What roman church court was established to investigate matters of alleged heresy?

The rooting-out-of, suppression, and prosecution of heresy was the domain of the Inquisition; more popularly known as the Spanish Inquisition for their geographic success in the Iberian Peninsula. More than likely, their harsh methods were one of the key underlying factors in the eventually Reformation movement, undercutting efforts to keep the faithful in the flock, as it were.