On conflicting descriptions for tor of a local cohomology group 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.
 A: 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 $\mathcal{O}_X/\mathcal{O}_X(-C)$.
