Alechine got a Death Sentence by Bolsheviks,  by mistake...

He was been arrested by Bolsheviks,  at the Odessa. And the next day he must be shot, like a spy...  

And Miracle Happened! 

Trotsky must sign the list of death sentences, but when he read it, he recognized the name, that  looks alike a name of famous chess player.

And Trotsky, Number 2  at the  Soviet Republic, stopped the death!

"Alekhine's personal life was restless and unhappy. He came from an aristocratic family in Russia that was dispossessed after the 1917 revolution. He was arrested and sentenced to death. But as the story goes, one of the five judges - a chess player - who had to sign the sentence, refused his signature. Alekhine survived the immediate aftermath as an interpreter and married a Swiss woman who had good connections with the Soviet government and knew Lenin personally. There is also a legend (discounted by Müller) that he played a game for his life with Trotsky, who was an accomplished player! He left Russia in 1921 for Berlin and then finally settled in Paris, where he received a doctorate in law at the Sorbonne university (the subject of his thesis: the prison system in China!). " From Hans Mullers book on Alekhine

No, i am not sure about playing the chess life game against Trotsky, I has no info about it. 

But, I am sure, that Trotsky at least know the famous name of the Alechine, who was been the rising star at that time! 

I left almost all my chess library at the Russia...

But what prove you ask? That the Trotsky knew Alechine?

 I think, that Trotsky was been intelligent! And he knew all famous Russians: poets, opera, ballet, dancers, singers, painters, musicians,.. 

Prove that alekhine was up for execution... prove that trotsky knew about it... prove that trotsky is the one that overrode the order...

I don't see why anyone would fabricate a story, but it does happen (even if you didn't do so, someone else might have).