I used to have a good relationship with an author, but now it's broken. However, my work is heavily based on the work of that author, and I'm not sure if citing it without informing them is OK here. Would there be any potential trouble from this? What if they explicitly tell me to not cite theirs? I just don't want to make the relationship worse, because they may not want to have any affiliation with me. But this is just my thought, I really don't know what they really think.
Yes, cite the other author's papers regarding any data or ideas on which your own work is based. It is crucial to do so; you MUST acknowledge the other author's contributions to your field if your own work builds upon those. As mentioned by others, unless you are writing a rebuttal, keep a neutral, dispassionate tone in your writing and refer to the other author's papers in neutral terms. Be specific in your citations. And yes, try to avoid having your paper reviewed by this author, at least for now.
The bigger problem was somewhat glossed over. Can you reach out and attempt to repair your relationship with the author with whom you've had the falling out? Ultimately that will be the best for your career, and maybe for your health. You can say, "I perceive we've had this 'time out' but I'd like to apologize if I've caused some problem. Can we meet for coffee and see if we can find some common ground?". Or something like that. It takes courage to go back to someone who has become hostile, but it could be that your load will be much lightened by such an effort. There's the possibility that some misunderstanding can be resolved. And even if you are rejected, you have at least tried. Life is short. I'm hoping for you!
If you feel that the work is relevant, you must cite it - not citing would be unethical.
Also, most people are happy if they are cited - regardless of their relationship to the person who cites them. Citing someone does not mean liking someone - even when criticizing someone's work, you have to cite.
And if you can think of no particular reason why they should not want to be cited by you, there probably is no such reason.
Yes, you should cite their work. The only reason I can think of for not wanting to cite it is if the other person believes that you have misinterpreted their work and have told you not to cite it.
Even then, if your work is based on theirs you need to cite them. I would be wary, of course of making disparaging comments about their work in a future publication. If you think you have done an improvement over what they have done, let your own work speak for itself. Of course, you shouldn't misrepresent what they have said, but that is true whether you have any personal relationship or not.
Who you cite is your business and nobody else's. You don't need to ask permission to cite(unless you're reproducing images or something*), and nobody can deny you the right to cite them, for whatever reason.
More importantly, if your work draws from someone's work, you must cite them, it's imperative. No personal equations can change that.
*In this case too, as pointed out by corey979, the permission is for copyright rather than citation, and the parties involved could include the publisher.
If you work is heavily based on their work, you have two choices:
- To publish your work, citing their work. If you publish your work without citing your main source, at best you will be asked by reviewers to add a reference to that. Worse outcomes may include your work being rejected as incompetent, you as an author getting on some sort of "black list" as dishonest, at worst, you may get accused of plagiarism and get banned from publishing. Seriously, is there a way for you to mention it somehow, like "well, I was going to cite that, but they asked me not to", so that it isn't perceived as you either unintentionally missing or intentionally hiding the relation of your work to their work?
- Not to publish your work, to avoid the consequences mentioned in p. 1. I can't imagine any possible legal or ethical reason for you to do so, but you may have your own personal reasons (like, not to alienate those people even more, etc). For all I know, citing a published scientific paper cannot be forbidden, so you're going to be doing them a personal favor, of a sort, at your own expense. Well, all kinds of inter-personal politics happen in scientific circles, and under some circumstances it may be worth it to have one less mortal enemy in exchange for one less published paper, but that's for you to decide.
I upvoted everyone else's answer.
I am not adding anything different, but want to tie it together concisely.
- Scholarly papers that draw on, reference, refine, or continue some prior scholarly work must cite specifically that prior work no matter who wrote it. That is your obligation and it has nothing to do with the source's preference. Indeed the source might be your own previous work. You cite it, you say what relevant concept or fact you draw from that work, you state clearly and dispassionately what its value is (from the perspective of your work) and you defend that evaluation, especially if it's negative or you say the prior work is incomplete. Be prepared to defend that in your scholarly paper. Good or bad or neither.
- Sometimes it is hard to decide whether to cite something or not. You do not want to slight another scholar by failing to cite relevant prior work, but you also should not meaninglessly cite work to support another author's vanity. (That can also include your own prior work.) You must also be prepared to say exactly what it is in the prior work that is salient from the perspective of your work and not inflate the number of citations just to make your work appear more scholarly or well-researched.
- Depending on the salience and relevance of the prior work, you may want to mention the surname of the author (or two authors or primary author followed by "et.al" if more than two) along with the reference number that is usually in square brackets. Such as:
Shannon and Hartley  derive the rate of information capacity of a channel in terms of its bandwidth and signal-to-noise ratio as ...
- But if it's a more minor or fleeting reference, it could be just:
In  the channel capacity is given as ...
- If some scholar has done incomplete work on a subject you are writing, tell us. If some scholar has done crappy work on the subject, tell us. If some scholar has done unoriginal work on the subject, tell us (and make a reference to the original work, it's embarrassing but when someone reinvents a wheel, we need the full history spelled out). No scholar has any right nor expectation to block a negative citation, but the critic must be able to defend the negative expectation. Do this dispassionately. Don't say anything personally negative like "How could an expert make such a silly error?" Just say what it is and how it is in error.
- Likewise, if there some prior work that is praiseworthy that is relevant to your paper, cite it. Spell out the significance and worthiness of the specific fact or concept cited. And do that dispassionately.
Personally, I think there is a glut of PhDs out there writing a shitload of crappy papers. Even in my own field (the IEEE transactions are sometimes just full of over-mathematicalized useless crap that is being published solely to boost someone's publication record) I think that. Those papers should go unnoticed and uncited unless there is a glaring error.
But there are also the few gems out there and if your paper draws on any, you must cite them.
tl;dr: Just cite their work.
I'm not sure if citing it without informing them is OK here.
It's ok; in fact, it's your obligation to cite their work, since your is based on theirs.
Would there be any potential trouble from this?
No, absolutely not. If you fail to cite them - then you'll have trouble.
What if they explicitly tell me to not cite theirs?
- They can't.
- They won't. Really. It won't even cross their mind.
I just don't want to make the relationship worse, because they may not want to have any affiliation with me.
Citation is not perceived as affiliation. Just make sure you don't discuss their work in a way which exacerbates the animosity between you; refer to it positively, i.e. instead of saying "X is deficient, but we can fix it with Y", say "We improve on X by doing Y").