Hilbert Syzygies Theorem

In this video, we look at Hilbert's syzygies theorem, perhaps the first major result in homological algebra. Basically, it shows how modules over a polynomial ring can be studied inductively by comparing them to free modules. In the graded setting, this gives a very nice theory of Hilbert series for such modules, since these Hilbert series are tightly constrained to be rational functions. In particular, one can talk about the dimension of such modules.