Hilbert's law of syzygy

The Hilbert Syzygien theorem is a mathematical theorem of the invariant theory , which David Hilbert published in 1890 in his treatise "About the theory of algebraic forms" (Mathematische Annalen, Volume 36, 1900, pages 473-534). The reference is cited in the following with MA36. The syzygy theorem plays (in the various variations that it has since experienced) an important role in algebraic geometry , commutative algebra and computer algebra . It is the middle of the three famous sentences from Hilbert's time in Königsberg ( basic sentence , syzygian sentence and zero position sentence ).

introduction

Hilbert did not call any of his theorems Syzygiensatz. Depending on the focus of research, theorem III in MA36 or the sentence (which he does not further designate) on the last page in MA36 will be understood as the syzygy theorem. The last sentence is the only one in the treatise that contains the term syzygy . Theorem III, on the other hand, suits the modern understanding more. Hilbert's treatise in MA36 is 61 pages and consists of five sections. In the first, Hilbert's basic theorem is repeated (Theorem I) and in the second it is extended (Theorem II). The third contains the Syzygien theorem (in its “modern” version, Theorem III), the fourth deals with Hilbert functions (Theorem IV) and the fifth contains the Syzygien theorem in its invariant theoretical form (it is more special than Theorem V, the “only” the Finiteness of the full invariant system asserted).

text

Theorem III

MA36, page 492: “If a system of equations of the form (13) is presented [ ], where the terms are algebraic forms, the listing of the relations between the solutions of the same [syzygies] leads to a second system of equations of the same form; From this second derived system of equations, a third derived system of equations arises in the same way. The procedure started in this way always reaches an end when it is continued and the -th system of equations of that chain [n = number of variables of the polynomial ring] at the latest is one that no longer has any [non-trivial] solutions not to the original text. ${\displaystyle F_{t1}X_{1}+...+F_{tm}X_{m}=0,(t=1,...,m)}$ ${\displaystyle n}$

Syzygy theorem (invariant theory)

MA36, page 534: “The systems of irreducible syzygies of the first kind, second kind etc. form a chain of derived systems of equations. This chain of syzygies breaks off at the finite, and there are no syzygies of a higher kind than the th type when the number of invariants denotes the full system. " ${\displaystyle (m+1)}$ ${\displaystyle m}$

Explanations

Hilbert understands under an algebraic form a homogeneous polynomial in variables over a body (occasionally also a homogeneous polynomial with only integer coefficients), but also sums of products of the coefficients of the body, ´variables´ understood as parameters (eg determinants). ${\displaystyle n}$

An invariant is a completely homogeneous function of the coefficients of an underlying algebraic form, which remains unchanged compared to all linear transformations of the variables.

A syzygy (from the Greek sysygia = pair) is a tuple in a relation equation of the form , so that syzygy actually does not mean “pair”, but “pair mate” (namely only one m-tuple of two that are related ). Hilbert uses syzygies in Theorem III (the solutions to his systems of equations) without naming them that way. The term syzygy has many other meanings outside of mathematics. ${\displaystyle m}$ ${\displaystyle (X_{1},...,X_{m})}$ ${\displaystyle A_{1}X_{1}+...+A_{m}X_{m}=0}$

Modern formulations (examples)

• Klaus Altmann : "Every finally generated module has a projective resolution of the length ." ${\displaystyle C[x_{1},...,x_{n}]}$ ${\displaystyle n}$
• Uwe Nagel : "If a finitely generated module is over the polynomial ring , it has a finitely free resolution of the length " ${\displaystyle M}$ ${\displaystyle K[x_{o},...,x_{n}]}$ ${\displaystyle M}$ ${\displaystyle \leq \ n+1}$
• David Eisenbud : “Let be the polynomial ring in variables over a field . Any finitely generated graded S-modules has a finite free resolution of length at most . " ${\displaystyle S}$ ${\displaystyle r+1}$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle r+1}$
• “ ”, Which stands for “global dimension”, a paraphrase that is based on the term “projective dimension”, which in turn has something to do with projective resolutions. ${\displaystyle \mathrm {gldim} (k[X_{1},...,X_{n}])=n}$ ${\displaystyle \mathrm {gldim} }$

literature

• David Hilbert: About the theory of algebraic forms , Mathematische Annalen, 36, 1890, pp. 473-534, digitized
• Roger Wiegand: What is a Syzygy? , Notices of the AMS, 2006, No. 4, online
• David Eisenbud: The geometry of syzygies. A second course in commutative algebra and algebraic geometry . Springer-Verlag 2005,

Web links

• VI Danilov, Hilbert's syzygies theorem, Encyclopedia of Mathematics, Springer

Individual evidence

1. ^ Commutative Algebra, FU Berlin, 2006
2. ^ Algebraic Geometry, Kai Gehrs based on the lectures by Uwe Nagel, University of Paderborn, 2007
3. ^ Lectures on the Geometry of Syzygies, University of California