Here is the dvar torah JayMatt emailed me before Shabbos to post now:
The Machlokes of Korach
The Mishna in Pirkei Avos (5:17) compares the
machlokes, argument, of Korach to that of Shamai
and Hillel. The Mishna states that the difference
between the two was that the machlokes between
Shamai and Hillel was leshaim shomaim, for the
sake of heaven, while the machlokes of Korach and his people was
not.
1. The Midrash Shmuel asks why does the mishna say “Korach and his group” and not “Korach and Moshe” the way the mishna says “Shammai and Hillel”?
He answers that Moshe was l’shem Shamayim, it was only Korach and his people who were not l’shem shamayim.
2. Why does it say “Korach and his group”? This is because even within the group there was different kavanas for each person. There was no real unity, they just united for this single cause (to argue with Moshe). That itself shows us their debate was not l’shem shamayim. (Me’or V’Shemesh in the name of Noam Elimelech)
3. It says Shammai and Hillel as opposed to Beis Shammai and Beis Hillel since Hillel and Shammai were certainly argued l’shem shamayim without any other factor involved. Whereas the talmidim might have had minute amounds of other intentions (e.g. my Rebbe is greater than yours) nevertheless it was still l’shem shamayim, though not on the high scale of a Shammai and Hillel.
4. How do we know Korach’s debate was without the correct intentions? The were arguing on their Rebbe. And anyone who argues on their Rebbe it is as if they are arguing with the Shechina. Therefore its obvious it was not l’shem shamayim.