This entire first chapter is a cutscene, there is no gameplay involved.
Do it yourself
Once you get out, a cutscene will play showing Davian split up from the rest of the team (with Ivy if it is on co-op.). Follow the hallway downwards until you come to a room with a Purist and several Drones in it. Kill the Purist and tap out the Drones.
In the next big room begin this battle with a flourish by tossing a grenade down into the collection of Drones, which should kill most of them, if not, you can take care of them afterwards. Kill the Purist and deal with any remaining Drones.
In this room there is a single Purist Valiant along with several standard Purists. There is also a duo of Banshees in the corner, take care of them afterwards. Once you kill the Valiant he should drop a Bio Sword, swap out your EMP Gloves for it if you wish, take heed, the Bio Sword is a deadly weapon.
Dead Man's Chest
After you have found the Commander, escort her through the ship back into the room with the Valiant in it, Purist reinforcements will enter the room, kill them, and continue down into another room.
In this room there will be several more Purists, including a collection of Purist Hunters, take them out.
Upon entering the main grav-lift chamber, the final chapter of the level will commense.
Warning, violent updrafts may cause intestinal disfunction...
As soon as the chapter begins (resist the urge to enter the Interceptor), stay back and get a Energy Rifle from a Banshee's corpse. Use it to take out the enemies from a distance, as soon as it runs low on ammo, get out your Bio Sword and go berserker rage on them. When all enemies lie dead, the level ends.