1. A method for managing dynamic address translation (DAT) tables in a virtual storage environment, the method comprising: a computer determining a fault has occurred associated with an address of virtual storage that is shared by a base space and a plurality of subspaces of a subspace group, wherein the base space and each subspace of the subspace group include dynamic address translation (DAT) tables that translate addresses of the virtual storage that is shared, to corresponding locations of data in physical memory; responsive to determining that the fault has occurred, the computer updating information of the DAT tables of the base space that correspond to the fault and reflect a resolution of the fault; the computer determining whether a subspace of the subspace group is associated with application activity that has generated the fault; and responsive to determining that a subspace of the subspace group is associated with the application activity that has generated the fault, the computer copying the information of the DAT tables of the base space that correspond to the address of virtual storage associated with the fault, to the DAT tables of the subspace determined to be associated with the application activity that has generated the fault.