  In this paper, we discuss the noncommutative QED$_2$ in the S-matrix
framework. We are interested in perturbatively proving that the exact Schwinger
mass $\mu^2=\frac{e^2}{\pi}$ does not receive noncommutative corrections to any
order in loop expansion. In this sense, the S-matrix approach is useful since
it allows us to work with the effective action $\Gamma[A]$ (interaction term)
to compute the corresponding gauge field 1PI two-point function at higher
orders. Furthermore, by means of $\alpha^*$-cohomology, we generalize the
QED$_2$ S-matrix analysis in the Moyal star product to all
translation-invariant star products.
