Universal Coefficient Theorem for Cohomology

We present a direct proof of the universal coefficient theorem for cohomology. It is essentially dual to the proof for homology.

Theorem 1 Given a chain complex C in which each C_{n} is free abelian, and a coefficient group G, we have for each n the natural short exact sequence

0 − ; − ; → Ext(H_{n−1}(C), G) − ; − ; → H^{n}(Hom(C, G)) − ; − ; → Hom(H_{n}(C), G) − ; − ; → 0, (2)

which splits (but there is no natural splitting).

In particular, this applies immediately to singular cohomology.

Theorem 3 Given a pair of spaces (X, A) and a coefficient group G, we have for each n the natural short exact sequence

0 − ; − ; → Ext(H_{n−1}(X, A), G) − ; − ; → H^{n}(X, A; G) − ; − ; → Hom(H_{n}(X, A), G) − ; − ; → 0,

which splits (but there is no natural splitting).

We shall derive diagram (2) as an instance of the following elementary result.

Lemma 4 Given homomorphisms f: K → L and g: L → M of abelian groups, with a splitting homomorphism s: M → L such that s^{∘} g = id_{L}, we have a split short exact sequence

0 − ; − ; → Coker f − ; − ; →^{g′} Coker(g^{∘}f) − ; − ; → Coker g − ; − ; → 0.  (5)

Proof We write each cokernel, such as Coker f, as L/ Im f etc. Then the sequence (5) appears as the upper edge of the following diagram; we shall identify it with the bottom row, which is the canonical short exact sequence formed from the triple Im(g^{∘}f) ⊂ Im g ⊂ M.

\xymatrix{0 \ar[r] & \frac{L}{Im f} \ar[d] \ar@{-->}[dr]^{g′} \\ 0 \ar[r] & \frac{Im g}{Im(g^{∘}f)} \ar[r]^{⊂} & \frac{M}{Im(g^{∘}f)} \ar[r] & \frac{M}{Im g} \ar[r] & 0}

Since g(Im f) = Im(g^{∘}f), g induces a homomorphism g′. Since s(Im(g^{∘}f)) = Im(s^{∘}g^{∘}f) = Im f, s: M → L induces a homomorphism s′ which splits g′, s′ ^{∘} g′ = id; thus g′ is injective. But g′ clearly factors through a surjective homomorphism L/ Im f → Im g/ Im(g^{∘}f), which is therefore an isomorphism.

Preliminaries We consider a chain complex C as in Theorem 1. We adopt the usual notation: Z_{n} for the group of n-cycles, B_{n} for the group of n-boundaries, and H_{n} = H_{n}(C) = Z_{n}/B_{n}. The key idea in our proof of Theorem 1 is to express ∂: C_{n} → C_{n−1} as the composite

∂: C_{n} − ; − ; → \frac{C_{n}}{B_{n}} − ; − ; → \frac{C_{n}}{Z_{n}} − ; − ; →^{≅} B_{n−1} − ; − ; →^{⊂} Z_{n−1} − ; − ; →^{⊂} C_{n−1}.  (6)

By definition, we have the short exact sequence

0 − ; − ; → B_{n} − ; − ; →^{⊂} Z_{n} − ; − ; → H_{n} − ; − ; → 0  (7)

for any n. As B_{n} and Z_{n} are subgroups of the free abelian group C_{n} and therefore free abelian, we recognize (7) as a free resolution of H_{n}.

Also by definition, from diagram (6) we have the short exact sequence

0 − ; − ; → Z_{n} − ; − ; →^{⊂} C_{n} − ; − ; →^{∂} B_{n−1} − ; − ; → 0,  (8)

which splits because B_{n−1} is free abelian. Using (7) and (8), we may rewrite the canonical short exact sequence

0 − ; − ; → Z_{n}/B_{n} − ; − ; →^{⊂} C_{n}/B_{n} − ; − ; → C_{n}/Z_{n} − ; − ; → 0

as

0 − ; − ; → H_{n} − ; − ; → C_{n}/B_{n} − ; − ; →^{∂} B_{n−1} − ; − ; → 0,  (9)

where ∂ is a quotient of ∂. As B_{n−1} is free abelian, this short exact sequence also splits, and we may choose a splitting homomorphism s: B_{n−1} → C_{n}/B_{n}.

Proof of Theorem 1 We dualize diagram (6) and the above short exact sequences to form the following diagram of exact sequences. To simplify, we write A^{∗} for the G-dual Hom(A, G) of any abelian group A.

\xymatrix{ &  & 0 \\  &  & H_{n}^{∗} \ar[u] \\ B_{n}^{∗} & C_{n}^{∗} \ar[l] & _{(\frac{C_{n}}{B_{n}} )}_{∗} \ar[u] \ar[l] \ar[d]_{s^{∗}} & 0 \ar[l] & 0 \\ 0 & Ext(H_{n−1}, G) \ar[l] & B_{n−1}^{∗} \ar[u]_{_{∂_{∗}}} \ar[l] &  & Z_{n−1}^{∗} \ar[u] \ar[ll] \\  &  & 0 \ar[u] &  & C_{n−1}^{∗} \ar[ull] \ar[u]}  (10)

Both vertical sequences are exact, since (9) and (8) split. The upper horizontal sequence is exact, because the functor Hom(−, G) takes cokernels to kernels. The lower horizontal sequence is exact, as it defines Ext(H_{n−1}, G).

As usual, we write Z^{n} for the n-cocycles, etc., in the cochain complex C^{∗}. Diagram (10) shows that Z^{n−1} = Ker[∂^{∗}: C_{n−1}^{∗} → C_{n}^{∗}] = Ker[C_{n−1}^{∗} → B_{n−1}^{∗}]. On replacing n−1 by n, we see that Z^{n} = Ker[C_{n}^{∗} → B_{n}^{∗}] = (C_{n}/B_{n})^{∗}. Also, B^{n} = Im ∂^{∗} ≅ Im[Z_{n−1}^{∗} → (C_{n}/B_{n})^{∗}], so that H^{n} = Z^{n}/B^{n} ≅ Coker[Z_{n−1}^{∗} → (C_{n}/B_{n})^{∗}].

We now apply Lemma 4 with K = Z_{n−1}^{∗}, L = B_{n−1}^{∗} and M = (C_{n}/B_{n})^{∗}, and use the splitting s^{∗}: M → L. Diagram (10) identifies Coker[K → L] with Ext(H_{n−1}, G) and Coker[L → M] with H_{n}^{∗}. These identifications reduce the split short exact sequence (5) to the desired (2).