From expatica.com,
According to the French criminal code (code penal) every crime that is committed abroad by a French national is punishable in France. This has allowed the government to prosecute individuals for crimes committed by French citizens regardless of geography. This has included prosecuting individuals for genocide. The law also gives France jurisdiction in cases of genocide where the crime is committed by a non-French national against a French citizen. However, in these cases criminal proceedings can only be initiated by the public prosecutor following a complaint by the victim or their legal successors. Additionally, France’s criminal jurisdiction has international reach based on territorial impact, national security or protection of currency against counterfeiting.
American law prevents double jeooardy, I.e. being tried twice for the same crime. If a French national has been extradited and sentenced for a crime committed on American soil, can they also be prosecuted for the same crime by the American government if they return to the US?