It is interesting that a previous poster pointed out the fact that FDR trusted Stalin more than he trusted Hitler. I would in fact argue that it isn't clear who he trusted more. Hitler certainly thought that the U.S. and the U.K. would eventually come over to join him in his fight against those he considered less than human. He felt they had a shared heritage and there were certainly many in the U.S. that felt positively about the Nazis. (read about Lindberg for one) You might also look at the difference in the way we treated prisoners of war from the German army compared to how we treated Japanese American citizens if you want a clear picture of who we were really friendly towards.
But the U.S. and the Soviet Union had no chance of working together after the war as even during the war they clearly began to compete for the pieces of Europe and influence after the war.
The alliance between the U.S., Great Britain, and the Soviet Union was born as a result of Nazi aggression towards Great Britain (which the U.S. was allied to) as well as Nazi aggression towards the Soviet Union. The U.S. and Great Britain were willing to ally with Soviet Union against Nazi aggression because although the Soviet Union was a communist country all three were fighting Hitler's totalitarian aggression. Unfortunately, after Germany's surrender in 1944 the plans for peace discussed at the Yalta Conference were stalemated by Soviet aggression in Eastern Europe. However, both Roosevelt (U.S) and Churchill (Great Britain) believed they had paved the path, getting Stalin to agree to free elections in the Soviet conquered territories in Eastern Europe as well as Germany.
The final wartime summit at Potsdam (1945) would reveal the Soviet Union's true intentions. Roosevelt had died in April (1945) and Harry Truman was the Commander-in-Chief at the conference. It was clear that the Soviet Union had never intended to allow free elections in previous Nazi occupied Germany. Since the war ended without a general peace agreement and/or treaty, it paved the way for growing tensions between the Soviet Union and the United States; communism verses democracy...otherwise referred to as the Cold War.
The point that the U. S. and the U.S.S.R. were never really allies is absolutely true. Russia probably would not have bothered to get involved in war, but as the above post mentions, Hitler's having violated the German/Russian non-aggression pact brought the U.S.S.R. over to the Allied side because they were attacked by Germany. General George S. Patton always knew the Russians were, and never would be true allies. How could they be when they were Communist and were scrambling to increase their land mass? Patton called Eisenhower at the end of the war, asking if he could go in and more or less "finish off" Russia to prevent them from threatening the U.S.
Fundamentally, the reason for the forged alliance in World War II had more to do with convenience than anything else. Both nations still distrusted one another. Yet, FDR understood that as bad as Stalin was, Hitler was far worse. The Nazi threat scared America more than the Communist one. For his part, Stalin could have cared less. When he signed the non- aggression pact with Hitler, Stalin was fairly happy with a Nazi Germany causing fear to the West and running around annexing countries to do it. Yet, when Hitler violated his own accord with the Soviets, spelling his own doom, Stalin immediately forged the alliance with the Allied forces. The fear and mistrust of both sides became clear when the war was ending and settling the terms for peace proved that there would be no continued reconciliation. Stalin wanted all of the nations Hitler had in order to create a buffer between he and the West. Truman had used the atomic bomb against Japan, and the escalation of the arms race had begun. Both sides were left as the dominant forces in the world, and with world control hanging in the balance, the adversarial relationship resumed and intensified with greater magnitude than before the war.
You have to remember that the Soviet Union and the United States were never on friendly terms, never trusted each other. World War II found them both on the same side of the war against Hitler, and that was literally the only thing they had in common.
It's also important to point out that the "alliance" was not much of an alliance at all. We shipped aid to the USSR as part of the Lend-Lease Act, but not very much compared to Britain, and the only real effect the aid had was to make the Red Army more mobile (lots of trucks). This angered Stalin, as did the fact that we waited until 1944 to reinvade Europe and open a second front against Germany. He believed (and there might be some merit to the argument) that the other allies were waiting intentionally to bleed the USSR further.
So the enmity was always there. We set it aside partially in the name of defeating Germany, but we were never allies in the true sense of the word, so turning to enemies after the war was not a giant stretch from the status quo.
Basically, this is because of ideological mistrust between the two countries.
On the part of the Soviets, their communist ideology stated that capitalist countries would surely try to destroy them. They believed this even more because of the fact that the US and England, among others, had been very opposed to the creation of the Soviet Union.
On the part of the US, they believed that the communists wanted to take over the world. They thought that the Soviets' desire to control Eastern Europe was simply the first step in a communist effort to spread their ideology over the whole world.
Because the two sides distrusted each other's ideologies, they felt the need to defend against one another. Any move one side took to defend itself would be seen as aggressive by the other side. Once the common enemy (the Nazis) had been defeated, there was nothing to prevent this suspicion from causing enmity.
Besides the general aspirations of both the USSR and the USA to establish their position in the world as the dominant super power, the main cause of conflict between the two countries was the communism. USSR openly expressed its intention to convert as many countries as possible to the political ideology of communism, and did not hesitate to use force and coercion to achieve these objectives. On the other hand, USA did not want communism. They feared the opposition from a united block of communist countries led by USSR. They were also apprehensive of the Communist ideology spreading to the USA and upsetting its economic and political systems.