Gemara

(Mishnah): If a Yoledes brought her Chatas (which permits her to Kodshim) and died (before bringing her Olah, which is Mechaper), her heirs bring her Olah;

If she brought her Olah and died, her heirs do not bring her Chatas.

Horayos 6a - 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: Indeed, some of the sinners were alive - "many of the Kohanim... (who saw the first Mikdash)..."

Question: Perhaps they were the minority!

Rejection: "The joyful sounds of the trumpets could not be heard, due to the crying (of those who had seen Bayis Rishon, i.e. they were the majority)."

Bava Basra 9b (R. Yitzchak): One who gives a Perutah to a poor person gets six blessings. "Paros la'Ra'ev Lachmecha... (Kevod Hash-m Ya'asfecha... his Tzedakah will bring him to Gan Eden..."

10b (R. Chiya bar Aba): It says "Lo Yo'il Hon b'Yom Evrah u'Tzdakah Tatzil mi'Maves"; and "Lo Yo'ilu Otzeros Resha u'Tzdakah Tatzil mi'Maves." It mentions two Tzedakos, because one saves from a harsh death, and one saves from judgment in Gehinom.

Sanhedrin 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.

Kesuvos 37b (Beraisa - R. Yishmael son of R. Yochanan ben Berokah): We find that people liable to death at the hands of Shamayim get atonement through money - "If ransom is put on him."

"Cherem Min ha'Adam Lo Yipadeh" teaches that money does not atone for one facing execution. "Kol Cherem" excludes even light capital sins, for which there is atonement for Shogeg (Korban Chatas)?

67b: When Mar Ukva was dying, he asked to see his Tzedakah records. He saw that he had given 7000 Si'anki Dinarim.

Mar Ukva: I have sparse provisions for a long journey!

He gave half his money to Tzedakah.

Yalkut Shimoni (Yeshayahu 494): A person has three lovers - his household, his money and his good deeds. When he dies, he gathers his household and asks them to save him from an evil death. They answer "there is no authority on the day of death" (Koheles 5:8) "a brother cannot redeem" (Tehilim 49:8). His money does not help him - "Lo Yo'il Hon b'Yom Evrah" (Mishlei 11:4). His good deeds precede him and save him.

Rishonim

Perush ha'Mishnayos (Avos 4:18): After death there is no perfection or addition. One can raise his level in this world. "Ein Ma'aseh v'Cheshbon v'Da'as v'Chachmah bi'Sh'ol Asher Atah Holech Shamah" (Koheles 10:9). A person will always remain at the level he was when he died.

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? "A brother cannot redeem!" (Tehilim 49:8)

Answer (Sefer Chasidim): 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. If one merited in his life and also sinned, and he is being punished for this, then it helps if people pray or give Tzedakah for him.

Roke'ach (217 DH u'Mah, cited in Beis Yosef ibid.): Why do we pledge Tzedakah for the dead only on Yom Kipur, but not on other Yomim Tovim? It says "ha'Kipurim once in a year will 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. It does not help for a Rasha. We give for the honor of the dead, for Tzadikim advocate for their descendants.

Mordechai (Yoma 727, cited in Beis Yosef OC 621 DH Kosav): The custom to pledge Tzedakah on Yom Kipur for Mesim is from the Sifri. "Atone for Your nation Yisrael" refers to the living. "That You redeemed" refers to the dead. This teaches that the dead need Kaparah.

R. Bechaye (Devarim 21:8): The Pesikta expounds "atone for Your nation Yisrael" refers to the living, who get Kaparah through their money. "That You redeemed" refers to the dead, who get Kaparah through the money of the living. This teaches that Hekdesh (Tzedakah) of the living helps the dead, especially if a son is Makdish for his father. It is merit for the father, and the son eats the Peros. The same applies to saying Kaddish or any Berachah in the Beis ha'Keneses.

Maharam Chalavah (17): Surely, vowing to Tzedakah cannot help for one who died. One is judged based on his level when he died. What happens after that cannot change things - "Ein Ma'aseh v'Cheshbon... bi'Sh'ol." 'Im Ein Ani Li Mi Li' (Avos 1:14) - I cannot perfect myself through someone else. This is why Tzadikim such as Mar Ukva inspected their deeds and made a calculation of Tzedakah when they were about to die. He said 'I have a long journey and sparse provisions', for what others give to Tzedakah for him will not help him. If the king's servants were given royal clothing, and they dirtied it and he called them, others who are not there cannot clean their clothing for them. When one dies, his deeds are detailed in front of him, and he signs - "b'Yad Kol Adam Yachtom" (Iyov 37:7). He will say 'they judged me well.' One cannot add to the Tzedakah he did, or straighten what he made crooked. Korbanos, Chesed and Tzedakah that others do for him will not help; Hash-m does not take bribes. The Sifri seems to contradict this. The conclusion in Horihyos 6a was that there is Kaparah after death only for a Tzibur, but not for an individual. Rav Papa learned this from the Sifri! Even the Rabim get Kaparah only along with the living, and only if the living were the majority. A Chatas slaughtered to atone for a dead person is Kosher, for there is no Kaparah for the dead (Zevachim 9b). If one was going to give Gezel ha'Ger to the Kohanim, and he died, we give the money the thief's children, for there is no Kaparah for the dead (Bava Kama Bava Kama 110a, Rashi DH ha'Kesef). If one vowed Tzedakah before he died, we give it. It is like other Korbanos. If one vowed, his children bring them. Chatas and Asham are different. If one separated one of them and died, we do not offer it. We do not vow Tzedakah for one who died, just like we do not vow Korbanos for them. If one cites teachings in the name of his father in the first year after his father died, he says 'I am a Kaparah for him.' He brings merit to the Rabim (through his Torah), so it should save him, just like Tzedakah he vowed in his lifetime.

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.

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.

Rema (YD 249:16): It is an ancient custom to pledge Tzedakah for the deceased when we mention Neshamos (Yizkor), and this helps their Neshamos.

Toras Chayim (Zonenfeld) 86: A man commanded in his will that his son do for him 'Pidyon Nefesh.' The Torah discusses this regarding a Mu'ad ox (an established gorer) that killed. The owner is Chayav Misah b'Yedei Shamayim, and he redeems this through Kofer (money). Kesuvos 37b excluded various things that money does not redeem, but it redeems for a Chiyuv Misah b'Yedei Shomayim. According to the opinion that Kofer is the victim's value, this does not apply to others liable b'Yedei Shomayim! Tosfos (37b DH Misos) answers that the author of this teaching, R. Yishmael, holds that Kofer is the damager's value. Tosfos ha'Rosh connotes that the other opinion agrees that others liable b'Yedei Shomayim pays their own value. The Gemara connotes that this is the simple meaning of the verse; the opposing opinion learns from a Gezeirah Shavah that it is the victim's value regarding a Mu'ad ox. Presumably, elsewhere he agrees that it is the sinner's value. Tosfos ha'Rosh asked that according to the opinion that Kofer is a monetary payment, there is no source to redeem one who is Chayav Misah! He answers that R. Yishmael holds that Kofer is atonement. I would have said that the other opinion holds that Kofer is also a monetary obligation; all agree that it is an atonement, like the simple meaning of the verse. In any case the Gemara concludes that Kofer is an atonement.

Toras Chayim (ibid.): The Redak (Shmuel 1:14:45) brings an opinion that "va'Yifdu ha'Am Es Yehonason" means that they redeemed him through giving to Hekdesh, even though his sin was Shogeg (he did not hear of the Cherem not to eat that day). This is not a source that giving money to Tzedakah redeems. Tosfos brings from Eliyahu Rabah 10 that R. Yochanan ben Zakai told the family of Bnei Eli to give to Tzedakah and be saved. Is there a source that giving less than one's full slave-market value helps? We find the word 'Pidyon' regarding Erchin (one who pledges the Erech of Ploni gives to Hekdesh an amount based on Ploni's age and grender). Five Shekalim is the Erech of a male above a month, and it is the Shi'ur for Pidyon ha'Ben. This is even for one who (for whatever reason) is not redeemed until adulthood, for the Chiyuv came when he was a month old. Is there a source that Tzedakah atones after death? It saves from judgment in Gehinom, but perhaps this is only if he himself gave in his lifetime! Heirs do not bring a Chatas for the deceased, for there is no Kaparah for the dead, i.e. by themselves, but they do get Kaparah along with the living (Horayos 6a). R. Bechaye, Sefer Chasidim and the Taz say that Tzedakah does not help for a Rasha who would not have given. The Ohr ha'Chayim says that if one stipulates, it helps even for a Rasha, as long as he was not a Rasha in that matter.