Every finitely generated module with a grading over the polynomial ring has a finite graded free resolution of length at most . The Koszul resolution of has length , so for . In a minimal graded free resolution, this Tor functor is ; the graded Nakayama lemma forces for .
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Hilbert's Syzygy Theorem is a fundamental result in the field of commutative algebra and algebraic geometry that concerns the relationships among generators of modules over polynomial rings. It provides a deeper insight into the structuring of polynomial ideals and their resolutions. In simple terms, the theorem addresses the projective resolutions of finitely generated modules over a polynomial ring.