Show a sequence of R-modules and R-homomorphisms is exact.

Asked Feb 20 '19 at 16:29

Active Feb 20 '19 at 16:38

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Suppose given a exact sequence, $$0 \to A\xrightarrow{\enspace f\enspace} B\xrightarrow{\enspace g\enspace} C\to 0$$ I want to show the sequence,$\DeclareMathOperator{\Hom}{Hom}$ $$0→\Hom_R(N,A) \xrightarrow{\enspace f_*\enspace}\Hom_R(N,B)\xrightarrow{\enspace g_*\enspace} \Hom_R(N,C)$$ is also exact.

Here $f_*:\Hom_R(N,A)\to \Hom_R(N,B)$ is defined to be, $$f_*:f\mapsto h\circ f$$where $h$ is a $R$-homomorphism from $A$ to $B$.

I'm having trouble proving exactness at both points. Please help. Thanks!

edited Feb 20 '19 at 16:38

Bernard

179,256

asked Feb 20 '19 at 16:29

gws

@ArnaudD. Yes, I edited it. – gws Feb 20 '19 at 16:34
2

Possible duplicate of Hom is a left-exact functor. See also this – Arnaud D. Feb 20 '19 at 16:35
$f_$ should be defined as $$(h:N\to A)\longmapsto (f\circ h:N\to B)$$ and similarly for $g_$. – Bernard Feb 20 '19 at 17:45

Show a sequence of R-modules and R-homomorphisms is exact.

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