As we have studied today, after Lenin´s death there were some transition years in which Stalin and Trotsky were the main leaders of two factions in fight for power. This is what Lenin wrote about them in 1922 about Stalin and Trotsky:
Comrade Stalin, having become Secretary-General, has unlimited authority concentrated in his hands, and I am not sure whether he will always be capable of using that authority with sufficient caution. Comrade Trotsky, on the other hand (...) is distinguished not only by outstanding ability. He is personally perhaps the most capable man in the present Central Committee., but he has displayed excessive self-assurance and shown excessive preoccupation with the purely administrative side of the work.
Stalin is too rude and this defect, although quite tolerable in our midst and in dealing among us Communists, becomes intolerable in a Secretary-General. That is why I suggest the comrades think about a way of removing Stalin from that post and appointing another man in his stead who in all other respects differs from Comrade Stalin in having only one advantage, namely, that of being more tolerant, more loyal, more polite, and more considerate to the comrades, less capricious, etc. This circumstance may appear to be a negligible detail. But I think that from the standpoint of safeguards against a split, and from the standpoint of what I wrote above about the relationship between Stalin and Trotsky, it is not a detail, or it is a detail which can assume decisive importance.
Lenin´s testament (1922)
In these transition years the President of the USSR was Alexei Rykov, who supported Stalin against Trotsky, Zinoviev and Kamenev. Rykov, Kamenev and Zinoviev were executed in 1938 during the Great Purge. Trotsky was assassinated by Stalin´s order in Mexico City in 1940. The killer was Ramón Mercader, a Spanish communist who worked as a Soviet agent. Mercader used an ice-ax to kill Trotsky and the story of this murder is very curious. Mercader visited Trotsky several times and gained his confidence. On the 20th August 1940 Mercader brought an ice-ax hidden in his raincoat and waited for the right moment to kill Trotsky. Here you have a short video which reproduces the murder:
And this is a picture of the famous ice-ax:
On the link above you can find more information about the house where the murder was committed. Now it´s a museum about Trotsky.
And here you have another link about what happened to the ice-ax: