Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1\_C(\mathcal{O}\_X)$ the sheaf associated to the presheaf $U \mapsto H^1\_{C \cap U}(\mathcal{O}\_X|\_U)$. Let $j:X\backslash C \to X$ be the natural immersion. Using the local cohomology sequence (see Hartshorne Ex. III.$2.3$) and the fact $\mathcal{H}^0\_C(\mathcal{O}\_X)=0$, we have a short exact sequence,
$$0 \to \mathcal{O}\_X \to j\_\*\mathcal{O}\_{X\backslash C} \xrightarrow{\delta\_i} \mathcal{H}^1\_C(\mathcal{O}\_X) \to 0.$$


My question is about two conflicting descrptions of $\mathrm{Tor}^1\_X(\mathcal{O}\_C,\mathcal{H}^1\_C(\mathcal{O}\_X))$. I elaborate on them.


Since $C$ is a Cartier divisor, for any small enough open affine subset $U$ of $X$, $C \cap U$ is defined by exactly one equation, say $f\_U$, on $U$. Then, $\mathrm{Tor}^1\_U(\mathcal{O}\_{C \cap U},\mathcal{H}^1\_C(\mathcal{O}\_X)|\_U)$ consists of all elements of the form $\delta\_i|\_U(g/f)$, where $g \in \mathcal{O}\_X(U)$ and $\delta\_i|\_U:\Gamma(U,j\_\*\mathcal{O}\_{X\backslash C}) \to \Gamma(U,\mathcal{H}^1\_C(\mathcal{O}\_X))$ described above. Indeed, this follows directly from the above short exact sequence and the fact that $\mathrm{Tor}^1\_U(\mathcal{O}\_{C \cap U},\mathcal{H}^1\_C(\mathcal{O}\_X)|\_U)$ consists of elements of the form $\delta\_i|\_U(g/f\_U^r)$ such that $f.\delta\_i|\_U(g/f\_U^r)=0$ (use that $\mathcal{O}\_C(U)=\mathcal{O}\_X(U)/(f\_U)$).
It seems to me that there is a natural morphism from $\mathcal{O}\_X(C)$ to $\mathcal{T}or^1\_X(\mathcal{O}\_C,\mathcal{H}^1\_C(\mathcal{O}\_X))$ induced by $\delta\_i$, the kernel of which is $\mathcal{O}\_X$. To summarize, I suspect that there exists a short exact sequence of the form $$0 \to \mathcal{O}\_X\ \to \mathcal{O}\_X(C)\to \mathcal{T}or^1\_X(\mathcal{O}\_C,\mathcal{H}^1\_C(\mathcal{O}\_X)) \to 0$$ This is my first interpretation.


My second interpretation is as follows. By checking on open affine sets, we observe that $j\_\*\mathcal{O}\_{X\backslash C} \otimes\_{\mathcal{O}\_X} \mathcal{O}\_C=0$ as well as $\mathcal{T}or^1\_X(\mathcal{O}\_C,j\_\*\mathcal{O}\_{X\backslash C})=0$. This would imply by the above short exact sequence, after applying $- \otimes\_{\mathcal{O}\_X} \mathcal{O}\_C$, that $\mathcal{T}or^1\_X(\mathcal{O}\_C,\mathcal{H}^1\_C(\mathcal{O}\_X)) \cong \mathcal{O}\_C$.


But both the interpretations cannot be correct due to the short exact sequence $$0 \to \mathcal{O}\_X \to \mathcal{O}\_X(C) \to \mathcal{O}\_C \otimes \_{\mathcal{O}\_X} \mathcal{O}\_X(C) \to 0 $$


What am I doing wrong? Any idea or reference on this question will be most welcome.



On conflicting descriptions for tor of a local cohomology group
I believe I understand now your first construction. You are comparing the following two short exact sequences. $$\begin{array}{ccccccccc} 0 & \rightarrow & \mathcal{O}\_X & \rightarrow & \mathcal{O}\_X(C) & \rightarrow & \mathcal{O}\_X(C)/\mathcal{O}\_X & \rightarrow& 0 \\ & & =\downarrow & & \downarrow & &\downarrow \\ 0 & \rightarrow & \mathcal{O}\_X & \rightarrow & j\_\*\mathcal{O}\_{X\setminus C} & \rightarrow & \mathcal{H}^1\_C(\mathcal{O}\_X) & \rightarrow & 0\end{array}$$ This induces a morphism of Tor sheaves, $$ \textit{Tor}^{\mathcal{O}\_X}\_1(\mathcal{O}\_X/\mathcal{O}\_X(-C),\mathcal{O}\_X(C)/\mathcal{O}\_X) \to \textit{Tor}^{\mathcal{O}\_X}\_1(\mathcal{O}\_X/\mathcal{O}\_X(-C),\mathcal{H}^1\_C(\mathcal{O}\_X)),$$
and this turns out to be an isomorphism. Moreover, from my comment above, there is an isomorphism of $\textit{Tor}^{\mathcal{O}\_X}\_1(\mathcal{O}\_X/\mathcal{O}\_X(-C),\mathcal{O}\_X(C)/\mathcal{O}\_X)$ with the kernel of the morphism $$r:\mathcal{O}\_X(-C)\otimes\_{\mathcal{O}\_X}(\mathcal{O}\_X(C)/\mathcal{O}\_X) \to (\mathcal{O}\_X(C)/\mathcal{O}\_X).$$ Since $r$ is a zero morphism, this finally gives an isomorphism, $$\textit{Tor}^{\mathcal{O}\_X}\_1(\mathcal{O}\_X/\mathcal{O}\_X(-C),\mathcal{H}^1\_C(\mathcal{O}\_X)) \cong \mathcal{O}\_X(-C)\otimes\_{\mathcal{O}\_X}(\mathcal{O}\_X(C)/\mathcal{O}\_X).$$ Of course the second sheaf is isomorphic to $\mathcal{O}\_X/\mathcal{O}\_X(-C)$.


