Gemara

Question: How could those who returned from exile in Bavel offer Chata'os to atone for idolatry done in the days of Tzidkiyahu at the end of Bayis Rishon? The sinners died!

Answer #1 (Rav Papa): Regarding an individual's Chatas, if the owner died, the Chatas must die. This does not apply to a Tzibur.

We learn from "Atone for Your nation Yisrael that You redeemed, Hash-m" - Eglah Arufah (a calf beheaded when a murdered corpse is found) is worthy to atone for those that Hash-m redeemed from Mitzrayim.

Objection: There, the calf is beheaded primarily to atone for the living. It atones also for the dead. Here, it was not an atonement for the living!

Answer #2: This was a special ruling. We must say so, for they sinned b'Mezid, and we do not bring Chata'os for Mezid!

Sotah 10b: "(David) cried... my son Avshalom, my son, my son... He said "my son" eight times to save him from the seven chambers of Gehinom, and to get his head to rejoin his body, or bring him to the world to come.

Chagigah 15b: When Acher died, he was not judged (due to his Torah), but he was not brought to the world to come, because he sinned. R. Meir said 'after I die, I will raise smoke from his grave. It is better that he be judged and get the world to come.' R. Yochanan said 'after I die, I will stop the smoke from his grave.' (This was fulfilled.)

Sanhedrin 46b - Question: Is burial to avoid disgrace (of a rotting corpse), or for atonement?

If one asked (before he died) not to be buried, if we are concerned for disgrace, we do not comply. If it is for atonement, since he does not want it, we honor his request.

Answer #1: Tzadikim are buried, even though they do not need atonement!

Rejection: They do - "Adam Ein Tzadik ba'Aretz... v'Lo Yecheta."

Answer #2: "V'Sofdu (Aviyah)... v'Kovru Oso." (If it is for atonement, they should have buried all of Yarav'am's family!)

Rejection: Hash-m did not want them to get atonement, for they were Resha'im.

47a (Beraisa): Chizkiyah dragged his father's bones on a bed of ropes, in order that the disgrace would atone for his father.

(Beraisa - R. Noson): It is a good sign for the deceased if he is punished in this world, e.g. if he was not eulogized, was not buried...

104a: Achaz is not listed among those without a share in the world to come, for his son Chizkiyah was a Tzadik. Chizkiyah's son Menasheh was listed, for a son brings merit to his father, but a father does not bring merit to his son:

"V'Ein mi'Yadi Matzil" - Avraham does not save Yishmael (from punishment), and Yitzchak does not save Esav.

Yehoyakim is not listed among those who have no share. He got atonement through the disgrace of (his skull) not being buried.

R. Freida's grandfather found a skull by the gates of Yerushalayim. It said on it 'this and one other.' He buried it (twice), and (each time) it came up.

Rishonim

Tosfos and Tosfos ha'Rosh (Sanhedrin 46b DH Kevurah): Surely burial is Kaparah. The Gemara (47b) discusses when is the Kaparah! We ask which is the primary reason for burial. The Gemara says that there is a bigger Kaparah when one is not buried! This means that he was not buried honorably.

Note: Sanhedrin 104a connotes that not being buried at all is a Kaparah.

Sefer Chasidim (170): A Chasid used to pledge for the Neshamos of his (deceased) relatives, and afterwards he pledged everything Stam for all Neshamos. He explained 'I benefit from the merit of some of them, and I do not deny the good they do for me. Also, they will do good to me.' If a Chatas' owner died, it dies, for Chatas is to protect from afflictions, and the owner already died. Also, death atoned. Tzedakah is different. It is like Eglah Arufah, which can atone retroactively for those who left Mitzrayim.

Question: If the son was a Rasha, how can the father redeem him through Tzedakah? "One cannot redeem a man" (Tehilim 49:8).

Answer (Sefer Chasidim): We find that if one vowed to bring an Olah, his son brings it after his father died. Tzedakah is like an Olah. Both are voluntary. This is not an absolute proof. A son can bring his father's Olah only if his father was Makdish the animal. Once, a man lost in the forest at night saw a man who had died. The Mes said that because he stole a field, he has no rest, and they weary him in forests. A similar case occurred in which a dead Nochri said 'tell my wife to return what I stole, and then I will have rest.' Perhaps this is not true for Nochrim, but surely it is true for Yisre'elim. Why does it help? In his life he did not do or command to return it! Rather, the living do for him. There is a distinction. If one did not merit in his life, all Tzedakah will not help him. If he merited and also sinned, and due to his sins he was expelled from Gan Eden or he is punished in Gehinom or they weary him through thorns or wounding angels frolic with him, or a Tzadik is not allowed into the domain of Tzadikim due to sin, then it helps if people pray or give Tzedakah for him. This is also if he stole and the heirs return it, or if the Mes is disgraced to diminish from his sins. If one did not merit, everything people do for him will not help him, unless he commanded in his lifetime and they fulfill after death, like we find with Moshe and David. (Moshe commanded to bring Yosef's coffin to Eretz Yisrael for burial. Yehoshua and Yisrael did this. David wanted to build the Beis ha'Mikdash, but Hash-m did not allow him. David commanded Shlomo to build it. These Mitzvos were attributed to Moshe and David.

Poskim

Shulchan Aruch (OC 621:6): The custom is to pledge Tzedakah on Yom Kipur for Mesim.

Rema: And we mention their names, for also they get Kaparah on Yom Kipur.

Beis Yosef (DH Kosav): The Mordechai supports the custom to pledge Tzedakah on Yom Kipur for Mesim from the Sifri. "Atone for Your nation Yisrael" refers to the living. "That You redeemed" refers to the dead, and teaches that the dead need Kaparah.

Levush (284:7): All of Yisrael are Arevim (guarantors) for each other. Yisrael were redeemed from Mitzrayim on condition that they and their descendants not murder. Therefore, when one murdered, the dead are responsible and they need atonement.

Beis Yosef (ibid.): The Roke'ach (217 DH u'Mah) brings an Asmachta for pledging Tzedakah on Yom Kipur for Mesim - "ha'Kipurim once in a year wil atone", and two verses later it says "a man will give ransom for his soul to Hash-m." How does it help the Mes if a living person gives Tzedakah for him? Rather, Hash-m investigates the hearts of the living and the dead. If the Mes gave Tzedakah when he was alive, or if he was an Oni but he had a good heart and he would have given if he had the money, then it helps a little. The living can ask to ease the Mes' judgment, like David did for Avshalom, and R. Yochanan for Acher. It does not help for a Rasha. We give for the honor of the dead, for Tzadikim advocate for their descendants.

Mishnah Berurah (18): It is called Yom ha'Kipurim (plural) because it atones for the living and the dead (Mahari Weil).

Mishnah Berurah (19): Vowing Tzedakah for a dead Rasha does not help. If he confessed before he died, we can say that he got Kaparah and he is considered a Tzadik. Presumably, what a son gives for his father helps in every case, for a son brings merit to his father.

Kaf ha'Chayim (34): The custom is to do so also at other times and on Shabbos when we mention Neshamos of relatives. The Zohar supports this.

Rema (284:7): The custom is to mention Neshamos on Shabbos after Kri'as ha'Torah.

Bi'ur Halachah (DH v'Nahagu): It was fixed to do so on Shabbos for then everyone is in the Beis ha'Keneses, and the living will put it to heart (that they wil die).

Kaf ha'Chayim (282:72): Mesim are not punished on Shabbos, therefore, it is proper to mention them for rest and pray for them.