# $Ext_F$ 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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