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This is tricky because it depends what you're holding constant, i.e., what you mean by "column of air"

When air cools, it becomes denser because it compresses - this assumes no change in mass (or amount) of air. If you're talking about a discrete column of air that does not change in mass, then cooling it would cause to to compress and become denser because the volume is less (density = mass/volume). The pressure would remain the same because there is the same amount of air.

If you want to change the amount of air in the column as fluid dynamics would demand, then you'd be holding volume constant. So the column would cool, compress, and more air would move in to fill the spot left by its compression. So the column would increase in mass as well as density and pressure.

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Q: Is it true that the pressure of a column of air at sea level as it becomes cooler and subsequently became denser it would be heavier?
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