In the lysogenic cycle, does the provirus split from the cell's main genetic material for replication?
When the virus is integrated into the hosts genome (and becomes a provirus) it is replicated with the cell genome, since it is now part of it. When the provirus gets activated (this happens by changes in the host's environmental conditions or health), it will get transcribed.
This is followed by the translation of the viral proteins which then leads to a cell which exclusively produces viruses until its resources are exhausted and the cell gets destroyed.