The locking nut on my guitar is too high, and I want it lowered. My guitar came with a free setup within one year, which I haven't used yet. I was wondering if I brought it to the store, would they lower the nut as part of the setup?
Not likely; lowering the nut is by no means part of any usual setup.
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But doesn't the nut height affect the action as well? So when they adjust the action, wouldn't they need to lower it? Right now I have the action as low as I can get it without fret buzz, the only thing that's limiting the action is the nut height.

Anyways, how much do you think it would cost to have the nut lowered?
Nut work is usually separate from set up. There's not much you can do to lower a Floyd nut anyway unless it's already been shimmed. You could take the shim out to lower it but if a nut's been shimmed it's usually because it needed to be, you don't shim a nut for no reason. Floyd nuts generally don't sit that lower compared to the frets. If you do need to lower a Floyd nut that means filing down the nut shelf and that is tricky to get right and it's not reversable.

Your nut probably isn't forcing high action anyway, unless you have freakishly low fretwire though if you did then I don't know why fret buzz would be such a problem. Setting action is pretty much all down to the bridge, assuming the neck relief is set correctly. The nut should be high enough that the low E string clears the first four frets easily with no buzz, even on a neck that doesn't have enough relief. So what I'd suggest is you take a look at the neck relief and the bridge height. Or it might just be that you're trying to get lower action than is physically possible. Remember that bigger frets and thicker strings restrict how low your action can be, the bigger everything is the more of a gap you need for the string to vibrate properly. This is a common problem with the sorts of guitars that have Floyds. Also remember that Floyds never really have quite as low action as a hard tail version of the same guitar would have and most Floyd units are made to have a radius of 10" which is a little more curved than most modern guitars too so it can often seem that the strings are further off the fretboard than they should be when really it's just that the radius doesn't match. Or you might simply have the wrong idea about what low action is. Around 5/64"-3/32" (about 1.9mm-2.3mm) is low action depending on the string size and fretwire type. Some really finely crafted hardtail guitars can go as low as 1/16" (1.5mm) but that's rare. Unless your action measures more than 2.5mm I wouldn't worry about it.
probably wouldn't cost more than $30 however why do you really need it lower? your action can only go so low and if you have it set so it's just above the fret buzz point how is lowering the nut going to improve that?
I would be surprised if they did the nut on a routine setup. If its only the nut, and you like the rest of the setup, do it yourself. Or do it yourself, carefully so its not obvious, then get them to do the free setup. Be very cautious, its easy to take material off, but hard to put it back on. Instead of taking wood off, try filing the grooves down. Why? Cuz you can replace a nut for only a few dollars, but a messed up neck is painful.
Na it's not part of a normal setup. I'd ask for him to do it though, wouldn't cost much probably.
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The distance from the first fret to the strings is almost 1 mm, which is unnecessary and makes the action at the first 5 or so frets high. Since the string vibrates in an elliptical fashion, having that much clearance is pointless, and also makes chords a bit harder to fret, which is unnecessary effort. I'm sure it also affects the action for the rest of the guitar as well, so I want to lower the nut.

Anyways thanks for the input, I'll call the store tomorrow to see how much it would cost. I've always wondered what a real professional setup feels like compared to what I've been doing.
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I've always wondered what a real professional setup feels like compared to what I've been doing.

That's a really good question, and I'd love to hear about it when you have your answer.