This one is free:
A good definition of life is that it persists. For example, toss a cube of sugar in a bowl of water. Toss a fish in a bowl of water. Which persists longer?
This has to do with entropy. Life collects low entropy energy, uses that energy to maintain a lower level of entropy within itself, and expels high entropy energy (or matter — think feces).
So, a planet full of life will do the same thing: Expel more high entropy energy than a planet not full of life.
Detect that, and you have detected life. How do you do that? The problem is left as an exercise for the student. (IOW, I have no freaking idea.)
Incidentally, sunlight is surprisingly (Thank You, God — seriously) low entropy energy. Hence it is the fountain for life on Earth.
Maybe some form of spectrum analysis could be used to detect that the radiation emitted by an exoplanet has higher entropy vis-a-vis its star than would result from the presence of non-life processes?
Seems to me that we should be able to detect signs of biology from the atmospheres, even of exoplanets. High nitrogen and CO2 traces in spectra would appear a tipoff.
Where there is life, there is poop!
Or, as Solomon stated a bit more elegantly:
An empty barn is a clean barn,
But great is the strength of the Ox.”