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# 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 Aufl\"osung ist freie Aufl\"osung 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 S^3 in such a way that the quotient S^3/N is an ideal generated by three elements.

This package implements this method.

## Author

- David Eisenbud <de@msri.org>

## Version

This documentation describes version 2.0 of Bruns, released June 28, 2008.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{BrunsSource,
      title = {{Bruns: make a 3-generator ideal with an ``any" resolution. Version~2.0}},
      author = {David Eisenbud},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

## 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
- Methods
  - `bruns(Matrix)` -- see bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module
  - `bruns(Module)` -- see bruns -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module
  - `brunsIdeal(Ideal)` -- see brunsIdeal -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal
  - `elementary(Matrix,ZZ,ZZ)` -- see elementary -- Elementary moves are used to reduce the target of a syzygy matrix
  - `evansGriffith(Matrix,ZZ)` -- see evansGriffith -- Reduces the rank of a syzygy
  - `isSyzygy(Module,ZZ)` -- see isSyzygy -- Tests if a module is a d-th syzygy

## For the programmer

The object Bruns is a package, defined in Bruns.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Bruns.m2:222:0.