OK, If the Gauge or Wire or sender were "open" the computer would probably see an open loop situation..Setting off the Check engine light..So you need to troubleshoot and eliminate Those first..
The sender should have a resistance to ground when measured, (you'll have to look it up for your car or ask a parts guy..) remove the sender, get a pan of boiling water (212 degrees) and a accurate lab (or meat) thermometer, next, hook the sender up to It's plug, and tightly wrap a bare ended ground wire around the sender body, and ground it to the engine..
Immerse The sender in the water CAREFULLY, and monitor the gauge in the car..It should report..If not remove the sender, clip your DVOM, set to R X 1 , OHMS scale, calibrated to "000" . to ground and either pin on the sender, immerse it back in the water, (be sure it's still 212 degrees) and watch your meter..It should read at or above the resistance Specs for the sender..Watch as the water cools, it should decline or increase in resistance readings (depending on how the sender is orientated) .. Do the same for the other pin..If it HAS a reading, It's probably good, If it's infinite, (no reading changes) It's bad..Replace it.
If the sender checks good, find the wire that goes AWAY from it to the gauge bucket and/or computer, (follow the color code to the pins ) You SHOULD get a manual for your car (auto zombie..like $21.00) for the pin numbers assigned..Using your DVOM, set to OHMS, R X 1 , Calibrated to "000", measure the wire to BOTH points..it should read "000" , in no reading OR the reading fluctuates , repair or replace the wire..It's broken, burned or has a bad pin..CLEAN and burnish the computer plug pins AND backplane with a Good electrical cleaner, and burnishing tool..BE careful in this area and disconnect the battery cable.
Next Check the wire between the gauge and / or the computer or sender the same way, from the laminate card, and the sender or computer..It should read "000" if not its bad, repair or replace as required..
If that is good, remove the laminate card from the bucket (it should all come as a unit, gauges and all..Inspect it for burned traces..or bad pins in the plug well..measure between the "S" , "I", and "G" pins as before, for a "000" reading..If any are infinite, the pin, or trace is bad or burned open..a new laminate card is in order..
If all is good, the gauge is bad..replace it as needed.
Yes, a DVOM is a digital Volt ohm meter..
A "Bucket" is the metal well the gauge sub~cluster sits inside of..remove that and you should have access to the laminate card and gauges..
A Laminate card is a (in GM cases, anyway) thin "filmy" flexible card with printed circuit traces on TOP and BOTTOM,(and sometimes in the middle) of the laminate to direct the signal flow to the proper devices..(gauges, lights, ect..)BE CAREFUL with it, It is fragile!! If you bend it too far, it may snap a trace..It should BOLT right onto the gauges.
The Black wire is NOT a ground, but rather a low signal From the Computer to the sender (DO not ground it, it can damage the Quad buffer / Driver on the computer) It works by sending a "Clock pulse" or a series of low pulses to the sender as opposed to a hard ground, that operate sender, in "sync with the computer damands..The other side should be hard data..(or even a fused 5 volts)..
Hope it helps out..