I don't understand the MV = PY I'm not a native speaker so please excuse my English.

When they draw the result that the central bank's monetary supply determine the inflation rate, didn't they suppose many variables sticky?

With money supply matching up with the demand, they start their train of logic as follows. MV = PY, here the Y is fixed since it's exogenous, and V is fixed for the convenience of computation, so M controls the P... Okay I get that part, but what I wonder is that, when you fix the V, doesn't that make the k also fixed, which means kY, the money demand function is eventually fixed.

Then, how can they manipulate the M when they assumed M equals to the fixed demand of money?
Money demand is fixed because it depends on the real economy....that's an assumption. 
So when M changes only P can change. That's the theory. It's not really a theory. Just an accounting equality, since V is never defined exogenously. It's bad economics. :/