# Understanding a short exact sequence of sheaves associated to a divisor

Let $$M$$ be a complex manifold and $$D = \sum_i a_i V_i$$ is an effective divisor where $$V_i$$'s are irreducible analytic hypersurfaces. Let $$s_0 \in H^0(M,\mathcal{O}([D]))$$ be a holomorphic section of the bundle $$\mathcal{O}([D])$$ in which $$[D]$$ is defined to be the image of $$D$$ under the short exact sequence of cohomology $$H^0(M,\mathrm{Div}) \to \mathrm{Pic}(M) = H^1(M,\mathcal{O}^{\times})$$ associated to the short exact sequence of sheaves $$0 \to \mathcal{O}^{\times} \to \mathcal{M}^{\times} \to \mathrm{Div} \to 0$$ where $$\mathcal{O}^{\times},\mathcal{M}^{\times}$$ are sheaves of holomorphic and meromorphic functions nonzero everywhere, respectively.

If $$E$$ is any holomorphic vector bundle, write $$\mathcal{E}(D)$$ for the sheaf of meromorphic functions on $$E$$ with poles of order $$\leq a_i$$ along $$V_i$$. Then, tensoring by $$s_0^{-1}$$ gives us an identification $$\mathcal{E}(-D) \overset{\otimes s_0^{-1}}{\rightarrow} \mathcal{O}(E \otimes [-D])$$ In Griffiths & Harris, Principles of Algeraic Geometry, page $$139$$ the authors assert that

In particular if $$D$$ is a smooth analytic hypersurface, the sequence of sheaves $$0 \to \mathcal{O}_M(E \otimes [-D]) \overset{\otimes s_0}{\rightarrow} \mathcal{O}_M(E) \to \mathcal{O}_D(E_{\mid D}) \to 0$$ is exact.

My question is why we need the condition of $$D$$ to be smooth here? In that case, could someone clarify the above sequence in more details. Any concrete example would be approciate.

This is somehow more subtle than I thought. For simplicity, consider $$E$$ to be the trivial bundle and $$D$$ irreducible. The smoothness condition on $$D$$ appears to guarantee that the sheaf of holomorphic functions on $$D$$ is well-defined (otherwise, all we could say is just its regular locus). In that case, for each open subset of $$M$$, consider the sequence $$0 \to \mathcal{O}(-D)(U) \to \mathcal{O}_M(U) \to (i_*\mathcal{O}_D)(U) \to 0$$ where $$i: D \to M$$ is inclusion and $$i_*$$ is pushforward functor. We can rewrite the above sequence $$0 \to \mathcal{O}(-D)(U) \to \mathcal{O}_M(U) \to \mathcal{O}_D(U \cap D) \to 0.$$ Since $$D$$ is effective we hence can write $$D = (s_0)$$ ($$s_0$$ is irreducible) in which $$s_0$$ is holomorphic. The first map in the above sequence is just tensoring with $$s_0$$, that is, $$s \mapsto s \otimes s_0 = ss_0$$, the second map is just the restriction $$s \mapsto s_{\mid D}$$. The map $$\mathcal{O}(-D) \to \mathcal{O}_M$$ is clearly injective, for the exactness at the middle term, if $$s_{\mid D} = 0$$ then in a neighborhood of some point $$p \in M$$, weak nullstellensatz asserts that $$s_0$$ divides $$s$$.
For a proof of weak nullstellensatz, look at page $$11$$ in Griffiths & Harris, Principles of Algebraic Geometry.