My players are sending some forged letters, allegedly from a recently murdered Duke.  They are supposed to be "Send these in the event of my murder" letters and the players' intent is to frame one of their enemies.

Since they have been burned in the past by their lack of consideration of divination magic, they devised a plan... while the real dead Duke's body is on display and out of their control, they are going to break into a crypt and steal the hand of one of the Duke's ancestors (another former Duke).  That way they hope that even if someone magically examines their forgeries, they will register as "written by the hand of the Duke".

I pulled up my spell books to see what the RAW have to say about it and... I cannot find any spells that relate directly to examining items for forgeries or for getting the backstory on non-magical items.  I feel like I must be missing something.  I have no problem just giving my players the "win" for this given their complicated and crazy plan, but it is going to bother me now because I can't believe, in a world of magic, that nobody bothered to come up with a spell to detect forgery.

I guess it depends on what you're willing to consider relevant to determining if the letter is a forgery, and why you want divination magic to be involved at all. I can see Divination, Commune, and Contact Other Plane all potentially being used to determine the validity of a letter. None of them specifically state - this determines a forgery, but I think determining if its been tampered with or was written by a specific person rather than just "a duke" could be within the spells as RAW.

The question is why does that matter? PCs do not have access to everything that monsters and NPCs operate with - for example I don't believe there are any spells or magic items that replicates the Magic Immunity of Rakshasas. If its a good way to drive the plot or open up another potential avenue of story, you could always have the party succeed with consequences - their plan works but a third party NPC is able to discover what they did and uses that information to coerce them into doing a task for this NPC or maybe they trigger a conflict and the party now has to deal with the consequences of that conflict such as there is an active war going on or the remaining enemies realize they've been manipulated and gang up against the PCs later. Its reasonable to assume that some NPCs might be able to use divination magic in a way the party can't. 

If the issue is a player will argue with you about RAW or otherwise try to rules lawyer with you to get what they want - that's a bigger issue than the GM stretched a spell as a plot device.  

The closest spell to what you're looking for is Legend Lore, which I believe has been featured on the front page of DNDBeyond. Though, that's only for legendary lore about a thing.

Forgery checks are usually opposed by an investigation check and there is probably someone who is trained to verify official documents for just this kind of occasion. However, beyond that, official letters are often sealed or marked with a Signet Ring, which imparts a unique signature that is harder to forge. This Duke might have a magical Signet Ring that leaves a specific magical mark that can be detected with magic and the players would either need to steal the ring to pull off their deception, or give a good explanation for why the seal would be missing (i.e. Damaged during travel, dispelled, etc).

If either the forgery is suspicious or the Signet Mark is missing, the players would be at a distinct disadvantage.

Basically, magical encryption is a two-party process.

Further, modern solutions also include specially printed paper that has embedded security features, like watermarks, colored/silver threads, etc. You can make the difficult of this challenge as high as you want through magical or mundane means. Maybe the duke gets his parchment from a specific forest with unique properties that are easily missed by those outside his inner circle.

There is only two spells I can think of that would help: speak with dead and commune.

Speak with dead would be pointless for your situation, as the caster would need access to the corpse.

Using commune depends on the question the caster poses. If they ask "was the letter written by the duke's hand", your party has taken the right precaution. If the question was "did the duke write the letter", they are out of luck. However, that is a 5th level spell, and I'm not sure if I would use that without a prior suspicion, and it is not sure if the being contacted by the spell actually knows if the duke did or not.

You could homebrew something off zone of truth. By RAW, it only applies to creatures, but I could see a wealthy kingdom’s house wizard or cleric developing a spell that would apply to documents. At least, they might if it’s been a problem in the past. 

I don't think it necessarily needs to be something that is seen through via Divination magic, though it is really cool that your players are attempting to consider that. Usually making something a forgery would require a skill check then the person who receives the letter and is suspicious would make another skill check to see if they can determine if its real or not, with the DC being equal to that of the players forgery skill check role. I usually use investigation or history to determine forgery depending on how familiar the person is with the alleged creator of the thing and what the thing is (forgery for me goes beyond just written stuff, including art and other things that can be forged).

I think that is why there isn't really a divination thing for it RAW because it us usually solved via a mundane process. Though that is really dumb. Perhaps an "Inspect Validity" Spell will be added to my list of things to make...

There is the spell divination that can do something like that, but its god related and not really the same thing. I suppose if you warship a god of trickery and secrets if something is forged they could have knowledge of it. I really like the idea of someone asking their god and getting the response "Yeah it is, and its a damn fine forgery too. I should seek out this person, see if they are interested in working for me..."

I think looking for specific spells that deal with this specific question of detecting a forged document is looking at this from too low in the weeds.  As the DM, designer, creator and manager of the world, you need to ask yourself if any of this is relevant.

From the Duke's perspective, would he have been concerned about people forging documents in his name?  Or perhaps it wasn't him -- maybe it was an ancestor of the House who had dealt with the problem.  The current Duke may not even be aware that his Signet Ring has anti-forgery enchantment properties, if that were the case.  It would depend on how long ago, and how big of a problem it turned out to be for the House at the time, and whether or not it was something known to the public.

Also consider who these letters are being sent to.  Even if the Duke's ancestor did have an anti-forgery enchantment imparted onto the Signet Ring, the person receiving it would also need to be aware of it in order to look for it in the first place.  Depending on the society they, this could potentially be a huge systemic need, if there is a crime syndicate in operation there, or even several.  Forgeries might be extremely common, and so determining the validity of documents might be paramount.  Or it simply isn't a concern.  Maybe it has only happened once or twice across several hundred years (at least, that anyone is aware of...).

With regards to the "hand of the Duke" thing...  that's actually rather clever of the party.  Though, again, it would depend on the context of the individuals the party is intending to deceive, and the nature of the society they occupy as to whether this would work or not.  If the person..  scrying, I guess...  the document is vague enough, that would absolutely work.  The document(s) were in the hands of "the Duke".  But that depends on just how specific the individual was in asking the question.  There is a big difference between "the Duke" and "a Duke".  There's a lot of grey area to navigate through there, and we here on the forums don't have the context to help you chart your course.

All of that said...  I think most of the responses so far are approaching this from too forensic of a perspective.  All the responses have revolved around testing the documents themselves.  But the thing is that Divination magic also includes prophetic visions, the likes of which an Oracle might deliver.  Perhaps the intended recipient lives in a culture that borrows from the Greeks, and there is an Oracle's Temple in the center of whatever town or city the recipient resides in.  There may be a well known prophecy, uttered long ago by one of the Oracles of the temple, that is just vague enough that for generations nobody has derived its meaning.  Something about "when false words pass the eyes of the raven...", and it just so happens than the individual who would be checking the documents has some affiliation with ravens, either through their profession or their past (a bird messenger who uses ravens exclusively, or their childhood nickname was Raven, or their title is The Raven, etc).  Or perhaps the prophecy is far more specific.  Something about the untimely death being followed by treasonous words, with other references that make it clear it was the Duke.  But again, this is more of a worldbuilding concern that you may not have the option of making use of now.