My car turns over starts then cuts off - 1990 Oldsmobile Cutlass Supreme Conv

The fuel pump operates off two items upon starting the car. When you first turn the key to the ON position you can hear the fuel pump faintly running inside the fuel tank. If you can not hear it it seems it is working for you stated the engine starts then cuts off. That tells me the engines oil pressure switch is working the fuel pump. On the oil pressure switch this switch has a third wire which is directed to the fuel pump. When the engine reaches full oil pressure the computer shuts off the signal from the oil pressure switch and then the computer turns on the fuel pump relay which stays on until you shut the engine off. I do beleive the fuel pump relay is what you need for it is not working no longer. This will have many relays with the same part number on it. You can easily swap any relay that looks the same with the same numbers on it and try it once to see if the engine stays running.