Given an object, for instance an ideal IP, over a local ring (RP, P), this method returns the preimage of that object under the canonical map R > RP after clearing denominators of IP.
For matrices (hence most other objects as well), clearing denominators is performed columnwise.
In conjunction with pruneComplex, liftUp is used to implement many of the elementary operations over local rings such as syz.
Here is an example of computing the syzygy over a local ring using liftUp and pruneComplex:







This is the process for finding the syzygy of FM:




Now we prune the map h, which is the first map from the right:


Scale each row with the common denominator of the corresponding column in FM:


The syzygy of FM is:


This is NOT the same as lift. Not tested with quotients properly.
The object liftUp is a method function.