The Brothers Karamazov is about guilt, faith and what is done with an idea once it has been spoken: the book follows a father who did not raise his sons, and what each of them, and a servant, made of that inheritance.
The religious thread is that of Alyosha, the novice, and of Elder Zosima, who teaches that each person is guilty before everyone and for everything on earth, that faith is born of active love, and that no one can be the judge of anyone else. Against him stands Ivan, who accepts God but not the world God created, and returns his ticket of admission to a harmony paid for with the tears of a child. The book does not answer Ivan with an argument: it answers with a life, and the question stays open.
The criminal thread is that of Mitya, accused of killing his father without having done it. There the book shows what happens with a single phrase, the one that says that without immortality there is no virtue and everything is permitted: Ivan formulates it as a problem, and the servant Smerdyakov receives it as a license. The court that condemns Mitya is impeccable in its forms and wrong, because it knows only facts, and the truth lies with two men it does not hear: Ivan, taken for a madman, and Smerdyakov, dead.
The synthesis lies in the boys of the Epilogue, who decide to keep a good day as an anchor. The book ends without proving anything: it bets on memory, on active love and on resurrection as reunion.