If this is indeed possible, then it must also be possible to create simple hydrocarbons (CO2 + H2O => CxHx + O2) using a similar process.
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The problem is getting enough co2, as it's not particularly concentrated in our atmosphere. So the main ways they go about it are big fans, which is tons of energy, capturing at the source (in smokestacks, etc) which requires complex transport and management, or growing plants and pyrolysing the biomass.
The fundamental theory behind it is quite simple, it's really more of a logistics problem.