Crime will be easy to eradicate, racism and homophobia, not so much. I've heard people say that once the workers gain class-consciousness they will realise that the problems are caused by capitalism, as opposed to blacks, gays and Jews, but there are of course still a great number of people who hate just to hate.
I would say a lot of hate crime is to do with Capitalism, when we reach true Anarchism/Communism I don't think there will be many who hate just to hate. We have to ask if they hate just to hate why they do such things and where have these beliefs risen from, I think you will find that even these people are a product of their society.
"Direct Action is a notion of such clarity, of such self-evident transparency, that merely to speak the words defines and explains them. It means that the working class, in constant rebellion against the existing state of affairs, expects nothing from outside people, powers or forces, but rather creates its own conditions of struggle and looks to itself for its means of action. It means that, against the existing society which recognises only the citizen, rises the producer. And that that producer, having grasped that any social grouping models itself upon its system of production, intends to attack directly the capitalist mode of production in order to transform it, by eliminating the employer and thereby achieving sovereignty in the workshop – the essential condition for the enjoyment of real freedom.” Emile Pouget