Yes, they can lie but they cannot fore an official document and pass it off as original, true and accurate. That is a state and federal crime.
When I was an investigator I never lied to the suspect. I would phrase my questions as: what would you say if I told you I had information that shows you did such and such? I never said I had such information but let them think that I might have it.
That is not lying but is a hypothetical presented to the suspect to see his/her reaction and to see if they make a slip in their response that shows they were involved in the crime.
You were smarter than the average bear!