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Let $f:X\to Y$ be a morphism of schemes. Then there exists a functor $f_*:{Sh}X\to {Sh}Y$ with $f_*\mathcal{F}(U)=\mathcal{f^{-1}(U)}$ whenever $\mathcal{F}$ is asheaf on $X$.

It is proved that the direct image functor is a left adjoint functor. Now the question is which conditions are needed to impose on $f$, $X$, $Y$ or sheaves on $X$ to deduce that the direct image functor translate an exact sequence of sheaves on $X$ to an exact sequence of sheaves on $Y$? i.e. When the direct image functor is an exact functor?

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migrated from Aug 15 '13 at 17:08

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The direct image functor is a right adjoint functor, which means it is left exact. – Zhen Lin Aug 15 '13 at 14:12
Vote to close (too elementary for mathoverflow). Read about sheaf cohomology, higher direct images, etc. – Martin Brandenburg Aug 15 '13 at 14:30

If $f$ is the inclusion of a closed subspace of $Y$, then there is a natural isomorphism $f_* \cong f_!$, so $f_*$ would be exact.

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