Universal Coefficient Theorem

In Projective Resolution of Modules, we’ve introduced enough homological algebras. Now we can state and prove the Universal Coefficient Theorem, which relates homology and cohomology via the and functors.

Universal Coefficient Theorem

Let be a chain complex of free abelian groups and be an abelian group. Then for each , there is a short exact sequence: This sequence splits, but the splitting is not natural.

Proof Sketch

1. For the chain complex , we have the cycles and boundaries . By definition, the homology is .
2. These groups fit into two short exact sequences for each :
   • (from the definition of homology)
   • (from the definition of cycles and boundaries)
3. Since is a free abelian group, its subgroup must also be free. Because is a free (and thus projective) module, the second sequence splits.
4. Applying the functor is contravariant and left-exact. Applying it to the split sequence from step 3 gives a split short exact sequence of cochain complexes:
5. A short exact sequence of cochain complexes induces a long exact sequence in cohomology. This gives:
6. By analyzing the differentials and applying the long exact sequence for Ext derived from the sequence , one can identify the terms in the long exact sequence from step 5. This process yields the short exact sequence stated in the theorem.
7. To show the sequence splits, one constructs a splitting map . This is done by choosing a splitting of the sequence in step 3. For any homomorphism , the composition (where is the quotient map) defines a cocycle whose cohomology class gives the desired splitting.

Remark

For finitely generated abelian groups, detects only the free part, while is naturally isomorphic to the torsion subgroup of . Thus, In other words, going from homology to cohomology moves the torsion up one dimension, the opposite is true when going from cohomology to homology.

Properties of the Ext Functor

Lemma

The Ext functor has the following properties:

3. for any abelian group .
4. .

e.g. We compute the cohomology groups using the Universal Coefficient Theorem. Since the sequence splits, . We compute the two terms for each . The calculation is summarized in the following table:

0			0
1
2
3
4
5

Universal Coefficient Theorem for Homology

Similarly, we also have the universal coefficient theorem for homology, which relates homology with different coefficients:

Universal Coefficient Theorem for Homology

Let be a chain complex of free abelian groups and be an abelian group. Then the following short exact sequence splits: