Furthermore, by the twisting construction of [@Sh Lemma 2.4.1][^9], the elements $z_0$ of $S$ for which $\vr(z_0)\!=\!E_{z_0}^*$ can be omitted from $S$. In the general case, the proof of [@IvSh Lemma 2.3.2] shows that the homomorphisms D\_\^\*\_( D\_,), D\_\_( D\_\^\*,),induced by the pairings \_ref[Aadj\_e]{} are well-defined and injective. It follows that $$\ind\,D_{\vr}+ \ind\,D_{\vr}^*\le0$$ and equality holds if and only if the homomorphisms \_ref[serre\_e1]{} are isomorphisms. On the other hand, if $\ti{D}_{\vr}$ and $\ti{D}_{\vr}^*$ are the operators corresponding to $D_{\vr}$ and $D_{\vr}^*$ over the normalization $\si\!:\ti\Si\!\lra\!\Si$, dropping any matching conditions at the nodes and the other restricting conditions at the points of $S$, then $$\begin{split}
\ind\, D_{\vr}&=\ind\,\ti{D}_{\vr}-2k m-\|\vr\|, \\
\ind\, D_{\vr}^*&=\ind\,\ti{D}_{\vr}^*-2k m-2k|S|+\|\vr\|,
\end{split}$$ where $k$ is the complex rank of $E$, $m$ is the number of nodes in $\Si$, and $$\|\vr\|=\sum_{z_0\in S}\dim_{\R}\vr(z_0).$$ Since the kernel and cokernel of $\ti{D}_{\vr}^*$ are isomorphic to the kernel and cokernel of a CR-operator on $T^*\ti\Si\!\otimes\!\si^*E^*$ twisted by the preimages of the nodes and the elements of $S$, $$\ind\,\ti{D}_{\vr}^*=-\ind\,\ti{D}_{\vr}+4km+2k|S|.$$ It follows that $\ind\,D_{\vr}^*=-\ind\,D_{\vr}$ and thus the injective homomorphisms in \_ref[serre\_e1]{} are in fact isomorphisms.

Families of nodal Riemann surfaces {#RSF_subs}
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By a (), we will mean a topological space $\ov\M$ together with a partition $$\ov\M=\bigsqcup_{l=0}^{l=k}\M^{(l)}$$ 