I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71: $$H_{2n-q}(S)\rightarrow H_{2n-q}(V)\rightarrow H_{2n-q}(V\, mod \,S)\rightarrow H_{2n-q-1}(S)\rightarrow H_{2n-q-1}(V).$$ In the paper it is claimed that the above exact sequence is isomorphic to the exact sequence (14) as follows: $$H^{q-2}(R^1(S))\rightarrow H^{q-1}(d\Omega^0(*S))\rightarrow H^{q-1}(\Phi^1(*S))\rightarrow H^{q-1}(R^1(S))\rightarrow H^q(d\Omega^0(*S)).$$ In particularly, when $q=2$, $H^{0}(R^1(S))\rightarrow H^{1}(d\Omega^0(*S))$ is isomorphic to $H_{2n-2}(S)\rightarrow H_{2n-2}(V)$, where the second homomorphism is induced by the embedding $S\rightarrow V.$
I understand $H^{0}(R^1(S))\cong H_{2n-2}(S)$ and $H^{1}(d\Omega^0(*S))\cong H_{2n-2}(V)$. However I don't understand why the homomorphisms are also isomorphic. Can anyone show me the argument to prove this?
${\bf{Edit}}$: Here $\Omega^q (*S)$ is the direct limit of $\Omega^q(kS)$ as $k\rightarrow\infty$, where $\Omega^q(kS)$ is the sheaf of meromorphic $q$-forms having, as their only singularities, poles of order $k$ (at most) on the components of $S$.
$d\Omega^0(*S)$ is the sheaf of differentials of $\Omega^0(*S)$, $\Phi^1(*S)$ is a subsheaf of $\Omega^1(*S)$ consisting of closed 1-forms and $R^1(S)$ is the cokernel of $d\Omega^0(*S)\rightarrow\Phi^1(*S)$.
