Oh boy - Hope I haven't bitten off more than I can chew

I've got a couple of lessons on more complicated chords on GN:

Extended chords covers 9ths, 11ths, and 13ths
Altered States covers chord alterations, like raised fifths (as in D7+)

Your chord has:
F
C
F#/Gb
G#/Ab

Since a chord can only have one version of each letter name, we can't have both F and F#, or Gb and G#. These three notes must be either F, Gb, Ab... or E#, F#, G#. I'll go with the former, since it has fewer accidentals. Now we figure out how it looks against various roots:

F = 1-5-b9-b3
C = 1-4-b5-b6
Gb = 1-#4-7-b9
Ab = 1-3-6-b7

The chord sounds dominant, because you have a tritone between C and F#/Gb. So we're looking for a chord spelling that includes b7, or the harmonic equivalent #6. Only the Ab has this. And since 6 is considered 13 when a chord has a b7, you're right about the name - but probably wrong about it being G#. Unless there is a special reason to call this note G#, Ab is simpler.

That's because all chords are worked out from the major scale of the root. You'd have either:

Ab major = Ab-Bb-C-Db-Eb-F-G-Ab
or
G# major - G#-A#-B#-C#-D#-E#-Fx-G#

Given a choice between four flats or eight sharps (two of them on F), I'd go with Ab13.

By the way, I knew what the chord was at a glance - it's a pretty standard fingering for a 13th chord. In my lesson on extended chords, if you look at the very first 13th chord voicing I show, and move the note on the 1st string to the 6th string, you'll have the same fingering. But I thought it might be helpful to go through the logic of figuring it out from note names.

I am afraid I have to disagree with much that has been said here. IT IS ENTIRELY WRONG TO PLAY G MAJOR OVER Am - D7 -Gma7. To do that is to disregard the harmony completely is a lazy approach.

Much nonsense is spoken about modes. By and large most players I know disregard them.
The reason being is that when speaking of modes is gives a sense that "all not are equal" when they clearly are not.

Take Cma7 (In the key of C). You can doodle around on the mode (C major scale) over that, however if will sound much better if you are aware of the chord tones (R -3rd - 5th - 7th) because these will sound "stronger" within any phrase. Particularly the 3rd and 7th. The notes in-between the chord tones (2nd - 4th - 6th) provide more of a colour. The 2nd (or ninth) can sound particularly cool if you want to sound like G.Benson. The 6th is a very interesting tone but I would use if sparingly. The 4th however is pretty much always a "wrong" note to me (Except over a dominant chord).

So whereas it is important to be aware of the modes, it is far more important to be aware of the chord tones at any given point.

Because all notes in a scale ARE NOT EQUAL!

It is much more fun to experiment with non-diatonic notes (Like the blue note or flat 5th). That is where the fun begins and you can create some really interesting phrases.

Over a 2 - V - 1, you should be trying to hit the chord tones first off and NOT playing the major scale in the root position.
That is just wrong!

Andy, I'm not sure there's any disagreement.

I've been teaching guitar for almost 35 years. As a teacher you have to do three things:

1. Provide information that is correct
2. Sequence that information in a way that's actually useful to the student
3. Try not to overwhelm, or they'll miss important stuff.

You're absolutely correct that all tones are not created equal. In fact, even the strongest ones are not equal - over a 7th chord, the root will provide the most sense of finality, the fifth is solid, but less so than the root, and the 3rd and 7th beg for an addition to the phrase, because it won't sound complete to end on those. Each sound has its own character and role, not only over a chord, but over a progression, a section, and the piece as a whole.

But even that answer is incomplete. Musical lines are about MELODY, not chord tones. If you play only chord tones, you get arpeggio runs - pleasing, but miles from brilliant.

In this thread, I have been trying to present information in a logical sequence. If you step back several posts, you'll see that I addressed targeting chord tones as the fourth part of a general plan for learning to improvise solos. But there are additional steps beyond learning to use chord tones - once you can focus on those, you need to learn the role of the non-harmonic tones... you're no longer thinking in terms of scales or arpeggios, but of melody versus harmony in a broader sense. Moving beyond that, you want to learn compositional techniques, because improvising is really composing on the fly - and composing is about creating coherent melody, not about using this or that scale/mode/arpeggio.

Please view my comments in context. To you, playing a G major scale over a ii-V-I progression in G may be "a lazy approach". Understand that all instruction is geared to a specific audience - in this case, someone wanting to improve their soloing without any sense of how to solve that problem. If I lay out a dissertation on all the ways to do that, I might satisfy those who are looking for minute detail - but in the context of the audience - the background of the original question - that would be like throwing a dictionary at an aspiring writer searching for a word and saying "the answer is in there!"

The simple fact is: you need to be able to play in a scale over a progression BEFORE you can develop an awareness of the chords behind you. Likewise, you need to be able to play a scale over a "key of the moment" progression BEFORE you can adequately focus on the chord of the moment. We have to go from macro to micro in skill development.

Pedagogy requires a logical sequence of explanation, experience, and assimilation before you can tackle the next part. If you go into all the details at the outset, you risk losing the audience in the weeds instead of leading them to the flowers.

For example: when a student is a complete beginner, one of the concepts they need to learn is time signatures, and their relation to meter. It would be a disservice to the student to digress into a compound time signatures. What the student needs at the moment is "the top number is the number of beats in a measure" and exceptions are handled later on.

That's what I've tried to do in my responses here. You're not wrong. But neither is my advice.