# zbMATH — the first resource for mathematics

A syzygy of a module is a trivial relation among its generators. So it is measuring the non-freeness in terms of relations. Its iterative construction provides a chain of syzygy modules. The study of syzygies became a powerful tool at least with D. Hilbert’s work published in [Math. Ann. XXXVI. 473–534 (1890; JFM 22.0133.01), and Math. Ann. XLII. 313–373 (1893; JFM 25.0173.01)]. He proved that the chain of syzygies of a finitely generated graded $$S$$-module ($$S = k[x_1,\dots,x_n]$$) becomes trivial after at most $$n$$ steps.
While Hilbert himself used the canonical grading of $$S$$ this is the most important feature for applications in algebraic geometry. So the higher syzygies are finitely generated graded $$S$$-modules for the defining ideal of a projective variety. That is, the fine structure of an algebraic variety might be determined by the graded structure of the higher syzygy modules.
Any graded ideal $$I \subset S$$ has the Hilbert function of a monomial ideal. The Hilbert function is determined by the minimal free resolution. Therefore it is of some interest to investigate minimal free resolutions of monomial ideals. This is the subject of Chapter 3 “Monomial Resolutions”. In general there is no guide to understand the minimal free resolutions of monomial ideals. The Chapter covers – among others – results about the Scarf complex, Lyubeznik’s resolution and in particular the description of resolutions via the Stanlay-Reisner correspondance by topological methods. In the final Chapter “Syzygies of Toric Ideals” there is the application of some of the previous ideas to the study of the free resolution of toric ideals by using multi-gradings.