Hilbert's syzygy theorem

The Hilbert Syzygiensatz is a mathematical theorem of invariant theory, which David Hilbert published " the theory of algebraic forms About " ( Mathematische Annalen, Vol 36, 1900, pages 473-534 ), in his treatise in 1890. The references cited in the following with MA36. The Syzygiensatz plays ( in different variations, which he now learned ) play an important role in algebraic geometry, commutative algebra and computer algebra. He is the middle one of the three famous sentences from Hilbert Konigsberg time ( basis set, Syzygiensatz and Nullstellensatz ).

Introduction

Hilbert does not name his sentences Syzygiensatz. Depending on the research focus will be the theorem III in MA36 or phrase (which he no longer referred to ) understand on the last page in MA36 as the Syzygiensatz. The last sentence is the only one in the treatise containing the word syzygy. The theorem III, however, is contrary to the modern understanding more. Hilbert's treatise in MA36 includes 61 pages and consists of five sections. In the first, the Hilbert basis set repeatedly (Theorem I) and in the second extended (Theorem II). The third contains the Syzygiensatz (in its "modern" version, Theorem III ), the fourth is of Hilbert functions (Theorem IV) and the fifth contains the Syzygiensatz in his invariantentheoretischen expression ( he is more special than the theorem V, the "only" the finiteness of the full Invariantensystems claimed).

Wording

Theorem III

MA36, page 492: " If a system of equations of the form ( 13) submitted [, (t = 1, ..., m )], where the members are algebraic forms ], thus leading the establishment of relations between the solutions of the same [ syzygies ] to a second system of equations of the same form; from this second derived equations arises in the same way a third -derived system of equations. The process thus begun always achieved with further continuation to an end and that is later than the nth equation system that chain [n = number of variables of the polynomial ring ] is one which has no [ non-trivial ] solutions more. " The [] additions not part of the original text.

Syzygiensatz ( invariantentheoretisch )

MA36, page 534: "The systems of irreduciblen syzygies of the first kind, second kind, etc. form a chain of derived equations. This Syzygienkette aborts in the finite and that gives [sic ] if there are no syzygies if m is of higher than the mentioned type, the number of invariants of the full system. "

Notes

Hilbert meant by an algebraic form a homogeneous polynomial in n variables over a field (or occasionally a homogeneous polynomial with integer coefficients only ) conceived, but also sums of products of the coefficients of the body, ' Variable' as a parameter (eg determinants).

An invariant is a function of the coefficients of a homogeneous whole underlying algebraic form, which remains unchanged over all linear transformations of the variables.

A syzygy (from the Greek sysygia = pair) is an m- tuple in a relation equation of the form so that syzygy not really, " pair ", but " consort " means (namely only an m- tuple of two, the relative occurred). Hilbert syzygies used in Theorem III ( the solutions of his equations) without calling it that. The term syzygy has outside mathematics are many other meanings.

Modern formulations ( Examples )

• Klaus Altmann: " Each module has finally produced a projective resolution of length n "
• Uwe Nagel: " If M is a finitely generated module over the polynomial ring, then M has a finite free resolution of length n 1. "
• David Eisenbud: "Let S be the polynomial ring in r 1 variables over a field K. Any finitely generated graded S -modules M Has A finite free resolution of length at most r 1. "
• A short form comes from Thorsten Holm: " gldim () = n" ( gldim stands for " global dimension ", a concept which is based on the concept of " projective dimension ", which has previously been something to do with projective resolutions ).