$Ext_F$ vanishes if $F$ is a field?

by Yuyi Zhang   Last Updated September 12, 2019 04:20 AM

In the universial coefficient theorem for cohomology, we have split exact sequence $0 \to Ext(H_{n-1}(C),G)\to H^n(C;G)\to Hom(H_n(C),G)\to 0$, where $H_n(C)$ are homology groups for a chain complex $C$ and $G$ is a chosen group.

It is said that if $G=F$ is a field, then we have isomophism $H^n(C;F)=Hom(H_n(C),F)$ because $Ext_F$ vanishes since $F$ is a field. I don't know why $Ext_F$ vanishes under this circumstance? Hope someone could help. Thanks!



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