# Exact sequence involving spectral data for Higgs bundles

In Beauville, Narasimhan, Ramanan's Spectral curves and the generalized theta divisor, Remark 3.7, the following exact sequence is presented:

$$0 \rightarrow M(-\Delta) \rightarrow \pi^* E \xrightarrow{\pi^*\varphi - x} \pi^*(L \otimes E) \rightarrow \pi^*L \otimes M \rightarrow 0$$

where $$X$$ is a smooth curve, $$L$$ is a line bundle on $$X$$, $$(E, \varphi)$$ is a $$L$$-twisted Higgs bundle on $$X$$, $$\pi: X_s \rightarrow X$$ is the spectral curve associated to the characteristic polynomial of $$\varphi$$ (supposed to be integral), $$x$$ is the tautological section of $$\pi^*L$$, $$\Delta = \pi^*L^{\deg\pi-1}$$ is the ramification divisor of $$\pi$$, and $$M$$ is the unique torsion free sheaf of rank 1 on $$X_s$$ such that $$E=\pi_* M$$.

Do you have any suggestion about how to prove the exactness of the sequence? The intuition is that $$M(-\Delta)$$ must be the eigenspace of $$\pi^*\varphi$$ associated to the eigenvalue $$x$$, but it is not clear to me why it should be.