Everyone knows is ok for cops to lie trying to get confessions.
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.