Wikipedia suggests that Stalin had Vavilov imprisoned just because Vavilov criticized Lysenko's theories, and Stalin favored Lysenko. Why didn't Stalin just ignore Vavilov? Did Vavilov embarrass Stalin?
The English Wikipedia article greatly simplified the issue. It is true that the opposition to the anti-Mendelian theories brought forth by Trofim Lysenko could have gotten him in trouble, but the actual accusations were related not to this academic conflict but rather to the allegations of anti-Soviet activities that surfaced in early 1930s, when Vavilov was still working with Lysenko, seeing in him quite promising scientist.
Documents from the era do give much support for the assumption that Stalin himself was involved in Vavilov's imprisonment to any serious extent (although he most likely knew about the affair or facilitated it). The decision for arrest has been issued by NKVD 1st Lt. Vladimir Ruzin, sanctioned by Deputy Prosecutor of the USSR, Mikhail Pankratyev and approved by National Commissar of Internal Affairs, Lavrentiy Beria. In the document, Ruzin claimed that after analysis of the materials concerning Vavilov that have been collected by NKVD, he found Nikolai Vavilov to likely implicated in the anti-govrnment activity including, but limited to foreign espionage and sabotage and thus recommends arrest of the scientist.
The allegations against Vavilov were not limited to his immediate activities. As usual for the similar cases, they also included various informations that could have been treated as aggravating circumstances or at least construed as the proof of disloyalty to the incumbent leaders of the state, communist government as such or even to the country itself. For example, Ruzin notes that Vavilov's father was a merchant who left the country during the Revolution and settled in Bulgaria, while his son was meeting him on regular basis. In addition, Vavilov was mentioned to be critical of the communist government 'since the first days of its establishment'. Document quotes the interview Vavilov gave the Parisian newspaper Paris-Midi in February 1933, where he said 'I do not serve government, I serve my country. I was an academic in Tsarist Russia and continue to live in my country that is, as it always had been, Russia' (what could have been easily presented as the proof that Vavilov does not support the government and might be a monarchist or nationalist). Furthermore, he was found to be one of the leaders of ant-Soviet, counter-revolutionary organization 'Peasant Labour Party' responsible for the sabotage of the wheat cultivation programs in new agricultural areas of USSR.
Let's pause for a moment, because this last accusation is very telling, chiefly because the 'Peasant Labour Party' (rus. Трудовая крестьянская партия), although quite commonly referenced in the official documents of by the National Commissariat of Internal Affairs and National Commissariat of Justice between late 1920s and early 1940s, simply... never existed. It was a complete fabrication of the security services meant to facilitate repressions of the people critical of the policies regarding agrarian issues, especially those holding higher positions in academic and administrative institutions. Such cases, much like the Shakhty Trial of 1928 or Industrial Party Trial of 1930 were generally meant to eliminate people whose loyalty to the leaders of the Soviet Union (chiefly Stalin and his closest circle) was considered dubious. In a way, these fabricated trials were a prelude to the Great Purge of 1937.
Now, Ruzin's document mentions Viktor Pisaryev who, arrested for being a member of PLP, admitted that the members of the Party infiltrated the All-Russian Agricultural Institute in 1927-1928 to sabotage the efforts of the Soviet government and Communist Party regarding the development of agriculture in the country and named Vavilov as one of the ringleaders of the clandestine organization. Likewise, another biologist, Nikolai Kuleshov also admitted to be part of the latter. Another document implicating Vavilov in the anti-government conspiracy is the declaration of Nikolai Tulaykov, biologist and member of the National Commissariat of Agriculture who was arrested in 1937 during the Great Purge and died in prison or was executed few months later. During the interrogation, Tulaykov was said to mention that several Soviet academics, including Vavilov, conspired to 'mobilize the foreign public opinion against the ideas of collectivization' in an attempt to compromise the agricultural output of the USSR.
Allegations against Vavilov were not limited to the domestic issues. Alexandr Muralyov, President of the All-Union Academy of Agricultural Sciences and National Commissar of Agriculture in the years 1929-1934, also arrested in 1937 and executed in 1938, allegedly admitted during the interrogation that Vavilov was 'ostentatiously associated with the Fascist-German genetic society and actively sabotaged the development of modern Soviet science both in the country and abroad'. Nikolai Abdulov, Vavilov's coworker even admitted to being paid for sending the classified information abroad (in this case, to Poland) during his arrest in 1932. With allegations like these, it was easy to present Vavilov, who has been a member of many foreign scientific organizations in United Kingdom, Germany, USA, Mexico, Czechoslovakia, India, Argentina and Italy as a dangerous foreign spy and saboteur. It didn't help that Vavilov was opposing Lysenko who in turn, was painting genetics as a pseudo-science serving the political goals of German fascists. This could have given an impression that the former indeed supports the racial ideas rampant in Germany, even though his critique of Lysenko's work had completely different character. It also needs to be added that various high-ranking foreign nationals with whom Vavilov contacted in the past were presented almost uniformly as members of foreign intelligence services.
It is worth noting that as presented in an order for Vavilov's arrest, many people involved in the Vavilov's case has been earlier arrested between 1927 and 1932, but given relatively similar accusations of espionage, sabotage and anti-Soviet activity, all of them were sentenced to 3-8 years imprisonment or heavy labour, with some of them having their sentence changed to house arrest. In 1937, when the repressions became far more extensive and intensive, the same people were charged with the same crimes again and this time they were usually sentenced to death and promptly executed. This strongly suggests that Vavilov was not arrested and sentenced for opposing Lysenko, but rather Lysenko used his influence to have Vavilov arrested for his alleged sabotage of the Soviet agricultural infrastructure and academic endeavours, as well as for the support of the anti-government 'right-wing' conspiracy (the term 'right wing' was generally associated with Trotsky, Bukharin, Zinovyev or Kamyenyev). Of note is also a letter of Issac Prezent, Lysenko's supporter, who wrote to Vyacheslav Molotov, President of the Council of People's Commissars, denouncing Vavilov as a troublemaker whose presence at the VII International Congress of Genetics planned to be held in Moscow in 1937 might 'give an impression that there is no freedom of academic pursuit in USSR'. It is worth noting that Prezent, along with Alexandr Nurinov openly called for punishment of certain scientists as 'enemies of the people' in an article published in Socialist Agriculture, although he named Koltsov and Serebrovsky as the most 'dangerous' ones. This shows that Lysenko and his supporters saw Vavilov as a potential enemy and tried to denounce him to get him out of their way.
Thus, although the tensions between Lysenko and Vavilov certainly played a role in the arrest and subsequent death of the latter, these were definitely not the indirect factors. Having been implicated in various anti-government activities in the early 1930s, Vavilov was definitely not safe during the Great Terror and although he survived the initial wave of arrests and executions, his history was more than enough to warrant the arrest. It is also possible that Lysenko and his supporters held Vavilov responsible for the VII International Congress of Genetics to be eventually held in Edinburgh in 1939, making attendance impossible for Soviet participants, including Vavilov himself who was selected to preside over the congress, and the arrest was a sort of personal revenge. This angle is all the more possible that Lysenko was in Stalin's good graces, as was Molotov, making such a scenario all the more possible. And once the proper authorities were persuaded to follow with Vavilov's arrest, his previous run-ins with the Soviet government, even though fabricated, could have been easily used against him.
Krementsov, N., Stalinist Science, Princeton University Press, Princeton 1997.
Soyfer V. N., Власть и наука. Разгром коммунистами генетики в СССР [Government and Science. Destruction of genetics by communists in USSR], CheRo, Moscow 2002.