Short answers, pulled from the story.
Trotskyism is a political ideology developed by Leon Trotsky to challenge traditional Marxist views on how socialism could emerge in backward economies. It originated in 1905 when Trotsky declared that the working class must lead the revolution against the Tsarist regime before the St. Petersburg Soviet of Workers' Deputies.
Leon Trotsky was expelled from the All-Union Communist Party (Bolsheviks) in November 1927 and exiled first to Alma-Ata in January 1928, then removed entirely from the USSR in February 1929. He died on the 21st of August 1940 after being attacked by Ramón Mercader in his Mexico City home on the 20th of August 1940.
Leon Trotsky founded the Fourth International in Paris in 1938 to replace the Third International because he believed it had become irretrievably lost to Stalinism. The new organization aimed to unite revolutionary forces globally under a program combining democratic demands with transitional measures leading toward proletarian seizure of power.
Trotskyist groups emerged as significant political forces across Latin America, Asia, Africa, and Europe throughout the twentieth century. Notable examples include Bolivia where the POR became a mass party in the late 1940s, Argentina where the Workers' Revolutionary Party formed in 1965, and Venezuela where Hugo Chávez declared himself a Trotskyist on the 10th of January 2007.
Internal divisions led to multiple splinter groups forming competing international committees throughout the mid-twentieth century. Michel Pablo's thesis caused a major rift resulting in the creation of the International Committee of the Fourth International (ICFI) in 1953, while other factions split over support for Algerian War independence movements in 1953.