p+1 p HΦ U , = ker δ / im δ . p X Then HΦ (U , ) is called the p− th cohomology module of the covering U with coePﬃcients in the presheaf and supports in the family Φ. P If a Φ- covering W is a reﬁnement of U , for each choice of the function τ : W → U , τ(V∗) = U∗. We then have the mapping (Lecture 9) + p p τ : CΦ U , → CΦ W , (p ≧ 0). X X τ+ induces the homomorphism

p p τW U : HΦ U , → CΦ W , X X = = with τU U identity, and τW W τW U τW U if U < W < W . Thus p {HΦ (U , ) , τW U } is a direct system of A-modules. Let P p = p p HΦ (X, ) direct limit HΦ (U , ) , τW U . HΦ(X, ) is Ω U ,W ∈ ∗ called the pP-th cohomology module of theP space X with coeﬃcientsP in the presheaf and supports in the family Φ. The resultP analogous to Proposition 7 (Lecture 11) is true in this case. o Proposition 7-a If S is a sheaf of A-modules, HΦ(X, S ) =ΓΦ(X, S ).

o o Proof. We can identify HΦ(U , S ) = ZΦ(U , S )(see Lecture 11) with the submodule of ΓΦ(X, S ) consisting of the sections f with supp f ⊂ X − U∗ U¯ ∈ Φ; then supp f ∈ Φ. If W is a reﬁnement of U , U∈S −(U∗) S since V∗ ⊂ U∗, we have X − U∗ ⊂ X − V∗, hence

o o τW U : HΦ(U , S ) → HΦ(W , S ) 126 Lecture 21

o is the inclusion homomorphism. Thus HΦ(X, S ) can be identiﬁed with o a submodule of ΓΦ(X, S ) i.e., HΦ(X, S ) ⊂ ΓΦ(X, S ). If f ∈ ΓΦ(X, S ), let U∗ = X − supp f , and let U be an open set containing supp f with U¯ ∈ Φ. Clearly U = U, U∗ is a Φ-covering (with special set U∗) and the cochain g deﬁned by

g(U) = f |U; g(U∗) = 0 o o is the cocycle in ZΦ(U , S ) = HΦ(U , S ) which is identiﬁed with f ∈ o o ΓΦ(X, S ). Thus ΓΦ(X, S ) ⊂ HΦ(X, S ), hence HΦ(X, S ) =ΓΦ(X, S ).

139 Given a sequence of homomorphisms of presheaves q−1 q q+1 dq dq+1 ···→ −−→ −−−→ →···

+ X X X with im dq ⊂ ker dq 1(i.e., d2 = 0) for each Φ-covering U the system {kp,q = C p , q with the homomorphisms U Φ U P q q p p δ : C (U , ) → C U , Φ Φ   X  X q−1  q  p p   and d :C U , → C U , Φ   Φ    X  X       =  q forms a double complex denoted by KU CΦ U , . d P Proof. Any homomorphism ′ −→ of two presheaves induces a ho- p ′ d p momorphism of CΦ U , −→P CΦ (PU , ) commuting with the cobou- 2 2 ndary operator δ. We haveP δ = 0, andP by hypothesis d = 0. Fur- ther dδ = δd; so we have the commutative case of a double complex, q.e.d.

For each pair U , W of Φ- coverings for which W is a reﬁnement of U choose τ : W → U with V ⊂ τ(V); if W = U , let τ : U → U be the identity and let q q + p p φ = τ : C (U , ) → C W , . W U Φ Φ   X X     Lecture 21 127

= CΦ( ) CΦ U , , φW U Ω X X U ,W ∈ ∗ is a direct system of double complexes. = + (If U < W , φW U τ , for an arbitrary but ﬁxed choice of τ : W → U if W , U , and τ : U → U is the identity.) 140

Proof. Since δ and d commute with φW U , φW U : CΦ(W , ) → CΦ(U , ) is a map of double complexes, and by constructionPφU U is the identity.P Φ If W is a - reﬁnement of W , φW U φW U corresponds to a possible choice of τ : S → U . Since for all possible choices of τ, τ(W∗) = U∗, the homotopy operator k (see Lecture 9)

q q k : C p U , → C p−1 W ,      X  X     1 q 1  q  o  q  maps CΦ U , = C U , into CΦ W , . Thus we have two p q p−1 q homomorphisms P k : CΦ U ,P → CΦ W P, and the trivial ho- p q p q−1 momorphism CΦ U , → CPΦ W , suchP that P P + = δk kδ φW W φW U − φW U , and further, d commutes with k. Hence φW U φW U is homotopic to

φW U , hence CΦ(U , ), φW U is a direct system. P 

Lecture 22

Proposition 8-a. If is a presheaf which determines the zero sheaf, 141 p = then HΦ(X, ) 0 forP all p ≧ 0. P p Proof. Let f ∈ CΦ (U , ). Then U¯ ∈ Φ, and hence has U∈U −(U∗) neighbourhoods G, H withP G¯ ⊂ H; G¯,SH¯ ∈ Φ. Shrink the covering ′ = ¯ ¯ ′ = U U − (U∗), U∗ ∩ H of H to a covering W WU U∈U ′ with W¯ U ⊂ U. For each x ∈ H choose a neighbourhood Vx ofx such that a) if x ∈ U, Vx ⊂ U ∩ H, b) if x ∈ WU, Vx ⊂ WU ∩ H, c) if x < W¯ U, Vx ∩ WU = φ, = = d) if x ∈ Uo ∩ ... ∩ Up Uσ, ρVxUσ f (σ) 0, and let V∗ = X − U¯ . U∈U −(U∗) S

Then V∗, Vx is a reﬁnement of U . Choose τ : H ∪ (∗) → U x∈H + such that x ∈ Wτ(x) and τ(∗) = U∗. Then it can be veriﬁed that τ f = 0.

The covering Vx ∩G¯ of G¯ has a locally finite refinement {Yi}i∈I. x∈H Let U1 be the proper covering consisting of V∗ together with all V such that V = Yi ∩ G for some i ∈ I. Then W1 is a Φ- covering which is a refinement of V∗, Vx . Hence there is a function τ1 : W1 → U with x∈H + = p = V ⊂ τ1(V) and such that τ1 f 0. Therefore HΦ(X, ) 0, q.e.d. P 129 130 Lecture 22

142 Let + dq dq 1 + ···→ S q−1 −−→ S q −−−→ S q 1 →··· be a sequence of homomorphisms of sheaves of A-modules with im dq ⊂ + + ker dq 1. Let Bq = im dq, zq = ker dq 1 and H q = zq/Bq. There is an induced sequence of homomorphisms of presheaves

+ dq dq 1 + ···→ S¯q−1 −−→ S¯q −−−→ S¯q 1 →···

q ¯q q q+1 q with im d = Bo ⊂ B¯ and ker d = z¯ . Also there is an induced sequence of homomorphisms

q d+1 p q−1 d p q d p q+1 ··· CΦ U , S −−→ CΦ U , S −−−→ CΦ U , S →··· p q p ¯q q p q where CΦ (U , S ) = CΦ U , S , with im d = CΦ U , B¯ and q+1 = p q p,q = p q p ker d CΦ (U , z ). Then H2 CΦ (U , S ) CΦ (U , z ) /CΦ q U , B¯ o . Letψ : zq → H q be the natural homomorphism. There is an in- q q q duced homomorphism ψ : z¯ → H with ψ(B¯ o) = 0. Hence there is an induced homomorphism

p q p q ψ : CΦ(U , z ) → CΦ(U , H ),

p Bq = which commutes with δ and φW U such that ψ(CΦ(U , ¯ )) 0. Hence there are induced homomorphisms :

p,q p q ψ : H2 CΦ(U , S ) → CΦ(U , H ), p,q p q Φ ψ∗ : H21 C (U , S ) → HΦ(U , H ), p,q p q Φ ψ∗ : H21 C (S ) → HΦ(X, H ). p,q p Φ q 143 The homomorphism ψ∗ : H21 C (S ) → CΦ(X, H ) is an isomorphism.

Proof. The exact sequence

q q q q 0 → β¯o → z¯ → z¯ /B¯ o → 0 Lecture 22 131 gives rise to an exact sequence

p q p q p q q 0 → CΦ(U , B¯ o) → CΦ(U , z ) → CΦ(U , z¯ /B¯ o) → 0.

Hence the induced homomorphism

p,q = p q p B¯ q pΦ q B¯ q H21 CΦ(U , S ) CΦ(U , z )/CΦ(U , o) → C (U , z¯ / o) is an isomorphism. Therefore

p,q p q q Φ B¯ H21 C (U , S ) → HΦ(U , z¯ / o) and hence the homomorphisms

p,q p q q Φ B¯ (1) H21 C (S ) → HΦ(X, z¯ / o) ··· are isomorphisms.

The exact sequence

q q q q 0 → B¯ o → B¯ → B¯ /B¯ o → 0 gives rise to an exact sequence

p q i p q j p q q 0 → CΦ(U , B¯ o) → CΦ(U , B¯ ) → CΦ(U , B¯ /B¯ o) → 0 and i, j commute with δ. Hence there is an exact cohomology sequence 144

∗ ∗ p−1 q q δ p q i ···→ HΦ (U , B¯ /B¯ o) −→ HΦ(U , B¯ o) −→ ∗ p q i p q q HΦ(U , B ) −→ HΦ(U , B¯ /B¯ o) → .

Since i, j and δ commute with φW U , there is an exact cohomology sequence of the direct limits

p−1 q q p−1 q ···→ HΦ (X, B¯ /B¯ o) → HΦ (X, B¯ o) → p q p q q HΦ(X, B¯ ) → HΦ(X, B¯ /B¯ o) →··· 132 Lecture 22

q q p The presheaf B¯ /B¯ o determines the 0-sheaf and hence HΦ(X, q q B¯ /B¯ o) = o for all p. Hence, by exactness,

∗ p q p q i : HΦ(X, B¯ o) → HΦ(X, B¯ o)

is an isomorphism. From the exact sequences of homomorphisms

B¯ q q q B¯ q 0 / 0 / z¯ / z¯ / 0 / 0

 0 / B¯ q / z¯q / z¯q/B¯ q / 0,

one obtains exact sequences of homomorphisms

p B¯ q p zq p zq B¯ q p+1 B¯ q p+1 q HΦ(X, 0) / HΦ(X, ) / HΦ(X, ¯ / 0) / HΦ (X, ) / Hφ (X, z ) 0 p B¯ q p q p q B¯ q p+1 q p+1 q HΦ(X, ) / HΦ(X, z ) / HΦ(X, ¯z / ) / HΦ (X, B¯ ) / HΦ (X, z )

where four of the vertical homomorphisms are isomorphisms. Hence, by the “ﬁve” lemma (see Eilenberg-Steenrod, Foundations of Algebraic Topology, p. 16), the homomorphism

p q p q q (2) HΦ(X, zB¯ o) → HΦ(X, z B¯ ) ······

145 is an isomorphism. Next, the exact sequence

0 → Bq → zq → H q → 0

gives rise to an exact sequence

0 → B¯ q → z¯q → H¯ q.

¯ q q ¯ q Let Ho be the image of z¯ in H . Then there is an exact sequence q q ¯ q ¯ q ¯ q 0 → z¯ /B¯ → H → H /Ho → 0 Lecture 22 133

¯ q ¯ q p and the presheaf H /Ho determines the 0-sheaf. Hence HΦ(X, ¯ q ¯ q H /Ho ) = 0, and in the exact cohomology sequence, the homomorphism

p ¯ q q p ¯ q (3) HΦ(X, H /B¯ ) → HΦ(X, H ) ······ is an isomorphism. Then ψ∗ is the composite isomorphism

p,q p q q p q q p q Φ B¯ B¯ ¯ H21 C (S ) → HΦ(X, z¯ / o) → HΦ(X, z¯ / ) → HΦ(X, H ).

Deﬁnition. We say that the degrees of {S q} are bounded below if there is an integer m such that S q = 0 for q < m.

Lecture 23

Deﬁnition. The Φ- dimension of a space X, Φ − dim X, is sup dim F. 146 F∈Φ Φ − dim X ≦ n if and only if every Φ- covering has a Φ- reﬁnement of order ≦ n.

Proof. Necessity. Let Φ − dim X ≦ n and let U be a Φ-covering of X. Let G be a neighbourhood of U¯ with G¯ ∈ Φ. Then U ′ = U∈U −(U∗) ∗ S (U −(U ))∩G¯, U∗ ∩G¯ forms a locally finite covering of G¯. Since G¯ is ′ normal and dim G¯ ≤ n, the covering U (see Lecture 11) has a (locally ′ finite) proper refinement W of order ≤ n. Let V∗ be the union of X − G¯ ′ and those elements W which are contained in U∗ ∩ G¯ together with V∗ form a Φ- covering W of order ≤ n which is a Φ- refinement of U .

Sufficiency. Let F ∈ Φ and let U be a finite proper covering of F. Let G be an open set with F ⊂ G and G¯ ∈ Φ. Extend each U ∈ U to an open set V of G with V ∩ F = U. These sets together with V∗ = X − F form a Φ- covering W of X. Then W has a Φ− refinement W of order ≦ n and {W ∩ F}W∈W is a refinement of U of order ≦ n. Thus dim F ≦ n, and hence Φ − dim X ≦ n. Note. The paracompactness of the sets of Φ was not used in this proof.

Example . In Example M (see C. H. Dowker, Quart. Jour. Math 6 147 (1955), p. 115) let Φ be the family of all paracompactsets of M (the space is also denoted by M). Then Φ − dim M = 0 and dim M = 1. Remark. It is always true that Φ − dim X ≤ dim x.

135 136 Lecture 23

Proposition 17. Let

+ dq dq 1 + ···→ S q−1 −−→ S q −−−→ S q 1 →···

be a sequence of homomorphisms of sheaves of A− modules with im dq = + ker dq 1 for q , 0 and im d0 ⊂ ker d1 and let G = ker d1/ im d0. If Φ − dim X is ﬁnite or if the degrees of {S q} are bounded below there is p p an isomorphism η : H CΦ(S ) → HΦ(X, G ). If + ′ dq ′ dq 1 ′ + ···→ S q−1 −−→ S q −−−→ S q 1 →···

is another such sequence (with G ′ isomorphic to G and identiﬁed with G ) and if h : S ′q → S q are homomorphisms commuting with dq such that the induced homomorphism h : G ′ → G is the identity, then there ∗ ′ are induced homomorphisms h : HPCΦ(S ) → HPCΦ(S ) with commutativity in p ′ η p H CΦ(S ) / HΦ(X, G )

h∗ h∗= identity p η p H Cφ(S ) / Hφ (X, G ).

q p q Proof. If the degrees of {S } are bounded below, then CΦ(U , S ) = 0 Φ for q < n for some n, and so the system CΦ(U , S ), W U is Ω U ,W ∈ ∗ 148 bounded on the left. If Φ−dim X ≤ m, then there is a coﬁnal directed set ′ ′ P q Ω∗, consisting of Φ−coverings of order ≦ m. If U ∈ Ω∗, CΦ(U , S ) = Φ 0 for p > m; thus the system CΦ(U , S ), W U is bounded Ω′ U ,W ∈ ∗ below.

If q , 0, we have H q = 0, and by a result in Lecture 22, we have p,q p p,q Φ = Φ = H21 C (S ) ≈ HΦ(χ, H ) 0, hence H21 C (S ) 0. Therefore by Proposition 16-a, there is an isomorphism

p p,o Φ Φ θ : H C (S ) → H21 C (S ). Lecture 23 137

Also for q = 0, there is an isomorphism (see Lecture 22)

∗ p,o p Φ ψ : H21 C (S ) → HΦ(x, G ).

Let η be the composite isomorphism

∗θ P P η = ψ : H CΨ(S ) → HΨ(X, G ).

Next, the homomorphisms h : S ′q → S q induce homomorphisms P ′q P q h : CΦ(U , S ) → CΦ(U , S ) which commute with d, δ and φW U , ′ and hence give rise to maps h : CΦ(U , S ) → CΦ(U , S ) which ∗ commute with φW U . Therefore, there are induced homomorphisms h which commute with θ,

p ′ θ p,0 ′ Φ Φ H C (S ) / H21 C (S )

h∗ h∗ p θ p,0 Φ / Φ H C (S ) H21 C (S ).

Since h commutes with dq, h maps z′q into zq and B′q into Bq, hence 149 induces a homomorphism h : H ′q → H q, and there is commutativity in p ′q ψ p ′q CΦ(U , z ) / CΦ(U , H )

h h p q ψ p q CΦ(U , z ) / CΦ(U , H ), where ψ is the homomorphism induced by the natural homomorphisms z′q → H ′q and zq → H q. Therefore there is commutativity in

∗ p,q ′ ψ p ′ Φ q H21 C (S ) / HΦ(X, H )

h∗ h∗ ∗ p,q ψ p Φ q H21 C (S ) / HΦ(X, H ). 138 Lecture 23

Therefore, taking q = 0, we see that h∗ commutes with η = ψ∗θ.

∗ p ′ θ p,0 ′ ψ p Φ Φ H (X, ) H C (S ) / H21 C (S ) / Φ G

h∗ h∗= identity ∗ θ p,0 ψ p p Φ H (X, ). H C (S ) / H21 CΦ(S ) / Φ G Lecture 24

Every Φ covering is shrinkable as is shown by the following result. 150 Let {Ui}i∈I be a locally finite covering of space X with some i∗ ∈ I such that U¯ i is normal for i ∈ I − (i∗). Then there is a refinement {Vi}i∈I with V¯ i ⊂ Ui. ¯ Proof. The union of the locally finite system {Ui}i∈I−(i∗) of normal closed ¯ sets is normal and closed and X − Ui∗ ⊂ Ui ⊂ Ui with X − Ui∗ i,i i,i S∗ S∗ closed and Ui open. Hence there are open sets G, H with X−Ui∗ ⊂ G, i,i∗ ¯ ¯ S = ¯ ¯ G ⊂ H, H ⊂ Ui. Let Vi∗ X − G, then Vi∗ ⊂ X − G ⊂ Ui∗ . i,i S∗ ¯ ¯ ¯ Since {H ∩ Ui}i∈I−(i∗) is a covering of H and the closed subset H of ¯ ¯ ¯ ¯ Ui is normal, there is a covering {Pi}i∈I−(i∗) of H with Pi ⊂ H ∩ Ui. i,i S∗ Let Vi = H ∩ Pi for i ∈ I − (i∗). Then Vi is open, V¯ i ⊂ Ui and Vi = H. i,i S∗ Then {Vi}i∈I is a covering of X and V¯ i ⊂ Ui for all i ∈ I q.e.d. If S is a fine sheaf and if C ⊂ U ⊂ X with C closed, U open and U¯ normal, then the restriction of S to C is fine.

Proof. This result is proved in the same way as in the case (see Lecture 15) that X is normal except that the open set H is to be replaced by its intersection with U if necessary, so that H ⊂ U.

Proposition 11-a If {Ui}i∈I is a locally ﬁnite covering of a space X with 151 some i∗ ∈ I such that U¯ i is normal for i ∈ I − (i∗), and if S is a sheaf whose restriction to each closed subset C of each Ui, i ∈ I − (i∗) is ﬁne

139 140 Lecture 24

(in particular this is true if S is ﬁne), then there is a system {li}i∈I of homomorphisms li : S → S such that

(i) for each i ∈ I there is a closed set Ei ⊂ Ui such that 1i(S x) = 0x if x < Ei,

(ii) li = 1. i∈I P Proof. Shrink to a covering {Wi}i∈I with W¯ i ⊂ Vi, V¯ i ⊂ Gi, G¯i ⊂ Ui, where Wi, Vi and Gi, are open. Using the ﬁneness of the restriction of S to G¯ i. one constructs homomorphisms hi : S → S i , i∗, (actually → the homomorphisms are hi : SG¯i SG¯i , and we extend these by zero ¯ ¯ outside Gi; SG¯i denotes the restriction of S to Gi) with

hi(s) = s if π(s) ∈ W¯ i,

= 0π(s) if π(s) ∈ X − V¯ i.

Let the set I − (i∗) be well-ordered and deﬁne

li = (1 − h j) hi (i , i∗), Yj

152 (In a neighbourhood of each point of X, li, i ∈ I, is only a ﬁnite product.) Then li : S → S is a homomorphism. Let Ei = V¯ i for i , i∗; then if π(s) ∈ X − V¯ i, we have li(s) = 0π(s) since hi(s) = 0π(s). Let = Ei∗ X − Wi; then Ei∗ ⊂ Wi∗ ⊂ Ui∗ . If π(s) ∈ X − Ei∗ , then, for i∈I−(i∗) S = = some i ∈ I − (i∗), π(s) ∈ Wi and hence hi(s) s; so li∗ (s) 0π(s).

If π(s) = x, choose a neighbourhood Nx of x meeting at most a ﬁnite number of the sets Ui, i ∈ I−(i∗), say, for i = i1,..., iq with i1 < ··· < iq. Then

= + li(s) hi1 (s) (1 − hi1 )hi2 (s) Xi∈I + (1 − hi1 ) ... (1 − hiq−1 )hi1 (s) Lecture 24 141

+ (1 − hi1 ) ... (1 − hiq ) (s) = s, and this completes the proof. Let + dq dq 1 + ···→ S q−1 −−→ S q −−−→ S q 1 →··· be a sequence of homomorphisms of sheaves with dq+1dq = 0. Such a sequence of sheaves is called a complex of sheaves.

Definition. A complex of sheaves {S q} is called homotopically fine, if, for each locally finite covering {Ui}i∈I with some i∗ ∈ I such that U¯ i is q−1 q q−1 normal for i ∈ I − (i∗), there exist homomorphisms h : S → S 153 q q and a family {li }i∈I of endomorphisms of S such that q q q = (i) for each i ∈ I there is a closed set Ei ⊂ Ui such that li (S x) 0x if < q x Ei , q = + q q−1 + q q+1 (ii) li 1 d h h d . i∈I P If each S q is fine, then the sequence {S q} is homotopically fine. Proof. Taking hq = 0, this result follows immediately from Proposition 11-a.

If the sequence {S q} is homotopically fine, and U is a locally finite covering satisfying the conditions of the previous definition, then p,q Φ = H12 C (U , S ) 0 for all p > 0. (This result is trivially true for p < 0) Proof. As in the proof of Proposition 12, there are induced homomor- q Γ q Γ q q phisms li (U) : (U, S ) → (U, S ) induced by li , and homomorphisms p−1 p q p−1 q k : CΦ(U , S ) → CΦ (U , S ) (p > 0) such that p p−1 + p p+1 = q δ k f (σ) k δ f (σ) li (Uσ) f (σ) Xi∈I + = f (σ) + dqhq−1 f (σ) + hqdq 1 f (σ) 142 Lecture 24

Thus δk f + kδ f = f + dh f + hd f . and hence 154

p q p q δk + kδ = 1 + dh + hd : CΦ(U , S ) → CΦ(U , S ).

Since d and h commute with δ, there are induced homomorphisms

q p q−1 p q d : HΦ(U , S ) → HΦ(U , S ), +q p q+1 p q h : HΦ(U , S ) → HΦ(U , S ).

p,q p Φ = q Now, H1 C (U , S ) HΦ(U , S ) and, from the homotopy k we have

+ + p,q p,q + = Φ Φ dh h d 0 − 1 : H1 C (U , S ) → H1 C (U , S ) p,q = is homotopic to zero and hence H12 CΦ(U , S ) 0. Lecture 25

Proposition 18. If the complex of sheaves {S q} is homotopically ﬁne, 155 there is an isomorphism ρ : HqCΦ(S ) → HqΓΦ(X, S ). If the complex of sheaves {S ′q} is also homotopically ﬁne, and h : S ′q → S q are homomorphisms commuting with dq, then h∗ commutes with ρ. q ′ ρ q ′ H CΦ(S ) / H ΓΦ(X, S )

h∗ d∗ ρ HqCΦ(S ) / HqΓΦ(X, S ).

p q Proof. The system CΦ(U , S ) = {(CΦ(U , S ))} of double complexes = p,q = is bounded above by p 0. Since (see Lecture 24) H12 CΦ(S ) 0 for p,q = p > 0, and H12 CΦ(S ) 0 trivially for p < 0, by Proposition 16, there is an isomorphism

q o,q Φ Φ θ : H C (S ) → H12 C (S ). p ′q p q Since h : CΦ(U , S ) → CΦ(U , S ) commutes with d, δ and ′ φW U , h : CΦ(U , S ) → CΦ(U , S ) is a map of double complexes ∗ which commutes with φW U . Therefore h commutes with θ.

q o q o q Since ΓΦ(X, S ) = HΦ(X, S ) = dir lim HΦ(U , S ) and the ho- o q Γ q q momorphism τU : HΦ(U , S ) → Φ(X, S ) commutes with d , there are induced homomorphisms

∗ q o q τ → ΓΦ U : H HΦ(U , S ) H ( (X, S ), ∗ q o q τ : dir lim H HΦ(U , S ) → H ΓΦ(X, S ).

143 144 Lecture 25

156 Since the operation of forming cohomology groups commutes with the operation of forming direct limits, (see Cartan - Eilenberg, Homo- logical Algebra, Proposition 9.3∗, p. 100), τ∗ is an isomorphism, and ∗ ∗ since h commutes with τU , h commutes with τ . Now,

o,q o q Φ = H1 C (U , S ) HΦ(U , S ), o,q = q o H12 CΦ(U , S ) H HΦ(U , S ), o,q = q o H12 CΦ(S ) dir lim H HΦ(U , S ). Thus we have an isomorphism

∗ o,q q Φ ΓΦ τ : H12 C (S ) → H (X, S ) which commutes with h∗. Let ρ = τ∗θ be the composite isomorphism then ρ commutes with h∗.

θ o,q ′ τ∗ q ′ Φ qΓ ′ H CΦ(S ) / H12 C (S ) / H Φ(X, S )

h∗ h∗ h∗

 ∗ q θ 0,q τ q Φ / Φ / ΓΦ , . H C (S ) H12 C (S ) H (X S ) Theorem 3 (Uniqueness Theorem). Let

+ dq dq 1 + ···→ S q−1 −−→ S q −−−→ S q 1 →···

157 be a homotopically ﬁne complex of sheaves of A− modules with im dq = + ker dq 1 for q , 0 and im do ⊂ ker d1 and let H o = ker d1/ im do. Let {S ′q} be another such complex with H ′o isomorphic to H o. If Φ − dim X is ﬁnite or the degrees of {S ′q} and {S q} are bounded below, then any isomorphism λ : H ′o → H o induces an isomorphism q ′ q φλ : H ΓΦ(X, S ) → H ΓΦ(X, S ) and if {S ′′q} is another such complex and µ : H o → H ′′o is an isomorphism, then

q ′ q ′′ φµλ = φµφλ : H ΓΦ(X, S ) → H ΓΦ(X, S ). Lecture 25 145

If h : S ′q → S q are homomorphisms (for each q) commuting with dq, and if the induced homomorphism h : H ′o → H o is an isomor- ∗ ′ phism, then the homomorphism h : HqΓΦ(X, S ) → HqΓΦ(X, S is the isomorphism φh. Proof. Since the hypotheses of Propositions 17 and 18 are satisﬁed, there are isomorphisms

q q o η : H CΦ(S ) → HΦ(X, H ), q q ρ : H CΦ(S ) → H ΓΦ(X, S ).

Since λ : H ′o → H o is an isomorphism, so is

∗ q ′o q o λ : HΦ(X, H ) → HΦ(X, H ).

−1 ∗ −1 ∗ ∗ ∗ Let φλ be the isomorphism ρη λ ηρ . Since (µλ) = µ λ , φµλ = 158 φµφλ.

q ′ ρ q ′ η q ′0 H ΓΦ(X, S ) o H CΦ(S ) / HΦ(X, H )

∗ φλ λ q ρ q η q 0 H ΓΦ(X, S ) o H CΦ(S ) / HΦ(X, H )

∗ φµ µ q ′′ ρ q ′′ η q ′′0 H ΓΦ(X, S ) o H CΦ(S ) / HΦ(X, H ).

If h : S ′q → S q are homomorphisms commuting with dq and inducing an isomorphism h : H ′o → H o, then it follows from the commutative diagram:

q ′ q ′ q ′0 H ΓΦ(X, S ) o H CΦ(S ) / HΦ(X, H )

∗ ∗ ∗ φh h h h q q q 0 H ΓΦ(X, S ) o H CΦ(S ) / HΦ(X, H ),

∗ q ′ q that φh = h : H ΓΦ(X, S ) → H ΓΦ(X, S ).

Lecture 26

Singular chains 159 Definition. Let Tp be the set of all singular p- simplexes in X. A singular p-chain with integer coefficients is a function c : Tp → Z such that ct = c(t) is zero for all but a finite number of t ∈ Tp. It is usually written as a formal sum ct · t. The support of c is the union of the supports t∈T t(sp) of those simplexesP t for which ct , 0. The boundary of the p - chain c is the (p − 1)-chain

p j ∂p−1c = ct (−1) ∂ jt. tX∈T p Xj=o

The singular p-chains of X form a free abelian group Cp(X, Z) and ∂p−1 : Cp(X, Z) → Cp−1(X, Z) is a homomorphism with ∂p−1∂p = 0; ∂p−1 decreases support, i.e., supp ∂p−1c ⊂ supp c . If W = {Vi}i∈I is a covering of X, let Cp(X, Z, W ) be the subgroup of Cp(X, Z) consisting of chains c such that ct = 0 unless supp t ⊂ Vi for some i ∈ I. Let j : Cp(X, Z, W ) → Cp(X, Z) be the inclusion homomorphism. Since ∂p−1 decreases supports, there is an induced homomorphism ∂p−1 : Cp(X, Z, W ) → Cp−1(X, Z, W ) which commutes with j. It is known (Cartan Seminar, 1948-49, Expos´e8, §3) that there is a subdivision consisting of homomorphisms r : Cp(X, Z) → Cp(X, Z, W ) such that

(i) rc = c if c ∈ Cp(X, Z, W ), 160

147 148 Lecture 26

(ii) supp rc = supp c.

Further, there is a homotopy hp+1 : Cp(X, Z) → Cp+1(X, Z) such that

(iii) ∂php+1c + hp∂p−1c = jrc − c,

(iv) hp+1 jc = 0,

(v) supp hp+1c ⊂ supp c.

Let I be well-ordered, let ji(t) = t if supp t ⊂ Vi but supp t 1 Vk for all k < i and let ji(t) = 0 otherwise. This deﬁnes a homomorphism

ji : Cp(X, Z, W ) → Cp(X, Z),

with j = ji and supp uic ⊂ Vi ∩ supp c. i∈I Let liP= jir : Cp(X, Z) → Cp(X, Z). Then supp lic ⊂ Vi ∩ supp c. Let 1 = li = jir = jr. Since r j = 1 : Cp(X, Z, W ) → Cp(X, Z, W ), i∈I l2 = j(r jP)r = jrP= l. Further, since jr = l, we have

∂php+1 + hp ∂p−1 = l − 1,

and supp lc ⊂ supp c. If U is open, let Cp(X, Z)U be the set of chains c ∈ Cp(X, Z) such that U does not meet supp c. Then Cp(X, Z)U is a subgroup of Cp(X, Z); let S pU = Cp(X, Z)/Cp(X, Z)U. Since ∂p−1, hp+1, li and l decrease supports, there are induced homomorphisms

∂p−1 : S pU → S p−1,U′ hp+1 : S pU → S p+1,U′

li : S pU, and l : S pU → S pU .

161 If V ⊂ U then Cp(X, Z)U ⊂ Cp(X, Z)V and there is an induced epi-

morphism ρVU : S pU → S pV which commutes with ∂p−1, hp+1, li and

l. Then {S pU , ρVU } is a presheaf which determines a sheaf Sp called the sheaf of singular p-chains. There are induced sheaf homomorphisms

∂p−1 : Sp → Sp−1, hp+1 : Sp → Sp+1, Lecture 26 149

li : Sp → Sp, and l : Sp → Sp, with 1 + ∂php+1 + hp∂p−1 = l = li. Xi∈I

If c ∈ Cp(X, Z) and U = X − V¯ i then, since supp li(c) ⊂ Vi, li(c) ∈ Cp(X, Z)U. Therefore li : S pW → S pW is the zero homomorphism for each open W ⊂ U. Hence li(S px) = 0x for all x ∈ X − V¯ i.

Definition. If G is a sheaf of A− modules, the sheaf Sp ⊗Z G is a sheaf of A-modules called the sheaf of singular p - chains with coefficients in G . p p+1 p Let S = Sk−p ⊗Z G for some fixed integer k and let d : S → p+1 p−1 p p−1 p p p p S , h : S → S , li : S → S and l : S → S be the homomorphisms induced by ∂k−p−1, hk−p+1, li and l. Then 162

p p−1 p p+1 1 + d h + h d = l = li, X p and li(S x )0x for all x ∈ X − V¯ i. + dp dp 1 + The sequence ···→ S p−1 −−→ S p −−−→ S p 1 →··· is homotopically ﬁne.

Proof. Let U = {Ui}i∈I be a locally ﬁnite covering with i∗ ∈ I such that U¯ i is normal for i ∈ I − (i∗). Shrink U to a covering W = {Vi}i∈I p−1 p p−1 with V¯ i ⊂ Ui. Construct the homomorphisms h : S → S , p p li : S → S (as above) such that

p p−1 p p+1 1 + d h + h d = li, Xi∈I p and li(S x ) = 0x if x < V¯ i. Thus we can take Ei = V¯ i. Φ = Γ Deﬁnition. Let Cp (X, G ) Φ(X, Sp ⊗Z G ); this A− module is called ﬃ Φ the module of singular p - chains of x with coe cients in G . Let Hp (X, G ) = ker ∂p−1/im ∂p in the sequence

Φ ∂p Φ ∂p−1 Φ ···→ Cp+1(X, G ) −−→ Cp (X, G ) −−−→ Cp−1(X, G ) →··· 150 Lecture 26

Φ Φ where the homomorphism ∂p : Cp+1(X, G ) → Cp (X, G ) is the one induced by the homomorphism ∂p : Sp+1 → Sp. The A− module Φ Hp (X, G ) is called the p-th singular homology module of the space X with coeﬃcients in the sheaf G , and supports in the family Φ.

163 An element of a stalk S px can be written uniquely in the form ct·t t∈T Pp where ct ∈ Z, ct = 0 if x is not in the closure of the support of t, supp t, and ct = 0 except for a finite number of t. An element of a stalk S qx ⊗Z Gx can be written uniquely in the form gt ·t where gt ∈ Gx, gt = 0 if x < supp t and gt = 0 except for a finite x∈T p numberP of t. Φ An element of Cp (X, G ) can be written uniquely in the form γt · t∈T Pp t where γt is a section of G over supp t, γt = 0 except for a set of simplexes whose supports form a locally finite system and the set of points x such that, for some t, γt(x) is defined and , 0x is contained in a set of Φ. (A section over a closed set E is a map γ : E → G such that πγ : E → E is the identity.)

Deﬁnition . An n - manifold is a Hausdorﬀ space X which is locally euclidean, i.e., each point x ∈ X has a neighbourhood which is homeomorphic to an open set in Rn.

If X is an n-manifold, then Φ − dim X = n.

Proof. Since X can be covered by open sets whose closures are homeomorphic to subsets of Rn, X is locally n-dimensional. Then each closed set E ∈ Φ is locally of dimension ≤ n and E is paracompact and normal, hence dim E ≤ n. Further, any non-empty set E ∈ Φ has a closed 164 neighbourhood V¯ ∈ Φ and V¯ contains a closed set homeomorphic to the closure of an open set in Rn. Hence dim V¯ ≧ n; thus Φ − dim X = n, and this completes the proof.

∂p ∂p−1 In the sequence ···→ Sp+1 −−→ Sp −−−→ Sp−1 →··· Let Hp = ker ∂p−1/ im ∂p; Hp is called the p - the singular homology sheaf in X. Lecture 26 151

If X is an n-manifold, the p-th singular homology sheaf in X is locally isomorphic with the p-th singular homology sheaf in Rn.

′ Proof. Let x0 ∈ U1 ⊂ X where U1 is open in X and let f : U1 → U1 ′ n be a homeomorphism onto an open set U1 ⊂ R . Choose an open set ¯ = ¯ ′ = n ¯ V with xo ∈ V, V ⊂ U1 and let U2 X − V, U2 R − f (V). Then {U1, U2} is a covering of X, and there is the homotopy deﬁned above,

∂h + h∂ = l1 + l2 − t, with l2(S px) = 0 for x ∈ V. Hence Hp(S ) is isomorphic with Hp(l1S ) ′ ′ ′ ′ in V. But f : U1 → U1 takes l1(Sp) into l1(Sp) where Sp is the sheaf n ′ of singular p− chains in R and l1 is the corresponding homomorphism ′ ′ n ′ ′ ′ for the covering {U1, U2} of R . In f (V), Hp(l1S ) is with Hp(S ). n n Using triangulations of R , one can verify, for R , that Hp = o for n n p , n and Hn is isomorphic with the constant sheaf (R × Z,π, R ). One uses a homotopy which does not decrease supports and which does 165 not induce a sheaf homotopy. The isomorphism is not a natural one but depends on the choice of an orientation for Rn. In an n-manifold X, Hp = 0 for p , n and Hn is locally isomorphic with Z. Let J = Hn; if J is isomorphic with Z the manifold is called orientable, otherwise the manifold is said to be non-orientable and J is called the sheaf of twisted integers over X. Example 2 is the restriction ot the M¨obius band of the sheaf of twisted integers over the projective plane. p p If S = Sn−p ⊗Z G on an n-manifold X, then H (S ) for p , 0 o and H (S ) = J ⊗Z G .

Proof. Since S px is a free abelian group, so are the subgroups Zpx and Bpx. Also, Hpx being either 0 or Z, is free. It is known (Cartan Seminar, 1948-49, Expose 11) that if 0 → F → F → F′′ → 0 is an exact sequence of abelian groups and F′′ is without torsion and if G is an abelian group, then the induced sequence 0 → F′ ⊗ G → F ⊗ G → F′′ ⊗ G → 0 152 Lecture 26

is exact. From the following commutative diagram of exact sequences, one can see that

Hp(S ⊗ G ) = ker ∂p−1/ im ∂p ≈ zp ⊗ G /Bp ⊗ G ≈ Hp ⊗ G .

p p 166 Thus H (S ) = Hn−p ⊗ G = O for p , O and H (S ) = J ⊗ G for p = O.

 0 o Bp ⊗ G o Sp+1 ⊗ G o zp+1 ⊗ G o 0

∂p 0

 | zp ⊗ G / Sp ⊗ G / Bp−1 ⊗ G / 0

0 vv vv ∂p−1 vv vv vz v Hp ⊗ G zp−1 ⊗ G qqq qqq qqq qx q 0 p Sp−1 ⊗ G q qqq qqq qx qq Bp−2 ⊗ G Hp−1 ⊗ G vv vv vv vv 0 {v 0

Q.e.d. Lecture 26 153

Proposition 19. If X is an n- manifold, there is an isomorphism:

−1 Φ p η : Hn−p(X, G ) → HΦ(X, J⊗Z G ).

Φ = Γ = Γ p Φ¯ Proof. Since Cn−p(X, G ) Φ(X, Sn−p ⊗z G ) Φ¯ (X, S ), Hn−p(X, p o p G ) = H ΓΦ¯ (X, S ). And, since H (S ) = J ⊗z G , HΦ(X, J ⊗z G ) = p o HΦ(X, H ). By proposition 17 and 18, there are isomorphisms

p p o η : H CΦ(S ) → HΦ(X, H ), p p ρ : H CΦ(S ) → H ΓΦ(X, S ).

Thus ηρ−1 is the required isomorphism.

This proposition is part of the Poincare duality theorem.

Lecture 27

Given any sheaf G of A-modules, there exists an exact sequences of 167 sheaves

e d1 dq O → G −→ S o −−→ S 1 →···→ S q−1 −−→ S q →··· where each S q(q ≥ O) is ﬁnite. q = Proof. For each open set U of X, let S U(q O, 1,...) be the abelian group of integer valued functions f (x0,..., xq) of q + 1 variables x0,... xq ∈ U. If V is an open set with V ⊂ U, the restriction of the functions q → q f gives a homomorphism ρVU : S U S V . There is a homomorphism q+1 q q+1 dU : S U → S U deﬁned by

q+1 q+1 j d f (xo,..., xq+1) = (−1) f (xo,..., xˆ j,..., xq+1). Xj=o

o If e : Z → S U is the inclusion homomorphism of the constant functions on U, the sequence

1 q e o d q−1 d q O → Z −→ S U −−→· · · → S U −−→ S U →··· q+1 = is exact (Cartan Seminar 1948-49, Expose 7, §8). Clearly ρVU d f q+1 q = d ρVU f and ρVU commutes with e. The presheaves {S U, ρVU }(q 0, 1,...) determine sheaves S q and there is an induced exact sequence

1 q e o d q−1 d q O → Z −→ S U −−→· · · → S U −−→ S →···

155 156 Lecture 27

p It is easily veriﬁed that each abelian group S U is without torsion, q and this property is preserved in the direct limit. Hence each stalk S x = 168 { q } q dir lim S U , ρVU x∈U is without torsion, i.e., the sheaves S are without torsion. Therefore the sequences of sheaves of A-modules

1 q e o d q−1 d q O → Z ⊗Z G −→ S ⊗Z G −−→· · · → S ⊗Z G −−→ S ⊗Z G →···

is exact, (this follows from the fact that if O → F′ → F → F′′ → O is an exact sequence of abelian group, F′′ is without torsion, and G is an abelian group, then the sequence

O → F′ ⊗ G → F ⊗ G → F′′ ⊗ G → O

is exact), that is, the sequence

1 q e o d q−1 d q (1) 0 → G −→ S ⊗Z G −−→· · · → S ⊗Z G −−→ S ⊗Z G →···

is exact. q We now show that each of the sheaves S ⊗Z G is a ﬁne sheaf. To q p do this, let E ⊂ G with E closed and G open and let h : S U → S U be the homomorphism deﬁned by

h f (xo,..., xq) = f (xo,..., xq) if xo,..., xq ∈ U ∩ G, = 0 otherwise.

q Then h commutes with ρVU and induces a homomorphism h : S → q q q q S for which hx : S x → S x is the identity if x ∈ G¯, and h(S x) = 0x if q x ∈ X − G¯. There is then an induced homomorphism h : S ⊗Z G → q q S ⊗Z which is the identity on the stalks S x ⊗Z Gx for x ∈ G and zero q q 169 on S x ⊗Z Gx for x ∈ X − G¯. Thus S ⊗Z G is fine, and the sequence (1) is a fine resolution of G . We now give a definition of the cochains of a covering of a space X coefficients in a sheaf G and support in a Φ-family and also give an alternative definition of the cohomology groups of X with coefficients in G and supports in the family Φ. We then prove Leray’s theorem on acyclic coverings. Lecture 27 157

p Definition. Let CΦ(U , G ), for an arbitrary covering U = {Ui}i∈I be the C p(U , G ) consisting of those cochains f such that the closure of the set supp f = {x : f (io,..., ip)(x) is defined for some σ = (io,..., ip) and , 0x}, belongs to the family Φ. (If p = 0, this definition does not agree with the previous one even when U is a Φ¯ -covering.) Then, since the homomorphisms

+ + δp 1 : C p(U , G ) → C p 1(U , G ) + τ : C p(U , G ) → C p(W , G ) decrease supports, they induce homomorphisms

p+1 p p+1 δ : Cφ(U , G ) → CΦ (U , G ) and + p p τ : CΦ(U , G ) → Cφ(W , G ) p respectively. Hence there are cohomology module HΦ(U , G ) and ho- p p = o momorphisms τW U : HΦ(U , G ) → HΦ(W , G ). For p 0, HΦ(U , 170 G ) =ΓΦ(X, G ) for every covering G , and Γ Γ τW U : Φ(X, G ) → Φ(X, G ) is the identity. Using the directed set Ω of all proper covering of p X, {HΦ(U , G ), τW U }U ,W ∈Ω is a direct system. The direct limit of this p system will be denoted by HΦ(X, G ). This module also is called the p- th cohomology module of the space X with coefficients in the sheaf G and supports in the family Φ. (This cohomology module is isomorphic with that previously defined by means of Φ-coverings, Lecture 21.) There are p p homomorphisms into the direct limits, τU : HΦ(U , G ) → HΦ(X, G ). Let X be a paracompact normal space, G a sheaf of A-modules over X, and let U = {U} be locally finite proper covering of X where each U ∈ U is an Fσ set. (U is indexed by itself.) Let Φσ be the set of all intersection E ∩ Uσ with E ∈ Φ. Since Uσ is an open Fσ set in X, Uσ is paracompact and normal. Hence each E ∩ Uσ ∈ Φσ is paracompact and normal. One easily verifies that Φσ is a Φ-family in Uσ. We now as sume the following conditions on the family Φ and the covering U . 158 Lecture 27

(i) For some inﬁnite cardinal number m, the union of fewer than m elements of Φ is contained in a set belonging to Φ. (If X ∈ Φ, choose m greater than the number of closed sets of X; if Φ is family of compact sets, let m = No.)

171 (ii) Each set in Φ meets fewer than m sets of the covering U .

(iii) Each U ∈ U is an Fσ sets, i.e., is a countable union of closed subsets of X.

p (iv) For each Uσ = U ∩··· U , H (Uσ, G ) = 0 for q > 0; here G o p Φσ denotes the restriction of G to Uσ (A covering U is called acyclic is conditions (iv) is satisﬁed.) Under the conditions (i), (ii), (iii), (iv) stated above, the homomor- p p phism τU : HΦ(U , G ) → HΦ(X, G ) is an isomorphism. Proof. Let

e d1 O → G −→ S o −−→· · · → S q−1 → dqS q →···

p q be any fine resolution of G . Then there is a system {(CΦ(U , S ))}U ∈Ω′ of double complexes, where Ω′ is the cofinal directed set of all locally finite proper coverings of X. This system is bounded above by p = 0 and on the left by q = 0.

Since X is normal, S q is ﬁne and U is locally ﬁnite, there is a homotopy (see Lecture 16)

kp−1 : C p(U , S q) → C p−1(U , S q) (p > 0).

Since kp−1 decreases supports, it induces a homomorphism

p−1 p q p−1 q k : CΦ(U , S ) → C (U , S ) (p > 0),

p p−1 p p+1 p q 172 with δ k + k δ = 1. Hence HΦ(U , S ) = 0 for p > 0, and for o q q p = 0, HΦ(U , S ) =ΓΦ(X, S ). Hence

O,q q Φ = ΓΦ H12 C (U , S ) H (X, S ). Lecture 27 159

Thus we have the isomorphisms indicated below (see Lecture 20):

q ≈ 0,q ≈ q ΓΦ , Φ Φ , H (X S ) o H12 C (U , S ) o H C (U S )

≈ ≈ 0,q ≈ qΓΦ q Φ H (X, S ) o H12 CΦ(S ) o H C (S ).

Since S q is fine and X is normal, S q is locally fine, hence its restriction to Uσ is locally fine. But Uσ is paracompact and normal, so q that the restriction of S to Uσ is fine. Hence there is an isomorphism (see Proposition 17 and 18, lectures 23 and 25 respectively),

−1 q q ηρ : H ΓΦ (Uσ, S ) → H (Uσ, G ). σ Φσ qΓ = oΓ Hence by condition (iv), H Φσ (Uσ, S ) 0 for q > 0, H Φ(Uσ, S ) ≈ o H (Uσ, G ) =ΓΦ (Uσ, G ). Φσ σ p q q+1 q+1 If f ∈ CΦ(U , S ) (q > O) and d f = O, then (d f )(Uo,..., = = qΓ = Up) O in each Uσ Uo ∩···∩ Up. Since H Φσ (Uσ, S ) 0(q > Γ q−1 0), there is a section g(Uo,..., Up) ∈ Φσ (Uσ, S ) with dg(Uo,..., Up) = f (Uo,..., Up) (choose g(Uo,..., Up) = 0 if f (Uo,..., Up) = 0. There is then a cochain g ∈ C p(U , S q−1)) with dg = f , (see p.57). o q Since f ∈ CΦ(U , S ), supp f is contained in a set belonging to Φ and 173 hence f (σ) is diﬀerent from zero on fewer than m sets Uσ. Then g(σ) is diﬀerent from zero on fewer than m sets Uσ and hence supp g is the union of fewer than m set {x ∈ Uσ : g(σ)(x) , 0, each of which is in Φσ and hence has its closure in Φ. Hence supp g is contained in a set Φ p q−1 p,q = belonging to and g ∈ CΦ(U , S ). Hence H2 Cφ(U , S ) 0(q > 0). Since the sequences

e d1 0 → C p(U , G ) → C p(U , S o) → C p(U , S 1)

p O 1 is exact, if f ∈ CΦ(U , S ) and d f = 0, then f = e(g) for some p p g ∈ C (U , G ) and clearly g ∈ CΦ(U , G ). Thus

p,o p p,0 p Φ Φ H2 C (U , S ) ≈ CΦ(U , G ) and H21 C (U , S ) ≈ HΦ(U , G ). 160 Lecture 27

Thus we have the isomorphism indicated below:

p ≈ p,0 ≈ p Φ , / Φ / H ( , ) H C (U S ) H21 C (U , S ) Φ U G

τU p ≈ p,0 ≈ p Φ / Φ / H (X, ). H C (S ) H21 C (S ) Φ G

Combining this diagram with the previous one, we see that the ho- p p momorphism τ : HΦ(U , G ) → HΦ(X, G ) is an isomorphism. Q.e.d In particular, we have proved the following proposition (Cartan Sem- inar, 1953-54, Expose 17, p.7).

174 Proposition 20. If U is a locally finite proper covering of a paracompact normal space X by open Fσ sets, and if G is a sheaf of A-modules q such that H (Uσ, G ) = O(q > O) for every Uσ = Uo ∩···∩ Uk(k = O, 1,...), then p p τU : H (U , G ) → H (X, G ) is an isomorphism. p In the case that Φ is the family of all compact sets of X, we write H∗ p instead of HΦ¯ . Proposition 20 (-a). If U is a locally finite proper covering of a locally compact and paracompact Hausdorff space by open Fσ sets with com- p pact closures, and if G is a sheaf of A-modules such that H (Uσ, G ) = O(q > O) for every

Uσ = Uo ∩···∩ Uk(k = 0, 1,...)

then p p τσ : H∗ (U , G ) → H∗ (X, G ) is an isomorphism.

Note. It is no restriction to assume that U is a proper covering. If W is any covering, there is an equivalent proper covering U with the open sets. Then τW U and τU ,W are isomorphisms. Lecture 28

Direct sum of modules 175 i Definition . The direct sum of a system {M }i∈I of A-modules is an A- i module whose elements are systems {m }i∈I, usually written as formal i i i i = sums i m where m ∈ M and m Q for all but a finite number of i. The operationsP in M are defined by i + i = i + i i = i m1 m2 (m1 m2), a m am , Xi Xi Xi Xi Xi

i i i where i m1 ∈ M, i m2 ∈ M, i m ∈ M and a ∈ A. P P P j i j Clearly there is a homomorphism p : i M → M deﬁned by j i = j j j i p ( i m ) m and a homomorphism h : M P→ i M deﬁned by P P pih j(m j) = 0 for i , j, p jh j(m j) = m j, m j ∈ M j.

A system of homomorphism (g′, gi) : (A, Mi) → (B, Ni), i ∈ I, in- ′ i i duces a homomorphism (g , g) : (A, i M ) → (B, i N ) where i = i i g( i m ) i g (m ). There is commutativityP in P P P j p j j h i j (A, M ) / (A, i M ) / (A, M )

(g′,g j) (g′,gP) (g′,g j) j p j j h i j (B, N ) / (B, i N ) / (B, N ). P 161 162 Lecture 28

i i i i i i Proof. Let i m1 ∈ i M , i m2 ∈ i M , i m ∈ i M and let a ∈ A. Then P P P P P P i + i = i + i = i i + i g( m1 m2) g( (m1 m2) g (m1 m2) Xi Xi Xi Xi 176 = i i + i i = i i + i i (g (m1) g (m2)) g (m1) g (m2) Xi Xi Xi = i + i g( m1) g( m2) Xi Xi g(a mi) = g( ami) = gi(ami) Xi Xi Xi g′(a)gi(mi) = g′(a) gi(mi) Xi Xi = g′(a)g( mi). Xi gi p j( mi) = g jm j = p j( gimi) Xi Xi = p jg( mi) Xi pkgh j(m j) = gk pkh j(m j) = gk(0) = 0 for k , j

and p jgh j(m j) = g j p jh j(m j) = g j(m j), hence gh j(m j) = h jg j(m j), m j ∈ M j. The operation of forming the direct limit commute with the operation of forming the direct sum. i ′ i Ω Proof. Let {Aλ, Mλ, φµλ, φµλ}λ,µ∈Ω ( a directed set) be a direct system for each i ∈ I. Let the direct limits be (A, Mi) with homomorphisms ′ i i i (φλ, φλ) : (Aλ, Mλ) −→ (A, M ). There are induced homomorphisms

′ i i (φµλ, φµλ) : (Aλ, Mλ) → (Aµ, Mµ) Xi Xi Lecture 28 163

177 for which φνµφµλ = φνλ,λ < µ < v and φλλ is the identity. Thus i ′ ′ {Aλ, i Mλ, φµλ, φµλ}λ∈Ω is direct system. Let its direct limit be (A, M) withP homomorphisms

′ i (φλ, φλ) : (Aλ, Mλ) → (A, M). Xi

φi φi i µλ i µ i (Aλ, Mλ) / (Aµ, Mµ) / (A, M )

i i hλ hµ

φµλ φµ i i i (Aλ, i Mλ) / (Aµ, i Mµ) / (A, M) h VVVV N ′′ VVV P NN φµ P VVVV NN φ′′ ′′VVVV NN φλ VVV NN VVVV N& } + i (A, i M ) i i P i i The system of homomorphisms (φλ, φλ) : (Aλ, Mλ) → (A, M ) in- ′ ′′ i i duces a homomorphism (φλ, φλ ) : (Aλ, i Mλ) → (A, i M ). Since for λ<µ, P P ′′ i = ′′ i i = i i i φµ φµλ( mλ) φµ ( φµλmλ) φµφµλmλ Xi Xi Xi = i i = ′′ i φλmλ φλ ( mλ), Xi Xi ′′ = ′′ ′′ therefore φµ φµλ φλ and there is an induced homomorphism φ : M → i i M . P i i i = If i m ∈ i M then m 0 except for a ﬁnite number of i, say i i i j j j i1,...,Pik. Then,P for some λ, each m has a representative mλ ∈ Mλ ; let i = < i i i = ′′ i = mλ 0 for i (i1,..., ik). Then i mλ ∈ i Mλ and i m φλ i mλ ′′ i ′′ φ φλ i mλ. Thus φ is an epimorphism.P P P P 178 ′′ ′′ ToP show that Φ is a monomorphism, let m ∈ M and φ (m) = 0. i i = Choose a representative i mλ of m. Then mλ 0 except for a ﬁnite = number of i, say i i1,...,P ik. Since = ′′ = ′′ i = i i 0 φ (m) φλ ( mλ) φλmλ, Xi Xi 164 Lecture 28

i j i j = i j i j = therefore each φλ mλ 0. Now choose µ so that each φµλmλ 0. i = ′′ Then m is represented by 0 in i Mµ and hence m 0. Thus φ is an i ′′ isomorphism. We identity M withP i M under the isomorphism φ . P Direct sum of sheaves i Let a be a sheaf of rings with unit and let {S }i∈I be a system of sheaves of a-modules. Then there is a unique sheaf S whose stalks i j j are the direct sums S x such that the homomorphisms hx : S x → S x i∈I determine a sheaf homomorphismP h j : S j → S . This S is called the direct sum of the sheaves S i and is denoted by S = S i. i∈I P

= i i i = Proof. Uniqueness. If s i s ∈ i S k with s 0 except for i1,..., ik, i j = choose a neighbourhood UPof x andP sections f j : U → S , j 1,..., k = i j = i = such that f j(x) S . Let f : U → S ( S x) be deﬁned by f (y) x i∈I i j i j S P j h f j(y). Since h is a sheaf homomorphism, the composite function P i f j h j U −→ S i j −−→ S

179 is a section and hence f is a section. Since

i j i j i j i f (x) = h f j(x) = h s = s = s, Xj Xj Xi

the section goes through s. Thus, since such section cover S , they uniquely determine the topology of S .

Existence From the presheaves of sections (¯a, ¯i) = {A , S i , ρ′ , ρi } S U U VU VU = Γ , i = Γ , i { , i , ρ′ , ρ } where AU (U a) and S U (U S ). Then AU i S U VU VU (where U runs through the directed set of all the openP sets of X) is a presheaf. It determines a sheaf (a, S ) and there is a homomorphism Lecture 28 165

j j i hU : S U → i S U with commutativity in P h j j U , i (AU, S U ) / (AU i S U) P (ρ′ ,ρ j ) (ρ′ ,ρ ) VU VU VU VU h j j V , i . (AV , S V ) / (AV i S V ) P 180 Hence there is an induced sheaf homomorphism h j : S j → S . The stalk (Ax, S x) of (a, S ) is the direct limit of the direct system { , i , ρ′ , ρ } AU i S U VU VU x∈U which is identiﬁed with the direct sum of the , i = { , i , ρ′ , ρi } = i directP limits (Ax S x) dir lim AU S U VU VU x∈U . Thus S x i S x and the required sheaf exists, q.e.d. P i i j There is also a homomorphism pU : i S U → S U (as deﬁned in the beginning of the lecture) with commutativityP in

p j , i U j (AU i S U ) / (AU , S U )

′ P (ρ ,ρ ) (ρ′ ,ρ j ) VU VU VU VU p j , i V j (AV i S V ) / (AV , S V ) P Hence there is a sheaf homomorphism p j : S → S j which is = i j = j clearly onto. If s i s ∈ S , it is easily veriﬁed that p (s) s , hence j j j j j i j p h (s ) = s and p Ph (s ) = 0 if i , j.

Lecture 29

i Given homomorphisms gi : S → T (i ∈ I), there is an induced homo- 181 i i morphism g : S → T with gh = gi. i∈I P = i = i Proof. If s i s ∈ S x, let g(s) igi (s ) ∈ Tx.Then g|S x : S x → Tx is clearly a homomorphism.P Choose anP open set U and sections f j : U → S i j so that the section deﬁned by f (y) = hi j f (y) goes through s. Then = g f (y) gi j f j(y) and g f being the sumP of a ﬁnite number of sections, is a section.P Thus g is continuous and is a sheaf homomorphism.

Note. Since a itself is a sheaf of a-modules, there is, for any I, a direct sum a where each direct summand is a. This is again a sheaf of a i∈I -modules.P

If S is a sheaf of a-modules over a space X, and aY is the restriction of a to a subset Y, then clearly the restriction SY of S to Y is a sheaf of aY -modules.

Deﬁnition. The following properties of sheaves S of a- modules over a space X are called property (a1) and property (a) :

Property (a1). There is a covering {U j} j∈J of X, and for each j ∈ J, there is an index set I j and an epimorphism φ j : aU j → SU j . i∈I Pj Property (a). There is a covering {U j} j∈J of X, and for each j ∈ J, k j a → there is a natural number k j and an epimorphism φ j : i=1 U j SU j . 182 P 167 168 Lecture 29

S has property (a1) if and only if each point x ∈ X has a neighbourhood U such that the sections in Γ(U, S ) generate SU . (That is, = k j for each y ∈ U and s ∈ S y, s j=1 ay f j(y) for some ﬁnite number k of j Γ elements a j ∈ Ay and sections fPj ∈ (U, S ).)

Proof. Necessity. Let S have property (a1). Since {U j} is a covering, x ∈ U j for some j; let U = U j. Then there is an index set I and an epimorphism φ : aU → SU. i∈I P i i Let f = φh 1 : U → S where 1 is the unit section in aU ,

1 hi φ U −→ aU −→ aU −→ SU. Xi∈I = i i i = Then if s ∈ SU , s φ( i ay) for some ay ∈ Ay with ay Oy except = for a ﬁnite number of i, sayPi i1, i2,..., ik. Then k k k i i i j = j i j = i j j ay fi j (y) ay φh 1y φh ay 1y Xj=1 Xj=1 Xj=1 k i = i j j = i = φ( h ay ) φ( ay) s. Xj=1 Xi∈I

Suﬃciency. For each x ∈ X, there is a neighbourhood Ux of x such

that the sections over Ux generate SUx . Then {Ux}x∈X is a covering of X. Let Ix be the set of sections Γ(Ux, S ). For each section i ∈ Ix, i there is a sheaf homomorphism φx : aUx → SUx given, for a ∈ Ay, by i = φx(a) a · i(πa). 183 Then there is an induced homomorphism

φx : aUx → SUx . XiεIx

Then for s ∈ S y and y ∈ Ux,

k k = j = i j j s ayi j(y) φx (ay) ∈ φx( aUx ). Xj=1 Xj=1 Xi∈Ix Lecture 29 169

Thus φx is an epimorphism. S has property (a) if and only if each point x has a neighbourhood U such that some ﬁnite number of sections fi ∈ Γ(U, S ) (i = 1,..., k) generate SU .

Proof. Similar to the proof given above.

It is clear that (a) implies (a1), i.e., each sheaf with property (a) has k property (a1). The sheaf a has property (a1) and i=1 a has property i∈I (a). In particular, the sheafPa of a-modules has propertyP (a). If Si, i = 1,..., k are sheaves of a- modules with property (a1) k (resp. (a)), then the direct sum i=1 ai has property (a1) (resp (a)). P Proof. Clear. Statements (1), (2), (3), (4), (5), (6), (Lectures 29, 30, 31,) are required to prove Serre’s theorem on coherent sheaves, (see the next lec- 184 ture for the deﬁnition of coherent sheaves), i.e., if 0 → S ′ → S → S ′′ → 0 is an exact sequence of sheaves, and if two of them are coherent, then the third is also coherent. ′ ′ (1) If f : S → S is an epimorphism and S has property (a1) (resp (a)), then S has property (a1) (resp (a)).

Proof. Clear.

Example . If M is a ﬁnitely generated A- module, the constant sheaf M has property (a) with respect to the constant sheaf A. If a constant sheaf is a sheaf of a- modules, then it has property (a1). If X is the unit segment 0 ≤ x ≤ 1, the subsheaf S of the constant sheaf Z2 obtained by omitting (1, 1) does not have property (a1) either as a sheaf of Z- modules or as a sheaf of Z2 -modules. With the same X, the sheaf S of germs of functions f : X → Z2, considered as a sheaf of Z2 -modules, has property (a1) but not property (a). But, considering S as a sheaf of rings with unit, it has property (a) with respect to itself, The sheaf a of germs of analytic functions in the complex plane has property (a) as a sheaf of a- modules, but as a sheaf of C- modules (where C is the ﬁeld of complex numbers) it does not even have property (a1). For, there 170 Lecture 29 are natural boundaries for analytic functions, e.g., let f be an analytic function in |z| < 1, with |z| = 1 as natural boundary. If a has property (a1) as a sheaf of C-modules, then by considering a point on the boundary, we see that f can be continued to a neighbourhood of this boundary point and this is a contradiction. Lecture 30

Deﬁnition. The following properties of sheaves S of a- modules over a 185 space X are called property (b1) and property (b).

Property (b1). For each open set U of X and each homomorphism f : aU → SU , ker f has property (a1) as a sheaf of aU- modules. i∈I P Property (b). For each open set U of X and each homomorphism f : k i=1 aU → SU, ker f has property (a) as a sheaf of aU - modules. P Note. Since (a1) and (a) are local properties, properties (b1) and (b) are ′ ′ equivalent to the following properties (b1) and (b ) respectively.

′ Property (b1). For each neighbourhood V of each point x ∈ X, there exists an open set U, x ∈ U ⊂ V, such that for each homomorphism f : aU → SU , ker f has property (a1) as a sheaf of aU - modules. i∈I P Property (b′). For each neighbourhood V of each point x ∈ X, there exists an open set U, x ∈ U ⊂ V such that for each homomorphism k f : aU → SU, ker f has property (a) as a sheaf of aU -modules. i=1 PThus (b1) and (b) are also local properties. The sheaf ker f is called 186 the sheaf of relations between the sections fi : U → SU, where fi = f hi1, 1 hi f U −→ aU −→ aU −→ SU. Xi∈I The sheaf of relations between the sections fi is described by the following result.

171 172 Lecture 30

The element of (ker f )x for x ∈ U are the elements i ai of which = i ai fi(x) 0. P P Proof. For each element ai, only a ﬁnite number of the ai being dif- ferent from zero, we haveP q = i j ai h ai j and X Xj=1 q q = i j = i j f ( ai) f ( h ai j ) f h (ai j ) X Xj=1 Xj=1 q q = i j = ai j f h (lx) ai j fi j (x) Xj=1 Xj=1

= ai fi(x), Xi and this completes the proof.

If we start with a system { fi}i∈I of sections of S over an open set U, then each fi deﬁnes a homomorphism (again denoted by fi) fi : aU → SU where fi(a) = a · fi(x), a ∈ Ax. Then the system { fi} of homomorphisms deﬁnes a homomorphism f : aU → SU , and the sheaf ker f i∈I is called the sheaf of relations betweenP the sections fi : U → SU . 187 (2) If S has property (b1) (resp (b)), then every subsheaf of S has property (b1) (resp (b)).

Proof. Clear.

Deﬁnition. A sheaf S of a- modules is called coherent if it has properties (a) and (b).

Note. If S is a coherent sheaf, then SU is coherent for each open set U. Coherence is a local property, i.e., if each point has a neighbourhood U such that SU is coherent, then S is coherent. If we define a sheaf S to be of finite type if, for each x ∈ X, there is an open set U, x ∈ U, such that each stalk of SU is generated by the same finite number of sections Lecture 30 173

f1,..., fk over U; it is then easily veriﬁed that conditions (a) and (b) for coherence are equivalent to the following conditions: (i) The sheaf S is of ﬁnite type.

(ii) If f1,..., fk are any finite number of sections of S over any open set U, then the sheaf of relations between these finite number of sections is of finite type.

Example 30. Let a be the constant sheaf Z2 on 0 ≤ x ≤ 1, let R be the subsheaf obtained by omitting (1, 1) and let S be the quotient sheaf with stalks Z2 at 1 and zero elsewhere. Then the natural homomorphism a → S has kernel R and R does not have property (a1). Hence S has neither (b1) nor (b).

Example 31. Let A be the ring of Example 5, with elements 0, 1, b, c, 188 such that b2 = b, c2 = c, bc = cb = 0. Let a be the subsheaf of the constant sheaf A on 0 ≤ x ≤ 1 obtained by omitting (1, b) and (1, c). Let S be the constant sheaf Z2 on which 1 and c of A operate as the identity and b operates as zero. The sheaf R of relations for the section 1 of Z2 consists of 0 and (x, b) for x < 1. If U is a connected neighbourhood of 1, the only homomorphism of aU into RU is the zero homomorphism. Thus R does not have (a1) and S has neither (b1) nor (b).

Example 32. Let A be the ring Z[y1, y2,...] of polynomials in infinitely many variables with integer coefficients. Let a be the constant sheaf A on 0 ≤ x ≤ 1 and let S be the constant sheaf Z on which all the in- determinates y1, y2,..., operate as zero. The sheaf of relations for the section 1 of Z is the constant sheaf formed by the ideal of all polynomials without constant terms. This ideal is not finitely generated, hence S does not have (b). However, for every homomorphism f : aU → SU, i∈I ker f is constant on each component of U and hence has propertyP (a1). Thus S has (b1) but not (b).

f g (3) If 0 → S ′ −→ S −→ S ′′ → 0 is an exact sequence of sheaves ′′ of a - modules such that S has (a1) (resp (a)) and S has (b1) (resp ′ (b)), then S has (a1) (resp (a)). 174 Lecture 30

189 Proof. If S has (a1), each x ∈ X has a neighbourhood U for which ′′ there is an epimorphism φ : aU → SU. Since gφ : aU → SU is a i∈I i∈I ′′ homomorphism and S hasP (b1), ker gφ has (a1). HenceP for some open set V with x ∈ V ⊂ U, there is a homomorphism

ψ : aV → aV Xj∈J Xi∈I

such that im ψ = (ker gφ)V . Hence, since φ is an epimorphism im φψ = (ker g)V , and therefore im φψ = (im f )V . Then, since f is a monomorphism, there is an epimorphism

−1 ′ f φψ : aV → SV . Xj∈J

′ ′′ ′ Thus S has (a1). Similarly if S has (a) and S has (b), then S has (a).

a ψ V / aV j∈J i∈I P P C CC gφ φ CC CC f g C! 0 / S ′ / S / S ′′ / 0 Q.e.d. f g (4) If 0 → S ′ −→ S −→ S ′′ −→ 0 is an exact sequence of ′ ′′ sheaves of a- modules such that S and S have (b1) (resp (b)) then S also has (b1) (resp (b)).

′ ′′ Proof. Let S and S have (b1) and let φ : aU → SU be a given i∈I ′′ 190 homomorphism. Since S has (b1), ker gφ hasP (a1) and hence for each x ∈ U, there is an open set V with x ∈ V ⊂ U and a homomorphism,

ψ : aV → aV Xj∈J Xi∈I Lecture 30 175

such that im ψ = (ker gφ)V . Then im φψ ⊂ (ker g)V = (im f )V and hence there is a homomorphism

′ θ : aV → SV Xj∈J ′ with f θ = φψ. Since S ahs (b1), ker θ has (a1) and there is an open set W with x ∈ W ⊂ V, and a homomorphism

η : aW → aW Xk∈K Xj∈J such that im η = (ker θ)W .

aW k∈K mm P η mm mmm mmm mmm mmm aV mv j∈J ψη

ÓP BB Ó BB ÓÓ BB ÓÓ ψ BB θ ÓÓ ! Ö ÓÓ aU ÓÓ i∈I Ó φ P GG gφ ÓÓ xx GG ÓÓ xx GG Ó xx GG ÓÑ Ó f |xx g G# 0 / S ′ / S / S ′′ / 0

To show that S has (b1) it is enough to show that im ψη = (ker φ)W . For any element r ∈ aW , k∈K P φψη(r) = f φη(r) = f (0) = 0.

191 Thus im ψη ⊂ (ker φ)W . Next, for any element p ∈ (ker φ)W , we have p ∈ (ker gφ)W = (im ψ)W . choose q ∈ aW such that ψ(q) = p; j∈J then P θ(q) = f −1φψ(q) = f −1φ(p) = f −1(0) = o. 176 Lecture 30

Thus q ∈ (ker θ)W = im η, and hence p = ψ(q) ∈ im ψη. Thus im ψη = ′ ′′ (ker φ)W and S has (b1). Similarly if S and S have (b), it can be proved that S also has (b). Lecture 31

′′ k ′′ If f : S → S is an epimorphism, φ : i=1 aU → SU a homomor- 192 phism, and if x is a point of the open set U,P then there exists an open set V with x ∈ V ⊂ U and a homomorphism k k ψ : aV → SV such that f ψ = φ| aV . Xi=1 Xi=1 ′′ Proof. Since f : S → S is an epimorphism, each of the sections θi = φhi1 : U → S ′′, i = 1,..., k, is locally the image of a section in S . Hence there is an open set Vi with x ∈ Vi ⊂ U and a section ηi : Vi → S such that f ηi θi|Vi. Let ψi : aVi → SVi be the homomorphism deﬁned by ψi(a) = a·ηiπ(a) These homomorphisms ψi induce a homomorphism k ψ : aV → SV Xi=1 = k k k where V i=1 Vi. Then, if i=1 ai ∈ i=1 aV , we have T k kP P k f ψ( ai) = f ( ai · ηiπ(ai)) = ai · θi · π(ai) Xi=1 Xi=1 Xi=1 k k i i = ai · φh 1π(ai) = φh ai Xi=1 Xi=1 k = φ( ai), Xi=1 = k i.e., f ψ φ| i=1 aV . P 177 178 Lecture 31

f g 193 (5) If 0 → S ′ −→ S −→ S ′′ → 0 is an exact sequence of sheaves of a- modules such that S ′ has property (a) and S has property (b), then S ′′ has property (b).

k ′′ Proof. Let U be an open set and let φ : i=1 aU → SU be a homomor- ′′ phism. Since g : S → S is an epimorphism,P by the result proved above, if x ∈ U, there is an open set V with x ∈ V ⊂ U and a homo- k = k ′ morphism ψ : i=1 aV → SV such that gψ φ| i=1 aV . Since S has property (a), thereP is an open set W with x ∈ WP⊂ V and an epimor- 1 ′ phism η : i=k+1 aW → SW . Then ψ and f η induce a homomorphism 1 θ : i=1 aWP→ SW . P l k We also have the projection homomorphisms p : i=1 aW → i=1 ′ 1 l = + ′ aW and p : i=1 aW → i=k+1 aW such that θ ψp fPηp . P P P ′ l p l p k i=k+1 aW o i=1 aW / i=1 aW KK t P KK f η P ψ ttP η KK θ tt φ KK tt KK tt ′ f % ty g ′′ 0 / S W / SW / SW / 0

The second square forms a commutative diagram since gθ = gψp + g f ηp′ = gψp = φp and hence p maps ker θ into ker φ. Actually, p maps ker θ onto ker φ, for, if a ∈ ker φ, then gψa = φa = 0 and by exactness ′ = there exists b ∈ SW such that f b ψa. Since η is an epimorphism, there l = + = + = exists c ∈ i=k+1 aW such that ηc −b. Then θ(a c) ψa f ηc 0 194 and p(a + cP) = a. Thus p maps ker φ. Since S has (b) ker θ has (a), and since p| ker θ : ker θ → ker φ is an epimorphism, ker φ has (a). Hence S ′′ has (b). The corresponding statement, with (a1) and (b1) in place of (a) and (b), is not true as the following example shows.

Example 33. Let X be the union of the sequence of circles Cn = (x, y) : 2 2 x + y = x/n , n = 1, 2,.... let each stalk of a be the ring Z[x1, x2,...] of polynomials in inﬁnitely many variables, with coeﬃcients in Z, and Lecture 31 179

let a be constant except that, on going around the circle Cn, xn and -xn interchange.

More precisely, let Tn be the automorphism of the ring Z[x1, x2,. which interchanges xn and −xn. If U is open in X and U ⊂ Cn, let AU be the ring of functions f deﬁned on U and with values in Z[x1, x2,...] such 1 that f is constant on each component of U not containing , 0 and, if n ! 1 1 a component W of U contains , 0 , f (x, y) = f , 0 for (x, y) ∈ W n ! n ! 1 and y < 0 and f (x, y) = Tn f , 0 for (x, y) ∈ W and y > 0. If U is n ! not contained in any Cn let AU be the ring of functions, constant on each component of U, with values in

Z[xn1 , xn2 ,...] ⊂ Z[x1, x2,...]

1 where n1, n2,..., are those values of n for which , 0 < U. n ! = If V ⊂ U let ρVU : AU → AV be given by ρVU f f |V. Let a be the sheaf of rings determined by the presheaf {AU, ρVU }. Let I be the sheaf of ideals formed by polynomials with even coef- 195 ﬁcients, then I is generated by the section given by the polynomial 2. Let ′′ S = a/I = Z2[x1, x2,...], then 0 → I → a → S ′′ → 0 is exact. Then, as sheaves of a-modules, I has properties (a) and (a1), ′′ a has (b) and (b1) and S has (b) but not (b1). f g (6) If 0 → S ′ −→ S −→ S ′′ → 0 is an exact sequence of sheaves of a-modules such that S ′ and S ′′ have property (a), then S has property (a).

Proof. Since S ′′ has property (a), for each point x there is a neighbour- k ′′ hood U of x and an epimorphism φ : i=1 aU → SU . There is an open P 180 Lecture 31

k set W with x ∈ W ⊂ U, a homomorphism ψ : i=1 aW → SW , an epi- k ′ ′ morphism η : i=k+1 aW → SW and homomorphismsP θ, p, p as in the k = previous proof.P If s ∈ SW , there is some a ∈ i=1 aW such that φa gs. = ′ = Then g(s − ψa) 0, and by exactness, for someP b ∈ SW , s − ψa f b. k = = + = + Then for some c ∈ i=k+1 aW , b ηc and s ψa f ηc θ(a c). l 196 Hence θ, i=1 aW →PSW is an epimorphism, and hence S has property (a). P

The corresponding statement, with (a1) in place of (a), is not true as the following example shows. = ∞ = Example 34. Let X n=1 Cn as in Example 33. Let a Z and let Sn be a sheaf which is locallyS Z4, but on going around the circle Cn, 1 and ′ 3 interchange. Let Sn be the subsheaf with stalks consisting of 0 and 2; ′′ ′ it is the constant sheaf Z2. Let S n = Sn/S n; this is also Z2. Then the sequence ′ ′′ 0 → S n → Sn → S n → 0 is exact. Let

′ =Σ∞ ′ =Σ∞ ′′ =Σ∞ ′′ S n=1S n, S n=1Sn, S n=1S n. Then the sequence

0 → S ′ → S → S ′′ → 0

is exact. Since S ′ and S ′′ are constant sheaves, they have property (a1) but S does not have property (a1).

Statements (1), (2), (3), (4), (5), (6) (Lectures 29, 30, 31) prove the following proposition due to Serre.

f g Proposition 21. If 0 → S ′ −→ S −→ S → 0 is an exact sequence of sheaves of a-modules and if two of them are coherent (i.e., possess property (a) and (b)), then the third is also coherent.

197 Corollary. If Si, i = 1,..., k are coherent sheaves of a- modules, then k i−1 Si is coherent P Lecture 31 181

Proof. Since the sequence of sheaves

Σk Σk−1 0 → Sk → i−1Si → i=1 Si → 0 is exact, the result follows by induction.

Lecture 32

Let S ′ and S be sheaves of a- modules and let S ′ have property (a). 198 If f and g are two homomorphisms of S ′ → S , the set of points x for ′ = ′ which f |S x g|S x is an open set. ′ = ′ Proof. Let W be the set of points x for which f |S x g|S x and let xo ∈ W. Since S ′ has property (a), there is an epimorphism

k ′ φ : aU → S U Xi=1 j = j for some neighbourhood U of xo. Then since xo ∈ W, f φh lxo gφh lxo ( j = 1,..., k), and hence for some open set V j, with x0 ∈ V j ⊂ U,

j j f φh 1 = gφh 1 : V j → S.

f j 1 h k φ ′ % V / aV / = aV / S SV i 1 V 9 P g = k k k Let V j=1 V j. If i−1 ai ∈ i−1 Ai with x ∈ V, then T P P k k k i i f φ( ai) = f φ( h ai) = ai f φh 1x Xi=1 Xi=1 Xi=1 k k i = aigφh 1x = gφ( ai). Xi=1 Xi=1

183 184 Lecture 32

k = k Hence f φ| i=1 aV gφ| i=1 aV and, since φ is an epimorphism, ′ = ′ f |S V g|S PV . Thus xo ∈ VP⊂ W with V open. Hence W is an open set. 199 The following result deals with the extension of stalk homomorphisms. Let S ′, S be sheaves of a- modules such that S ′ is coherent and ′ S has property (a). At a point x ∈ X, let f : S x → S x be a homomorphism of Ax modules. Then there is a neighbourhood U of x and a ′ ′ homomorphism f : S U → SU whose restriction to S x is fx. Proof. Since S ′ and S have property (a), there is a neighbourhood W of x and epimorphisms

p ′ φ : aW → S W , Xj=1 q and θ : aW → SW . Xj=1

= q j q For each j 1,..., p, choose an element i=1 ai ∈ i=1 Ax such that q P P θ a j = f φh jl′  i  x x Xi=1    and choose sections   q

η j : V j → aV j Xi=1 = q j = with x ∈ V j ⊂ W, V j open, such that η j(x) i=1 ai , j 1,..., p. Let the homomorphism P q

g j : av j → av j Xi=1

be deﬁned by g j(a) = a · η j(π(a)). Then, for a ∈ Ax,

q = = j θg j(a) θ(a · η j(x)) a · θ( ai ) Xi=1 Lecture 32 185

j j = a · fxφh 1x = fxφh (a).

200 Then the homomorphisms {g j} j = 1,..., p, induce a homomorphism p q g : av → av Xi=1 Xi=1 = p p = where V ∩ j=1V j, such that θg| i=1 Ax fxφ. P r ψ p φ ′ k=1 aU / j=1 aU / SU / 0 P P g x ff q θ i=1 aU / SU P Since S ′ has property (b), there is an open set Y with x ∈ Y ⊂ V, and homomorphism r p ψ : aY → aY Xk=1 Xj=1 such that im ψ = kerφ. Then

r r = = θgψ Ax fxφψ Ax 0 Xk=1 Xk=1

by exactness. Hence by the previous result, there is an open set U with r x ∈ U ⊂ Y, such that the homomorphism θgψ k=1 aU coincides with the zero homomorphism. Therefore θg induces aP homomorphism

p f :  aU  / im ψ → SU. Xj=1        p ′ We can identify this quotient ( j=1 aU )/ im ψ with S U so that φ 201 becomes the natural homomorphism;P then f is a homomorphism f : 186 Lecture 32

′ = p ′ S U → SU with f φ θg j=1 au. If s ∈ S x since φ is an epimorphism p p p there is some element = Pa j ∈ = Ax such that s = φ( = a j). Then j 1 j 1 j 1 P P P p p f (s) = f φ( a j) = θg( a j) Xj=1 Xj=1 p = fxφ( a j) = fx(s); Xj=1

′ i.e., the restriction of f to S x is fx, q.e.d.

Let Y ⊂ X and either, let x be paracompact and normal and Y closed, or, let X be hereditarily paracompact and normal. Let S ′, S be sheaves of a- modules over X such that S ′ is coherent and S has ′ property (a). If f : S y → Sy is a homomorphism of sheaves of ay modules, there exists an open set U with Y ⊂ U, and a homomorphism g : ′ → such that g|S ′ = f. SU SU y

Proof. By the previous result, for each point y ∈ Y there is an open set ′ Vy in X with x ∈ Vy and a homomorphism φy : S vy → SVy such that ′ = ′ φy S y f S y. Then by the ﬁrst result of this lecture, the set of points of V ∩ Y at which φ = f is open in Y. Hence there is a set W open in X y y y ′ = ′ with y ∈ Wy ⊂ Vy such that φy S Wy∩Y f S Wy∩Y .

202 We now show that there are systems {Gi}i∈I and {Hi}i∈I of open sets of X such that, if G = Gi, i∈I S

(i) H¯i ∩ G ⊂ Gi,

(ii) the system {Gi} is locally ﬁnite in G,

(iii) Y ⊂ Hi, i∈I S (iv) each Gi is contained in some Wy. Lecture 32 187

(1) If X is paracompact and normal and Y is a closed subset of X, then the covering {X − Y, Wy}y∈Y of X has a locally finite refinement {G j} j∈ j, and the covering {G j} j∈J can be shrunk to a covering {H j} j∈J with H¯ j ⊂ G j. Let I ⊂ J be the set of indices for which G j ∩Y is not empty. Then, for i ∈ I, Gi is not contained in X −Y and hence is contained in some Wy. Clearly Y ⊂ Hi and conditions i∈I (i), (ii), (iii), (iv) are satisfied. S

(2) If X is hereditarily paracompact and normal, then G = Wy is y∈Y paracompact and normal. Then there is a locally finite refinemS ent {Gi}i∈I of the covering {Wy}y∈Y of G. Since G is normal, the covering {Gi}i∈I of G can be shrunk to a covering {Hi}i∈I with H¯i ∩ G ⊂ Gi. The sets Gi, Hi being open in G are open in X. Conditions (i), (ii), (iii), (iv) are thus satisfied.

Since Gi is contained in some Wy, there are homomorphisms ψi : ′ ′ = ′ S Gi → SGi such that ψi S Gi∩Y f S Gi∩Y . Let Ei j be the set of 203 ¯ ¯ ′ ′ points x ∈ Hi ∩ H j ∩ G at which ψi|S x , ψ j|S x; then Ei j is closed in G.

Let E = Ei j; it is the union of a locally finite system of closed sets in i, j G, henceS is closed in G. Let U = G − E, then U is open in G, hence open in X, and Y ⊂ U. ′ Let g : S U → SU be defined as follows: For x ∈ H¯i ∩ U let ′ = ′ g|S x ψi|S x. This gives a consistent definition of g, and g is continuous on each closed set S ′ of a locally finite system in S ′ (These closed H¯ ∩U U ′ i ′ = sets cover S U). Thus g is a sheaf homomorphism and g|S Y f . The above result for the case when X paracompact and normal is more useful in applications. In particular, the above results are when both S ′ and S are coherent sheaves of a−modules.

Example 35. Let T be the space of ordinal numbers ≤ ω1 with the topology induced by the order, let Q be the space of ordinal numbers ≤ ωo and let X = T ×Q. Then X is compact Hausdorﬀ and hence paracompact normal. Let Y1 = (T −(ω1))×ωo, Y2 = ω1 ×(Q−(ωo)) and Y = Y1 ∪Y2. ′ Let a = S = S = Z2, then S is coherent. Let f : SY → SY be the 188 Lecture 32

homomorphism which is the identity on Y1 and is zero on Y2, There is no extension of f over an open set containing Y.

Let Y ⊂ X, and let S be a coherent sheaf of a modules over X. Then the restriction SY is a coherent sheaf of ay modules.

204 Proof. Since S has property (a), for each y ∈ Y there is an open set W p of X with y ∈ W and an epimorphism θ : i=1 aW → SW . Then the p p restriction θ i=1 aW∩Y : i=1 aW∩Y → SW∩PY is an epimorphism. Thus

SY has property P (a). P

k To prove that SY has property (b), let θ; i=1 aW∩Y → SW∩Y be a homomorphism where W is an open set in X, andP let y ∈ W ∩ Y. Choose i a section fi : Vi → S, (i = 1,..., k)y ∈ Vi, through φh 1y ∈ S y and let = k ′ k V i=1 Vi. There is a homomorphism φ : i=1 aV → SV deﬁned by ′ k = k k k φ iT=1 ai i=1 ai fiπ(ai). If i=1 ai ∈ i=1 PAy, then ′ k = k = k i k i = k φ Pi=1 ai Pi=1 ai fi(y) i=P1 aiφh 1yφP i=1 h ai φ i=1 ai. ′ PSince theP set of pointsP of Y where φP = φ is openP in Y, there is ′ k = an open set G of X with y ∈ G ⊂ V ∩ W, such that φ i=1 aG∩Y

φ k a . Since has property (b), there is an open P set U with i=1 G∩Y S y ∈PU ⊂ G, and a homomorphism

1 k ′ ψ : aU → aU Xi=1 Xi=1 such that the sequence

1 k ψ′ φ′ aU −−→ aU −→ SU Xi=1 Xi=1 = ′ 1 is exact. Then if ψ ψ i=1 aU∩Y , the sequence P 1 k ψ φ aU∩Y −→ aU∩Y −→ SU∩Y Xi=1 Xi=1

is exact. Thus SY has property (b). Lecture 33

Deﬁnition . A sheaf a of rings with unit is called a coherent sheaf of 205 rings if it is coherent as a sheaf of -modules, i.e., it has property (b) (Property (a) is trivially satisﬁed.)

If a is coherent and S is a sheaf of a- modules, then S is coherent if and only if for each point x there is an open set U with x ∈ U, and an exact sequence l k ψ φ aU −→ aU −→ SU → 0. Xi=1 Xi=1 Proof. Necessity. If S is coherent, property (a) implies the existence of φ and property (b) implies the existence of ψ.

Suﬃciency. Since a is coherent, aU is coherent in U for each open l k l set U and so are i=1 aU and j=1 aU . As the image of i=1 aU , im ψ k has property (a),P and as a subsheafP of j=1 aU , im ψ hasP property (b). Thus im ψ is coherent, and there is an exactP sequence

k 0 → im ψ → aU → SU → 0. Xi=1 Hence, since two of the sheaves are coherent, the third, S )U, is coherent for a neighbourhood of each point x and hence S is coherent.

Example 36. In the ring B = Z[y, x1, x2,...] of polynomials in inﬁnitely many variables with integer coeﬃcients, let I be the ideal generated 206 by yx1, yx2,... and let A = B/I. Then multiplication by y gives a

189 190 Lecture 33

homomorphism f : A → A whose kernel C consists of polynomials in Z[x1, x2,...] without constant terms. Then the A- module C is not ﬁnitely generated. Hence if X is a consisting of one point, the constant sheaf A is not a coherent sheaf of rings.

Example 37. Let X = {x : 0 ≤ x ≤ 1} and let F be the ring of functions f : X → Z4 for which f (x) = f (1) for x > 0 and f (0) = f (1) mod 2. There are eight functions in F. Let a be the sheaf of germs of function in F. Let g be the (constant) global section deﬁned by the function g : X → Z4 where g(0) = 2 and g(x) = 0 for x > 0. The sheaf R of relations for this section is obtained by omitting from a the germs at 0 of functions f with f (0) = 1 or 3. Then the sections of R over any connected neighbourhood U of 0 contain only the germs of even valued functions, hence do not generate RU . Thus R does not have (a1) and hence a has neither (b1) nor (b).

The sheaf of germs of analytic functions in the complex plane is a coherent sheaf of rings.

Proof. Let a be the sheaf of germs of analytic functions in the complex place, and let f1,..., fk be sections of a over a neighbourhood U of a point zo, i.e., fi is an analytic function in U. We can write fi(z) = n 207 (z − zo) gi(z) where gi does not vanish at zo and hence does not vanish in = K a neighbourhood Vi of zo, zo ∈ Vi ⊂ U. Let V i=1 Vi. Let R be the = sheaf of relations between the sections fi|V, i 1T,..., k. We will show that R is ﬁnitely generated in V.

Let P = C[z] be the ring of polynomials in z with complex coef- k ficient and let M be the submodule (over P) of the direct sum i=1 P k ni = consisting of elements (p1,..., pk) for which i=1 pi(z)(z − zo) P 0. k Since P is a euclidean ring and i=1 P is finitelyP generated over P, the submodule M is finitely generatedP over p. (See van der Waerden , Mod- j j = ern Algebra, Vol, I, p. 106). Let (p1,..., pk), j 1,..., l, be a system of j = j j = generators for M. Let ri (z) pi (z)/gi(z); then each ri (z), i 1,..., k, Lecture 33 191 is an analytic function in V, and

k k j = j ni = = ri (z) fi(z) pi (z)(z − zo) 0 ( j 1,..., l). Xi=1 Xi=1 j j Thus for each j, the section determined by (ri ,..., rk) is in R. We will show that these sections generate R. k = Let i=1 tiz1 , fiz1 0 be a relation between germs at z1 ∈ V, i.e., let = k = ti(z), i P1,..., k, be analytic functions at z1 such that i=1 ti(z) fi(z) 0. k ni = Then i=1 ti(z)gi(z)(z − zo) ≡ 0. Let n max(n1,...,Pnk) and suppose = that nkP n. We can write n ti(z)gi(z) = (z − zo) qi(z) + si(z), (i = 1,..., k − 1) where qi(z) is analytic at z1 and si(z) ≡ 0 of z1 , zo and is a polynomial 208 of degree less than n if z1 = zo. Then

(t1(z)g1(z),..., tk(z)gk(z)) = n n1 q1(z)((z − zo) , 0,..., 0, −(z − zo) + ··· n nk−1 + qk−1(z)(0,..., (z − zo) , −(z − zo) + (s1(z),..., sk(z)) where sk(z) is the analytic function deﬁned by

k−1 ni sk(z) = tk(z)gk(z) + qi(z)(z − z0) . Xi=1 n n Now, ((z − z0) , 0,..., 0, −(z − z0) 1 , etc. are in M and by direct k−1 ni veriﬁcation we have i=1 si(z)(z−z0) ≡ 0. Since s1(z),..., sk−1(z), are n polynomials, it followsP that sk(s)(z − zo) , is a polynomial.

(i) If z1 , z0, then s1(z),..., sk−1(z) are all zero and

k−1 ni sk(z) = tk(z)gk(z) + qi(z)(z − z0) , Xi=1

k−1 n n n ni (z − z0) sk(z) = tk(z)gk(z)(z − z0) + qi(z)(z − z0) (z − z0) Xi=1 192 Lecture 33

k−1 n ni = tk(z)gk(z)(z − z0) + ti(z)gi(z)(z − z0) Xi=1 ≡ 0,

hence sk(z) ≡ 0, = ∞ r (ii) If z1 z0, then sk(z) has a series expansion r=0 ar(z − z0) and on n multiplication by (z − zo) this series has a ﬁniteP number of terms. Hence sk(z) is already a polynomial.

209 In either case, (t1(z)g1(z),..., tk(z)gk(z)) is a linear combination of elements of M with coeﬃcients analytic at z1. Hence l = j j (t1(z)g1(z),..., tk(z)gk(z)) h j(z)(p1(z),..., pk(z)), Xj=1

where h j(z) is analytic at z1. Then l = j j (t1(z),..., tk(z)) h j(z)(r1(z),..., rk(z)). Xj=1 Thus the sheaf R of relations is generated by a ﬁnite number of sections, hence the sheaf a is coherent. This result is a special case of Oka’s theorem, Cartan Seminar, 1951- 52, Expose 15, §5. The following proposition on the extension of coherent sheaves is based on Expose 19, §1, of the same seminar. Proposition 22. Let Y ⊂ X and either, let X be paracompact and normal with Y closed, or, letX be hereditarily paracompact and normal. Let a be a coherent sheaf of rings with unit over X and let S be a coherent sheaf of aY - modules over Y. Then there is an open set U with Y ⊂ U, and a coherent sheaf J of aU - modules over U whose restriction to Y is isomorphic to S .

Proof. Since S is coherent, there is a covering {V j ∩ Y} j∈J of Y, where V j is open in X, and for each j ∈ J an exact sequence

l j k j ψ j φ j aV j∩Y −−→ aV j∩Y −−→ SV j∩Y → 0. Xi=1 Xi=1 Lecture 33 193

210 From the properties of X there exist system s{Gi}i∈I and {Hi}i∈I of open sets of X such that, if G = ∪iGi,

(i) H¯ i ∩ G ⊂ Gi,

(ii) the system {Gi}i∈I is locally ﬁnite in G,

(iii) Y ⊂ ∪iHi,

(iv) each Gi is contained in some V j,

For the ﬁrst case, we can assume that each Gi is an Fσ set in X, hence ∩k G and all intersections r=1Gir are Fσ-sets and hence are paracompact and normal. In the second case, all subsets of X are paracompact and normal. Since each Gi is contained in some V j, there are exact sequences

l j ki ψi φ j aGi∩Y −→ aGi∩Y −−→ SG j∩Y → 0. Xi=1 Xr=1

Either Gi is paracompact and normal with Gi ∩ Y closed in Gi or Gi li a is hereditarily paracompact and normal, and the sheaves r=1 Gi and ki a ′ r=1 Gi are coherent. Hence (see Lecture 32) there is anP open set Gi ′ Pwith Gi ∩ Y ⊂ Gi ⊂ Gi and an extension

li ki ′ ψ : a ′ → a ′ Gi Gi Xr=1 Xr=1

′ of ψi. For the ﬁrst case we may assume that Gi is also an Fσ- set. Let i ki ′ ′ = ( a ′ )/ im ψ . Then, if φ is the natural homomorphism, the S r=1 Gi sequenceP li ′ ki ′ ψ φ i a ′ −−→ a ′ −→ → 0 Gi Gi S Xr=1 Xr=1 is exact, i.e., 211 194 Lecture 33

ki ′ i 0 → im ψ → a ′ → → 0 Gi S Xr=1 is exact and hence S iis coherent. There is an isomorphism

i gi : S ′ → SG′∩Y . Gi ∩Y i ′ ¯ ′ ¯ ′ ′ ′ There are open sets Hi with Hi ∩ Y ⊂ Hi , Hi ∩ G ⊂ Gi where ′ = ′ ′ q ′ G Gi . For the ﬁrst case, G and all intersections ∩r=1Gi are Fσ i∈I sets, henceS are paracompact and normal. ′ ′ ′ ′ For i, j ∈ I, there is an open set Gi j, Gi∩G j∩Y ⊂ Gi j ⊂ Gi ∩G j, and a j j j homomorphism f : S → S i such that f S = g−1g S . i j Gij Gij i j Gij∩Y i j Gij∩Y For the ﬁrst case, we may assume that G is an F set. Then there is an i j σ ′ ′ ∩ ′ ∩ ⊂ ′ ⊂ ∩ j open set Gi j with Gi G j Y Gi j Gi j G ji such that fi j f ji SG′ ∩Y ij j ′ ′ is the identity and f ji fi j S . is the identity. Let Ei j = H¯ i ∩ H¯ j ∩ Gij∩Y ′ ′ ′ (G − Gi j), then Ei j is closed in G . ′ ′ ′ For i, j, k ∈ I there is an open set Gi jk, Gi ∩ G j ∩ Gk ∩ Y ⊂ Gi jk ⊂ ′ ′ ′ = = ¯ ′ ¯ ′ ¯ ′ Gi ∩ G j ∩ Gk, such that fi j f jk|Gi jk fik| Gi jk. Let Ei jk Hi ∩ Hi ∩ Hk ∩ ′ − ′ (G Gi jk), then Ei jk is closed in G . = = ′ ′ 212 Let E ( i, j Ei j) ( i, j,k Ei jk) and let U (G − E) ∩ ( i Hi ). ′ ′ Then E is closedS in G SandSU is open with Y ⊂ U. Over each HSi ∩ U i ′ ′ there is a sheaf S ′ ; over each H ∩ H ∩ U there is an isomorphism Hi ∩U i j

′ i j i = ′ ′ → ′ ′ fi j fi j SH ∩H ∩U : SH′∩H′∩U SH ∩H ∩U i j i j i j

′ = ′ −1 ′ ′ ′ and fi j ( fi j) . Further, over each Hi ∩ H j ∩ Hk ∩ U these isomor- ′ ′ = ′ phisms are consistent, i.e., fi j f jk fik. Then by identiﬁcation there is i determined a sheaf J over U such that JH′∩U is identiﬁed with S ′ . i H j∩U i Then the isomorphisms gi : S ′ → SH′∩Y induce an isomorphism Hi ∩Y i i g : JY → S . Since each S ′ is coherent. J is coherent. Hi ∩U Bibliography

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