1. The Tosefta discusses two Kiros that share a wall.
2. One may put food next to the wall of a Kirah on Shabbos.
3. If the coals cooled off on their own, it is similar to coals being taken away or having ash put on them.
4. There is a dispute about whether one may put an item that improves as it cooks on a place from which coals were removed.
5. The Gemara discusses what is considered an item that improves the longer it cooks.
A BIT MORE
1. The law is that before Shabbos one may put food on the Kirah that has its coals removed or ash put on it. There is a dispute among the Tana'im regarding what may be put on this Kirah. One may not put anything on the second Kirah.
2. This is permitted even if the Kirah did not have the coals removed or ash put on it before Shabbos.
3. Similarly, if one puts some flax pieces on the fire, it is considered like ash was put on the coals.
4. Rav and Shmuel: It is forbidden. Rebbi Yochanan and others: It is permitted.
5. There is a doubt about certain items, such as fish that has flour on it. On one hand, leaving it on the fire longer might be considered to improve it, since the fish becomes cooked better. On the other hand, flour should not be left on the fire for a long time, and thus perhaps leaving this food on the fire is not considered an improvement for the food. Rav Nachman concludes that anything that has flour is generally considered not to improve on the fire (after it has been cooked or baked).