Found a mistake in advisor's paper - what should I do?
First, I'd double-check your work. Are you certain this is a mistake? Is there something you're missing? Do the best you can to understand what's going on.
Second, I would definitely raise this with your supervisor, but do so from a perspective of questioning and seeking clarification. Rather than appearing in judicial robes with the paper in hand and declaring it flawed by the laws of humankind and the universe, approach it as a question: There's a step in the paper that you don't understand. You think the step should say &quot;XYZ&quot;, but the paper says &quot;XZY&quot;. You're not sure if you're making a mistake or if there is an error in the paper.
There's nothing disrespectful about this humble approach, and you get a double benefit that if it turns out you are the one that is wrong then you have no face to save, and hopefully learn something along the way.
Mistakes happen, and they don't indicate the person making the mistake is dishonest or flawed in some fundamental way. Next steps to take after this will depend on how crucial the flaw is and what other work it might impact. If you take these steps and there are still some negative impacts on your working relationship, well... probably that person wasn't worth working with anyways, and some rift was going to develop sooner or later anyways. It would not be your fault.