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Bruns :: Bruns

Bruns -- produces an ideal with three generators whose 2nd syzygy module is isomorphic to a given module

Description

Bruns is a package of functions for transforming syzygies.

A well-known paper of Winfried Bruns, entitled ”Jede” freie Auflösung ist freie Auflösung eines drei-Erzeugenden Ideals (J. Algebra 39 (1976), no. 2, 429-439), shows that every second syzygy module is the second syzygy module of an ideal with three generators.

The general context of this result uses the theory of ”basic elements”, a commutative algebra version of the general position arguments of the algebraic geometers. The ”Syzygy Theorem” of Evans and Griffiths (Syzygies. London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985) asserts that if a module M over a regular local ring S containing a field (the field is conjecturally not necessary), or a graded module over a polynomial ring S, is a k-th syzygy module but not a free module, then M has rank at least k. The theory of basic elements shows that if M is a k-th syzygy of rank >k, then for a ”sufficiently general” element m of M the module M/Sm is again a k-th syzygy.

The idea of Bruns’ theorem is that if M is a second syzygy module, then factoring out (rank M) - 2 general elements gives a second syzygy N of rank 2. It turns out that three general homomorphisms from M to S embed N in S3 in such a way that the quotient S3/N is an ideal generated by three elements.

This package implements this method.

Author

Version

This documentation describes version 2.0 of Bruns.

Source code

The source code from which this documentation is derived is in the file Bruns.m2.

Exports

  • Functions and commands
    • bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module
    • brunsIdeal -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal
    • elementary -- Elementary moves are used to reduce the target of a syzygy matrix
    • evansGriffith -- Reduces the rank of a syzygy
    • isSyzygy -- Tests if a module is a d-th syzygy