SHECHITAH FOR IDOL-WORSHIP
(Mishnah): If one slaughters to serve mountains, hills, seas, rivers or the wilderness, the Shechitah is invalid;
If two people slaughter together, and one intends to serve one of these things, and the other intends for proper Shechitah, the Shechitah is invalid.
(Gemara) Inference: The Shechitah is only invalid, but it is not Takroves.
Contradiction (Beraisa): If one slaughters to serve mountains, hills, seas, rivers, the wilderness, the sun, moon, stars, constellations, Micha'el (the great ministering angel), or the angel that oversees a worm, this is Takroves.
Answer (Abaye): Our Mishnah discusses Shechitah to serve the mountain (or sea...) itself. The Beraisa discusses Shechitah to serve its overseeing angel.
Support: Presumably, all cases of the Beraisa resemble Micha'el, i.e. an angel.
FORBIDDING ANOTHER'S PROPERTY
(Rav Huna): If Shimon's animal was crouched in front of an idol, once Reuven slaughters one Siman, he forbids it.
This is like Ula;
(Ula): Even though Reuven cannot forbid Shimon's cow through bowing to it, if he did an action to it, he forbids it.
Question (Rav Nachman - Beraisa): If one slaughters a Chatas outside the Mikdash, for idolatry, on Shabbos (all b'Shogeg), he brings three Chata'os to atone for this.
If the animal were forbidden after slaughtering one Siman, he would be exempt for Shechutei Chutz (slaughtering a Korban outside the Mikdash), for cutting the second Siman would be like cutting earth (since the animal is already forbidden. Shechutei Chutz applies only to Kosher Korbanos!)
Answer #1 (Rav Papa): The Beraisa discusses Chatas ha'Of. Since one needs to cut only one Siman, the liabilities come simultaneously (when the Siman is cut).
Rejection: Rav Huna holds like Ula, who says that even a small action forbids the animal (i.e. once a fragment of the Siman is cut)!
Answer #2: Rather, the case is, he said that he does not intend to serve idolatry until he completes the Shechitah.
Rejection: If so, why does the Beraisa discuss a Chatas? The same applies to any Korban!
Answer #3 (Mar Zutra): The Beraisa discusses Chatas ha'Of in which exactly half of the Kaneh was already cut. The moment he adds to this cut, the Shechitah is completed, so the liabilities come simultaneously.
(Rav Papa): Had Rav Huna not said that the animal is forbidden once one Siman is cut, we could not have challenged him from the Beraisa.
He could say like Ula, that a (major) action (i.e. complete Shechitah) to another's animal forbids it. (Since the animal is permitted until the Shechitah is completed, he is liable (also) for Shechutei Chutz.)
(Rav Papa): Had Rav Huna not said that one can forbid another's animal through an action, we could not have challenged him from the Beraisa.
He could say that one cannot forbid a Chatas, for it is not considered his animal (the meat goes to the Kohanim. Since his Shechitah for idolatry does not forbid the animal, he is liable for Shechutei Chutz.)
Question: This is obvious. What is Rav Papa's Chidush?
Answer: One might have thought that since one gets atonement from a Chatas, it is considered his animal. He teaches that this is not so.
(Rav Nachman, Rav Amram, R. Yitzchak): One cannot forbid another person's property (Rashi - even through a major action).
Question #1 (Beraisa): If one slaughtered a Chatas outside the Mikdash, for idolatry, on Shabbos, he brings three Chata'os.
We established that this discusses Chatas ha'Of in which exactly half of the Kaneh was already cut.
Inference: He is liable for all three only in this case, because the liabilities come simultaneously. Regarding an animal Korban, he would be exempt for Shechutei Chutz (because the animal becomes forbidden in mid-Shechitah, even though he does not really own his Korban)!
If one could not forbid property that is not his, we could establish the Beraisa to discuss even Chatas Behemah, since he cannot forbid it!
Answer: Since one gets atonement from a Chatas, it is considered his, so he can forbid it.