Both the pickup and module can cause that problem.
The pickup's actual name is the Hall Effect sensor. That is what sends inputs to the the module, which in turn send inputs to the computer as to what injector to fire next and for how long. Usually when the pickup fails, it tends to affect the module so the module is recommended to be replaced as well.
When the module fails, the results are the same as when the pickup fails, it's sending incomplete information and that causes the driveability problem. When it's just the module that failed, the pickup isn't affected.
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