Show that $L_0T$ is right exact.Here T is an additive functor from category of $\Lambda$-modules to abelian group.
I try to prove it,but I think the condition is too little.Any hints?
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By construction, if you have a short exact sequence $0\to X'\to X\to X''\to 0$, you get a long exact sequence whose initial terms look like $\cdots\to L_1TX'' \to L_0TX'\to L_0TX\to L_0TX''\to 0$ so $L_0T$ is right exact.
Pedro
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but by definition of right exact,the given sequence is $X' \to X \to X'' \to 0$ – Daniel Xu Jan 02 '19 at 03:39
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@DanielXu The definitions are equivalent, you can always assume the first map is a monomorphism. – Pedro Jan 03 '19 at 16:21