What do you do with the resulting algae? What's a good way to sequester that captured carbon? I guess on industrial scales we should really just pump an algae sludge right back into the holes we pumped them out of.
You can sink it in the ocean - beneath around 500m it just gets compressed and sinks to the bottom. So that would be "permanent" sequestration. The other option is to make it into "biochar" which is basically anaerobically combusting it to leave carbon residues, that you can then bury in the ground. This is scientifically accepted as also being permanently sequestered. This has the benefit of acting as a soil amendment also, increasing water retention and facilitating microorganism development.