Avoiding foreign key repetition on related tables in a relational DB
Looks like you can remove the Period FK entirely from the object. When assigned to a Region, it is already assigned to the Period assigned to the Region. Define a special Region -- a NULL Region, "not a Region", Pass-Thru Region, an all inclusive region, whatever -- to act as a placeholder for a Period when you mean "assigned only to Period rather than a Region."

This, I think, would involve the minimum amount of changes to your schema: drop the Period FK field from the Object table.