In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings. Hilbert's syzygy theorem concerns the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module. As the relations form a module, one may consider the relations between the relations; the theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in indeterminates over a field, one eventually finds a zero module of relations, after at most steps. Hilbert's syzygy theorem is now considered to be an early result of homological algebra. It is the starting point of the use of homological methods in commutative algebra and algebraic geometry. History The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890). The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem (Theorem III), which is used in part IV to discuss the Hilbert polynomial. The last part, part V, proves finite generation of certain rings of invariants. Incidentally part III also contains a special case of the Hilbert–Burch theorem. Syzygies (relations) Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring. Given a generating set of a module over a ring , a relation or first syzygy between the generators is a -tuple of elements of such that Let be a free module with basis The -tuple may be identified with the element and the relations form the kernel of the linear map defined by In other words, one has an exact sequence This first syzygy module depends on the choice of a generating set, but, if is the module which is obtained with another generating set, there exist two free modules and such that where denote the direct sum of modules. The second syzygy module is the