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## Exact functor

Homological Algebra
Papapetros Vaggelis
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### Exact functor

Let $\displaystyle{S}$ be a multiplicative subset of the ring $\displaystyle{A}$ and let $\displaystyle{M}$

be an $\displaystyle{A}$ - module.

The functor $\displaystyle{M\rightsquigarrow S^{-1}\,M}$ is exact. In other words, if the sequence of $\displaystyle{A}$ - modules

$\displaystyle{M' \xrightarrow{f} M \xrightarrow {g} M''}$ is exact, then so also is the sequence of

$\displaystyle{S^{-1}\,A}$ - modules

$\displaystyle{S^{-1}\,M' \xrightarrow{S^{-1}\,f} S^{-1}\,M \xrightarrow {S^{-1}\,g} S^{-1}\,M''}$