Ash may not be biochar, but it is quite a bit different from
charcoal, too. It is less inert in the soil. It does have some
available nutrients (wood ash has potassium and a little bit
of phosphorous and minerals; it is alkaline and will raise pH).
The nutrient content of some kinds of ash, mostly fruit and
vegetable rinds, is listed in the Primalseeds list of random
compostable materials:
http://www.primalseeds.org/npk.htm