Few things to keep in mind,
Yeah as someone above pointed out it could be a weak battery,,(since you said it seems to crank but then it stops) but then if u have tried jump starting,,this possiblity is ruled out.

Security system and engine immoblizers generally keeps the power from going to relay unit that powers the starter solenoid which causes the engine not even to crank in an event of security breach. But since you said it cranked during jump start, i dont think it is security issue.

If it is indeed the key being not recognized by the system it is as simple as trying your spare key.

Normally a car has three steps in key, accessory mode, ignition mode and starter mode. It sounds. You need to have working ignition circuit to keep the engine ruinning, which is different from cranking it which is done thru starter circuit, ( ie used just to power up the started motor momentarily until the engine is running, then ignition circuit sustains the engine run). It could mean that you ignition circuit fuse might be blown out, which causes the car to crank but not start.

And another couple of simple things that one can do are, try squirting some starter fluid into the intake manifold while cranking. The engines with fuel injection system tend to have memory which seems to get the engine to crank in cold mode even though it's warm outside,,if the engine was stopped in cold condition last time,,which causes the cylinders to flood with gas during cranking and make it hard to start. The solution to press the gas pedal all the way and crank and let the blast of air in the engine to clear off the flooded cylinder..

Welll good luck..Hope it keeps guys from worshipping the mechanics so much.