During the fight, Alex's detonater was destroyed. Knowing Barkov was on the site, Alex chose to fight for something he believed in and was willing to sacrifice himself by rigging the furnace from the inside, allowing Farah to pursue Barkov.
Alex Keller was one of the main characters in 2024's Modern Warfare, and I was a little disappointed that he didn't return in the campaign for Modern Warfare II, though I'm hoping we see him again in whatever game launches in 2024.