Let $X$ be a smooth projective algebraic variety over a field of characteristic zero and let $D$ be a simple normal crossing divisor on $D$. Put 


$j: U \hookrightarrow X$


for the inclusion of the complement $U=X-D$ on $X$. 


Are the functors $j\_\ast$ and $j\_!$ exact?


I think the answer is yes. 


I would be very grateful if someone could provide me with a proof. 



are these functors exact?
I assume you're asking about quasicoherent sheaves.


Since the inclusion of $U$ into $X$ is an open immersion, $j\_!$ is exact. You can find this in [Tag 03DJ](http://stacks.math.columbia.edu/tag/03DJ).


Since $j$ is an affine morphism, $j\_\*$ is exact (EGA 2 Corollary 5.2.2). $j$ is affine, because the affine property of morphisms is étale local on the target, and the inclusion of the complement of a principal closed subscheme (such as the locus defined by $x\_1 x\_2 \cdots x\_r = 0$ in affine space) is affine. See e.g., [Tag 07ZT](http://stacks.math.columbia.edu/tag/07ZT)


