Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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IV.1 Cohomology with Coefficients <strong>in</strong> G 157<br />
(IV.1.5) Lemma. ker(�<strong>in</strong>) ∼ = Hom(Hn(C),G).<br />
Proof. We def<strong>in</strong>e<br />
φ : ker(�<strong>in</strong>) → Hom(Hn(C),G)<br />
so that, for every x+Bn(C) ∈ Hn(C), φ( f )(x+Bn(C)) = f (x). This function is well<br />
def<strong>in</strong>ed and is a homomorphism. On the other hand, we def<strong>in</strong>e<br />
ψ : Hom(Hn(C),G) → ker(�<strong>in</strong>)<br />
by sett<strong>in</strong>g ψ(g) =gp,wherep: Zn(C) → Hn(C) is the quotient homomorphism.<br />
Also this function is a homomorphism; <strong>in</strong> addition, the compositions satisfy ψφ =<br />
1 ker(�<strong>in</strong>) and φψ = 1 Hom(Hn(C),G). �<br />
This result shows that ker(�<strong>in</strong>) is <strong>in</strong>dependent from the groups Hom(Bn(C),G) and<br />
Hom(Zn(C),G), and depends only on Hn(C), the cokernel of the monomorphism<br />
<strong>in</strong> : Bn(C) → Zn(C), and on G. As <strong>in</strong> the case of the functor −⊗G, we wonder<br />
whether the same is also true for coker(�<strong>in</strong>−1). In fact, we have the follow<strong>in</strong>g result<br />
which is dual to Proposition (II.5.4):<br />
(IV.1.6) Proposition. Let H be the cokernel of the monomorphism i: B → Zbetween<br />
free Abelian groups and any Abelian group G. Then, both the kernel and<br />
the cokernel of the homomorphism �i: Hom(Z,G) → Hom(B,G) depend only on<br />
H and G. Moreover, ker(�i) ∼ = Hom(H,G), while coker(�i) gives rise to a new contravariant<br />
functor<br />
Ext(−,G): Ab −→ Ab,<br />
called extension product.<br />
Proof. The method used <strong>in</strong> this proof is analogous to the one for prov<strong>in</strong>g<br />
Proposition (II.5.4), that is to say, merely replac<strong>in</strong>g the functor −⊗G with the<br />
functor Hom(−,G) and keep<strong>in</strong>g <strong>in</strong> m<strong>in</strong>d the contravariance of the latter. We leave<br />
the details to the reader. Nevertheless, we note that Ext(−,G) does not depend on<br />
the free presentation of H. �<br />
Let us compute Ext(Zn,G) for any Abelian group G. S<strong>in</strong>ceExt(−,G) does not<br />
depend on any particular free presentation of Zn, we choose the presentation<br />
Z ��<br />
n ��<br />
Z<br />
��<br />
��<br />
Z/n<br />
where n is the multiplication by n. Then, we have the exact sequence<br />
Hom(Zn,G)<br />
��<br />
Hom(Z,G)<br />
�n ��<br />
Hom(Z,G)<br />
��<br />
Ext(Zn,G)<br />
and s<strong>in</strong>ce Hom(Z,G) ∼ = G, we conclude that Ext(Zn,G) ∼ = G/nG. In particular,<br />
Ext(Zn,Z) ∼ = Zn.<br />
Due to Lemma (IV.1.4),ifH is free, Ext(H,G)=0.<br />
��<br />
0