what's with the gemstomes?spoilers

we fiest see the doctor handing over the red gemstone found on dutchess' body at the lake.then we see the green gemstone popping out of the locket?and the inspector puts it in the sword and it fits in it!wts up with these gems??
Also,how come locket is found in the stable?or eisenheim planted it there?and why the inspector suddenly feels to search the stable again?
Pls explain
Obsession is young man's game:Cutter(The Prestige)