So it's alternate cost of 0 strictly effects the colorless mana for {archive trap} right. I will still have to pay the blue in order to make them mill.
For what I read it's the total cost, given they searched for something in lib, so no blue
actually it is even better. the total cost becomes {0} to play it. If it said it costs {3} less to cast then you would still have to pay the blue
RESOLVED
Combo with {Ghost Quarter} targeting opponents land. They'll search not to fall behind. Drop all traps in hand.