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The amount of debt forgiven is reportable on 1099-C - http://www.irs.gov/pub/irs-pdf/f1099c.pdf - generally is taxable, unless any exemption apply -- you should file a form 982 - www.irs.gov/pub/irs-pdf/f982.pdf - to proof your exemption - and might exclude all or part of canceled debt from taxable income.
However - because the debt was canceled AFTER the death - that income belongs to the estate - and is not reported on the decedent's final tax return.
Thus - if the final tax return is filed jointly with the spouse - the spouse is not liable for the canceled debt of the deceased.
What you have - is an income in respect of a decedent (IRD).Income in respect of a decedent realized AFTER the death is taxable.. Income in respect of the decedent is gross income that the decedent would have received had death not occurred and that was not properly includible in the decedent's final income tax return.
Income in respect of a decedent must be included in the income of one of the following.
The decedent's estate, if the estate receives it.
The beneficiary, if the right to income is passed directly to the beneficiary and the beneficiary receives it.
Any person to whom the estate properly distributes the right to receive it.
If there is no estate - none will be responsible for the tax liability.
If the estate insolvent - you may file form 982 to exclude income realized because of canceled debt.
Please see discussion here - http://www.taxalmanac.org/index.php/Discussion:Debt_Cancellation_of_Deceased_Taxpayer_year_after_death
Please note that there is no clear indication of the IRS position toward such situation.
Let me know if you need any help.