Syzygy (mathematics)

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 polynomial rings are Noetherian, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings.

Hilbert's syzygy theorem concern 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; Hilbert's syzygy theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in n indeterminates over a field, one eventually finds a zero module of relations, after at most n 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.

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.

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