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I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of codimension $1$). We know that we have an exact sequence, $$H^0(\mathcal{O}_U) \to H^0(U-Z,\mathcal{O}_U|_{U-Z}) \xrightarrow{\delta} H^1_Z(\mathcal{O}_U)$$

My question is: Is $\delta$ a $\mathcal{O}_U(U)$-linear morphism? If so, then it seems that $H^1_Z(\mathcal{O}_U) \otimes_{\mathcal{O}_U(U)} \mathcal{O}_Z(Z)$ should be zero. Is this correct? Am I missing something?

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Please compute through an example before asking! What happens if $U = \text{Spec}(k[x])$ and $Z = V(x)$?

Later edit. The answer to your question is that if $A$ is a ring and $f \in A$ an element, and $M$ an $A$-module, then we can look at the sequence $$ 0 \to M[f^\infty] \to M \to M_f \to M_f/M \to 0 \to 0 \to \ldots $$ This sequence will be isomorphic, as a sequence of $A$-modules, to the sequence $$ 0 \to H^0_Z(X, \mathcal{F}) \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \to H^1_Z(X, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots $$ where $X = \text{Spec}(A)$, $Z = V(f)$, $U = X \setminus Z$, and $\mathcal{F}$ is the quasi-coherent $\mathcal{O}_X$-module associated to $M$. Since the functor $\otimes_A A/fA$ is right exact and since $M_f \otimes_A A/fA = 0$ we find that the tensor product you wrote down is zero.

In the above I am using cohomology with supports $H^i_Z$. This can be defined for an arbitrary closed subset $Z$ of a topological space $X$. This then gives me long exact sequences relating $H^i_Z(X, -)$, $H^i(X, -)$, $H^i(U, -)$. If $-$ is an abelian sheaf, then these are long exact sequences of abelian groups. If $-$ is a sheaf of $\mathcal{O}_X$-modules, then these are long exact sequences of $\Gamma(X, \mathcal{O}_X)$-modules. You can prove this by taking a resolution by injectives in the category of $\mathcal{O}_X$-modules and using general theorems saying that such resolutions compute the right thing (i.e., the same as what you get for abelian sheaves). Or you can use the functoriality of the construction (because multiplying by an global section of the structure sheaf commutes with everything in sight).

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  • $\begingroup$ Thank you. I am not very familiar with computations in local cohomology. Could you please suggest a hint/reference how to compute $H^1_Z(\mathcal{O}_U)$ in this case. In particular, I do not know of any particular injective resolution of $\mathcal{O}_U$, which could help in the computation. $\endgroup$ – user45397 Sep 12 '15 at 18:31
  • $\begingroup$ @user45397 In this case it's easy to figure out $H^0(U, \mathcal O_U)$ and $H^0(U-Z, \mathcal O_U)$, without knowing anything about local cohomology. So you just need to find the map between them. An injective resolution is not needed. $\endgroup$ – Will Sawin Sep 12 '15 at 18:57
  • $\begingroup$ @Sawin: I am sorry, I still do not understand how to show that $\delta$ is not $\mathcal{O}_U(U)$-linear. Could you please elaborate a bit. $\endgroup$ – user45397 Sep 12 '15 at 19:06
  • $\begingroup$ @Kohl: I am confused. The previous question is not a new question but rather a follow up of the answer above. There is nothing new in it. Am I missing something? $\endgroup$ – user45397 Sep 12 '15 at 20:12
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    $\begingroup$ @user45397 Stefan and Will might not have seen your comments because you need to write 'at StefanKohl' or 'at WillSawin' for messages to appear in their inboxes. Anyway, I think Stefan was addressing the answerer Vlad who ends his answer with a 'question'. It's a Socratic question, obviously. I would encourage Vlad to expand this into more of a detailed explanation, lest certain reviewers in the review queue have his answer deleted via Community vote. $\endgroup$ – Todd Trimble Sep 12 '15 at 21:07

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