Hilbert's Syzygy Theorem

Hilbert's Syzygy Theorem states that a finitely generated graded module over a polynomial ring in n variables over a field admits a free resolution of length at most n, meaning the process of writing the module as a quotient of a free module, then finding a free module mapping onto the relations among those generators, its syzygies, and repeating, terminates after at most n steps. Named for David Hilbert, it is a foundational result of commutative algebra guaranteeing that such resolutions cannot grow indefinitely long.