Even if he had hold luggage, he could ask it to be tagged only as far as FRA. Many people do that, and there is no problem.
The OP's problem isn't about luggage, but about breaking one of the conditions of travel, that of not completing every stage of an outward flight. OP said he bought a flight to Luxembourg because it was cheaper than a straight US to Frankfurt flight. Often a ticket from an airport (A) via a hub (B) to a destination in a third country (C) is cheaper than a straight A to B ticket, because the airline wants you to travel with them to get to C (rather than your booking with a competitor flying non-stop from A to C). For example, Lufthansa may offer a cheaper fare travelling between New York and London via Frankfurt than BA flying from JFK to LHR non-stop. And the Lufthansa fare is likely to be lower than straight JFK to FRA flight. I don't think there is a non-stop flight between US and LUX, but LH are in competition with a number of other carriers for that sector (AF, KLM, BA etc and US carriers using codeshare).
What OP is proposing is known as 'hidden city fare', in this case Luxembourg being the hidden city (or destination). It's hiden because it's only on the ticket in order to secure a cheaper fare to Frankfurt and the OP has no intention to travel to Luxembourg. Airlines have argued that hidden city fare violates their ticketing conditions. You will be caught out if you do it on a return ticket, but it will be difficult to see how the airline can catch you out on two, unrelated one-way fares (but such tickets would normally be very expensive, and in any case you cannot board your return flight at FRA if you haven't travelled from LUX - the airline computer will flag you up as no show for the LUX to FRA sector).