It's the writing itself which must be false, not the document. This element is intended to cover situations involving false signatures or improperly filling in blanks on a form. The document or writing has to at least look genuine enough to fool people. Forgeries of completely ridiculous things, like titles to swamp land in Florida, deeds to land on the moon, or a license to kill, do not qualify as having the apparent ability to fool most people. The document or writing has to have some legal significance affecting at least another person's right to something, usually some property right, broadly defined to include intellectual property like the form of a signature.
Legal significance is distinguished from social significance.
Similarly, if you found an old book by a famous author and wrote their signature inside the front cover to make it look like an autographed edition, you would not be guilty of forgery because this has social, and not legal significance. But, if you were to sell the autographed book at an auction, you would be guilty of false pretenses. The specific state of mind for forgery does not require an intent to steal, only an intent to fool people. The person must have intended that other people regard something false as genuine. A forgery is complete upon having created such a document with this requisite intent.