No, you must loosen the alternator. It would be impossible to apply the required pressure to the chain tensioner to refit it with the alternator in place.

It just requires one very large bolt (14 or 15 mm?) (the top bolt) to be undone and removed and then the alternator pivots and swings down out of the way (be careful that the alternator is not hanging full weight on the electrical harness that is attached to avoid damage to the harness and connections).