Wasn't it Ajax who killed Achilles by firing an arrow from the Towers of Illium and praying that it would hit the "unprotected" spot on his heal?Not in Homer"s "Illiad," but Mythology?
I am not certain which version of Greek Mythology you are using, but I am not able to corroborate your findings. From what I have understood, it was Paris who shoots the arrow, guided by Apollo, at Achilles. This makes sense because Paris was an archer, diverging from the honor code of the Greek warrior in that he preferred to attack from a distance. At the same time, I would say that Ajax and Achilles were on the same side in the Trojan War. Achilles, Odysseus, and Ajax are three of the primary reasons why the Greeks end up triumphing in the war. Achilles and Ajax held a great deal of respect for one another, as Achilles respected his strength and the fact that he was one of the only warriors who was unsupported by the Greek Gods. At the same time, Ajax is the one who approaches Achilles during his feud with Agamemnon in the hopes of reenlisting his contributions. When Patrocles dies, Ajax is the one who brings the body back to Achilles, fighting Hector for it. When Achilles dies, Ajax and Odysseus retrieve the body back from the battlefield and help to conduct its last rites. It seems unlikely that he would be the one to kill Achilles after doing all of these in support of him.