|MadSci Network: Genetics|
This is an interesting question. DNA is a fairly stable molecule and does not denature until 94C (under normal conditions). In a careful cremation, everything is completely incinerated, so you would not expect DNA to survive(a more accurate term than "live"). If the body were not completely cremated, you might expect some DNA to survive in well protected areas, such as inside teeth (see reference below). As for remaining airborne, DNA ought to be stable if airborne, although the mechanism of how the DNA would get airborne in the first place is unclear.
To get more information, I suggest consulting PubMed, which is a database of biomedical-related journal articles. Good luck in your search.
Your MAD Scientist,
Sweet DJ and Sweet CH. "DNA analysis of dental pulp to link incinerated remains of homicide victim to crime scene." (1995) J Forensic Sci 40(2):310-4
Try the links in the MadSci Library for more information on Genetics.