What is the proper preposition to follow the noun hatred?
Do we have a hatred for Buddhism?
Do we have a hatred of Buddhism?
Do we have a hatred against Buddhism?
These are all just examples. It’s not my actual sentence that I wish to form.
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COCA BNC hatred of 763 181 hatred for 382 57 hatred against 40 6 hatred toward 73 0 hatred towards 31 12 hatred at 11 14
Obviously these figures will include some noise, where the object of the preposition is not actually the object of the hatred (stuff like "hatred of epic proportions", "I traded my hatred for love", and so on), but for recognizing the overall trend they are sufficient: all three of your options are options, but if you want to be on the safe side, you should pick of over against — and when writing for a British audience, you might even wish to pick it over for.
Edit: I have now taken a look at every single cite (yes, all 1570 of them), and here are the cleaned-up figures:
COCA BNC hatred of 738 173 hatred for 380 58 hatred against 41 6 hatred toward 74 0 hatred towards 31 12 hatred at 3 2
As you can see, the picture does not really change. One reason for that being that quite often a false positive for one preposition was a false negative for another ("hatred of the American people for Iranians", "hatred of millions toward U.S.", "hatred of Arab against Arab"), which is how some of the numbers have actually managed to go up.
Another interesting thing to observe was parallel constructions whose authors went out of their way to use of: "hatred of, and shame for, sin"; "outright hatred of and violence against women"; "I didn't have any great love for or hatred of Shakespeare". (There were no examples of this behavior for prepositions other than of, but then again their sample sizes were considerably smaller.)