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.
Facts
StatementIf M is a finitely generated module over a polynomial ring in n indeterminates over a field k, then the nth syzygy module of M is always a free module. 1 Classification
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Source Hilbert's syzygy theorem (Wikipedia)
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1. Hilbert's syzygy theorem (Wikipedia)
Statement
Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring k[x₁,…,xₙ] in n indeterminates over a field k, then the nth syzygy module of M is always a free module.
History
The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890).
In Group: Ring Theory and Commutative Algebra, Lead section
is the starting point of the use of homological methods in commutative algebra and algebraic geometry.
In Branch: Algebraic Geometry, Lead section
in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem
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