The search for heretics ordered by the Pope primarily refers to the establishment of the Inquisition, which began in the 12th century. The Inquisition aimed to identify, investigate, and eliminate heretical beliefs that contradicted the teachings of the Catholic Church. This often involved trials, interrogations, and severe punishments for those found guilty of heresy. The most notorious of these efforts occurred during the Spanish Inquisition, which intensified in the late 15th century under King Ferdinand and Queen Isabella.
He usually reserves the power to excommunicate for heretics.
I am not sure what your question is in reference to but the pope usually limits his powers of excommunication to heretics.
Pope Alexander VI
the inquisition
Pope Gregory I
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which pope ordered all church to keep an ever lighted lamp in it
.Catholic AnswerPope Gregory IX (1227-1241) became pope when he was already an old man - over eighty when he was elected, and he reigned for fourteen years. Pope Gregory was particularly hard on heretics, especially those who proved obstinate, he took the trial of heretics away from the local Bishops and gave it to the Dominican Order, this was called the monastic crusade, especially as many penitents were incarcerated in Benedictine monasteries. From the Catholic Encyclopedia article (see link below): Gregory IX was very severe towards heretics, who in those times were universally looked upon as traitors and punished accordingly. Upon the request of King Louis IX of France, he sent Cardinal Romanus as legate to assist the king in his crusade against the Albigenses. At the synod which the papal legate convened at Toulouse in November, 1229, it was decreed that all heretics and their abettors should be delivered to the nobles and magistrates for their due punishment, which, in case of obstinacy, was usually death. When in 1224 Frederick II ordered that heretics in Lombardy should be burnt at the stake, Gregory IX, who was then papal legate for Lombardy, approved and published the imperial law.
According to the Catholic Church, members of the orthodox Churches are technically schismatics, because they do not recognize the pope but have the same basic beliefs. Protestants are technically heretics because they do not believe certain doctrines of the Catholic Church.
It is not a person, but rather a particular type of letters patent or charter issued by a pope (in this case Pope Innocent III). The Papal Bull Ad Abolendam basically was letter declaring war on heretics.