This is QuadraticIdealExamplesByRoos.info, produced by Macaulay2, version 1.24.11

QuadraticIdealExamplesByRoos -- Examples of Quadratic Ideals with Embedding Dimension Four by Jan-Erik Roos
***********************************************************************************************************

Description
===========

Quadratic ideals based on Main Theorem and Tables in "Homological properties of
the homology algebra of the Koszul complex of a local ring: Examples and
questions" by Jan-Erik Roos, Journal of Algebra 465 (2016) 399-436.

Authors
=======

  * David Eisenbud <de@msri.org>
  * Michael Perlman <mperlman@umn.edu>
  * Ritvik Ramkumar <ritvikr@cornell.edu>
  * Deepak Sireeshan <dsbx7@umsystem.edu>
  * Aleksandra Sobieska <asobieska@wisc.edu>
  * Teresa Yu <twyu@umich.edu>
  * Jacob Zoromski <jzoromsk@nd.edu>

Version
=======

This documentation describes version 0.1 of QuadraticIdealExamplesByRoos.

Source code
===========

The source code from which this documentation is derived is in the file
QuadraticIdealExamplesByRoos.m2 (see
common/share/Macaulay2/QuadraticIdealExamplesByRoos.m2 ).

Exports
=======

  * Functions and commands
      * almostKoszul -- Examples discovered by Jan-Erik
        Roos
  * Methods
      * "almostKoszul(Ring,ZZ)" -- see almostKoszul, --
        Examples discovered by Jan-Erik Roos
  * Other things
      * depthZeroTable -- Creates hashtable of Jan-Erik
        Roos' examples of quadratic ideals with depth zero
      * higherDepthTable -- Creates hashtable of
        Jan-Erik Roos' examples of quadratic ideals with positive depth
      * onedimToricIrrationalPoincare, --
        Produces the example of a one-dimensional toric ideal whose Poincar\'e
        series is irrational.
      * roosIsotopes -- Creates hashtable of Jan-Erik
        Roos' quadratic "isotopes"
      * roosTable -- Creates hashtable of Jan-Erik Roos'
        examples of quadratic ideals
      * twodimToricIrrationalPoincare, --
        Produces the example of a two-dimensional toric ideal whose Poincar\'e
        series is irrational.

For the programmer
==================

The object QuadraticIdealExamplesByRoos, is a package (Macaulay2Doc).

roosTable -- Creates hashtable of Jan-Erik Roos' examples of quadratic ideals
*****************************************************************************

Synopsis
========

  * Usage:
        H = roosTable ()
  * Outputs:
      * H, a hash table (Macaulay2Doc),

Description
===========

This is based on Main Theorem and Tables 3-7 in "Homological properties of the
homology algebra of the Koszul complex of a local ring: Examples and questions"
by Jan-Erik Roos, Journal of Algebra 465 (2016) 399-436. The ideals in this
table exemplify 83 known cases of bi-graded Poincar\'e series of quadratic
ideals of embedding dimension four in characteristic zero. The coefficient
field is QQ.

+-----------------------------------------------------------------------------+
|i1 : roosTable                                                               |
|                                                                             |
|o1 = HashTable{1 => ideal 0                                                  |
|                           2                                                 |
|               2 => ideal x                                                  |
|                            2   2                                            |
|               3 => ideal (x , y )                                           |
|                            2                                                |
|               4 => ideal (x , x*y)                                          |
|                            2   2   2                                        |
|               5 => ideal (x , y , z )                                       |
|                            2   2                                            |
|               6 => ideal (x , y  + x*z, y*z)                                |
|                            2    2   2    2                                  |
|               7 => ideal (x  + y , z  + u , x*z + y*u)                      |
|                            2   2                                            |
|               8 => ideal (x , y , x*z)                                      |
|                            2        2                                       |
|               9 => ideal (x , x*y, y )                                      |
|                             2                                               |
|               10 => ideal (x , x*y, x*z)                                    |
|                             2   2   2   2                                   |
|               11 => ideal (x , y , z , u )                                  |
|                             2         2         2               2           |
|               12 => ideal (x  + x*y, y  + x*u, z  + x*u, z*u + u )          |
|                             2    2    2   2                                 |
|               13 => ideal (x  + z  + u , y , x*z, y*u + z*u)                |
|                                  2   2    2                                 |
|               14 => ideal (x*z, y , z  + u , y*u + z*u)                     |
|                                  2         2                                |
|               15 => ideal (x*z, y , y*z + u , y*u + z*u)                    |
|                                   2         2                               |
|               16 => ideal (x*y + z  + y*u, y , y*u + z*u, x*z)              |
|                                             2                               |
|               17 => ideal (x*z, y*z + x*u, y , y*u + z*u)                   |
|                             2   2   2                                       |
|               18 => ideal (x , y , z , y*u)                                 |
|                                  2              2                           |
|               19 => ideal (x*z, y , y*u + z*u, u )                          |
|                                  2   2                                      |
|               20 => ideal (x*z, y , z  + y*u, y*u + z*u)                    |
|                                  2   2                                      |
|               21 => ideal (x*z, y , z , y*u + z*u)                          |
|                             2                         2                     |
|               22 => ideal (x  + x*y, x*u, x*z + y*u, y )                    |
|                                       2   2                                 |
|               23 => ideal (x*z, x*u, y , z )                                |
|                                  2         2                                |
|               24 => ideal (x*z, y , y*z + z , y*u + z*u)                    |
|                             2             2                                 |
|               25 => ideal (x , x*y, x*z, u )                                |
|                                  2                                          |
|               26 => ideal (x*z, y , y*u, z*u)                               |
|                                       2                                     |
|               27 => ideal (x*y, x*z, y , y*z)                               |
|                             2                                               |
|               28 => ideal (x , x*y, x*z, x*u)                               |
|                             2         2         2               2           |
|               29 => ideal (x  + x*y, y  + x*u, z  + x*u, z*u + u , y*z)     |
|                                   2        2    2    2   2                  |
|               30 => ideal (x*y + u , x*z, x  + z  + u , y , y*u + z*u)      |
|                             2    2   2    2   2    2                2       |
|               31 => ideal (x  - y , y  - z , z  - u , x*z + y*u, - x  + x*y |
|                             2    2        2              2                  |
|               32 => ideal (x  + z , x*z, y , y*u + z*u, u )                 |
|                             2         2         2         2               2 |
|               33 => ideal (x  + x*y, y  + y*z, y  + x*u, z  + x*u, z*u + u )|
|                             2                2   2        2    2    2       |
|               34 => ideal (x  + x*y + y*u + u , y , x*z, x  + z  + u , y*u +|
|                             2    2    2   2                                 |
|               35 => ideal (x  + z  + u , y , x*z, x*y + y*z + y*u, y*u + z*u|
|                             2    2   2   2                                  |
|               36 => ideal (x  + y , z , u , y*z - y*u, x*z + z*u)           |
|                             2   2                                           |
|               37 => ideal (x , y , x*y - z*u, y*z - x*u, x*z - y*z - x*u + y|
|                             2   2   2        2                              |
|               38 => ideal (x , y , z , z*u, u )                             |
|                             2          2         2                          |
|               39 => ideal (x  + y*z + u , x*z + z  + y*u, x*y, x*u, z*u)    |
|                             2           2         2    2   2    2           |
|               40 => ideal (x  - x*u, - y  + x*u, y  - z , z  - u , x*z + y*u|
|                                  2   2        2                             |
|               41 => ideal (x*y, y , z , z*u, u )                            |
|                             2              2                                |
|               42 => ideal (x  + x*y, z*u, y , x*u, x*z + y*u)               |
|                             2   2             2                             |
|               43 => ideal (x , y , y*z, z*u, u )                            |
|                                       2              2    2                 |
|               44 => ideal (x*z, y*z, y , y*u + z*u, z  + u )                |
|                                              2                    2         |
|               45 => ideal (x*y + y*z, x*y + z  + y*u, y*u + z*u, y , x*z)   |
|                             2                  2                            |
|               46 => ideal (x , x*y, y*z, z*u, u )                           |
|                             2         2                     2               |
|               47 => ideal (x  + x*y, y , x*u, x*z + y*u, - x  + x*z - y*z)  |
|                                  2                    2                     |
|               48 => ideal (x*y, z  + y*u, y*u + z*u, y , x*z)               |
|                                  2   2                                      |
|               49 => ideal (x*z, y , z , y*u, z*u)                           |
|                             2             2   2                             |
|               50 => ideal (x , x*y, x*z, y , z )                            |
|                                                  2                          |
|               51 => ideal (x*y, x*z, y*z + x*u, z , z*u)                    |
|                             2             2                                 |
|               52 => ideal (x , x*y, x*z, y , y*z)                           |
|                             2    2             2                            |
|               53 => ideal (y  - u , x*z, y*z, z , z*u)                      |
|                             2        2   2              2                   |
|               54 => ideal (x , x*z, y , z , y*u + z*u, u )                  |
|                             2                         2   2         2       |
|               55 => ideal (x  + x*y, x*z + y*u, x*u, y , z , z*u + u )      |
|                             2          2             2    2   2             |
|               56 => ideal (x  + x*z + u , x*y, x*u, x  - y , z , z*u)       |
|                             2          2        2                          2|
|               57 => ideal (x  + y*z + u , x*u, x  + x*y, x*z + y*u, z*u + u |
|                             2         2         2   2                       |
|               58 => ideal (x  + x*y, x  + z*u, y , z , x*z + y*u, x*u)      |
|                             2    2             2                            |
|               59 => ideal (x  - y , x*y, x*u, z , z*u, x*z + y*u)           |
|                             2          2                        2           |
|               60 => ideal (x  + y*z + u , x*z + y*u, z*u, x*y, z , x*u)     |
|                             2    2        2             2                   |
|               61 => ideal (x  - y , x*y, z , x*u, z*u, u )                  |
|                             2    2                        2                 |
|               62 => ideal (x  - y , x*y, x*u, y*z + y*u, z , z*u)           |
|                             2             2   2                             |
|               63 => ideal (x , x*y, x*u, y , z , z*u)                       |
|                             2    2        2                                 |
|               64 => ideal (x  - y , x*y, z , x*u, y*u, z*u)                 |
|                             2             2   2                             |
|               65 => ideal (x , x*y, x*z, y , z  + y*u, y*u + z*u)           |
|                                  2        2        2                        |
|               66 => ideal (x*z, y , y*u, z , z*u, u )                       |
|                                       2        2                            |
|               67 => ideal (x*y, x*z, y , y*u, z , z*u)                      |
|                             2             2        2                        |
|               68 => ideal (x , x*y, x*z, y , y*z, z )                       |
|                             2                             2                 |
|               69 => ideal (x , x*z, x*u, x*y - z*u, y*z, z )                |
|                             2        2      2                               |
|               70 => ideal (x , x*y, x z*u, y , y*z)                         |
|                             2   2   2   2                                   |
|               71 => ideal (x , y , z , u , x*y, z*u, y*z + x*u)             |
|                             2    2                  2                       |
|               72 => ideal (x  - y , x*y, y*z, z*u, z , x*z + y*u, x*u)      |
|                             2   2   2   2                                   |
|               73 => ideal (x , y , z , u , z*u, y*u, x*u)                   |
|                             2         2                       2             |
|               74 => ideal (x , x*y + z , y*z, x*u, y*u, z*u, u )            |
|                             2                  2        2                   |
|               75 => ideal (x , x*y, x*z, x*u, y , y*z, u )                  |
|                             2                  2                            |
|               76 => ideal (x , x*y, x*z, x*u, z , z*u, y*u)                 |
|                             2                  2                            |
|               77 => ideal (x , x*y, x*z, x*u, y , y*z, y*u)                 |
|                             2        2   2        2                         |
|               78 => ideal (x , x*y, y , z , z*u, u , x*z + y*u, y*z - x*u)  |
|                             2                  2        2                   |
|               79 => ideal (x , x*y, x*z, x*u, y , y*u, z , z*u)             |
|                             2             2             2                   |
|               80 => ideal (x , x*y, x*z, y , y*z, y*u, z , z*u)             |
|                             2   2   2   2                                   |
|               81 => ideal (x , y , z , u , x*y, x*z, y*z - x*u, y*u, z*u)   |
|                             2                  2        2                   |
|               82 => ideal (x , x*y, x*z, x*u, y , z*u, u , y*z, y*u)        |
|                             2   2   2   2                                   |
|               83 => ideal (x , y , z , u , x*y, x*z, x*u, y*z, y*u, z*u)    |
|                                                                             |
|o1 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|            }                                                                |
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|- y*z + x*u)                                                                 |
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| z*u)                                                                        |
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|)                                                                            |
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|*u)                                                                          |
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|)                                                                            |
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|   2    2                                                                    |
|, y  + z )                                                                   |
+-----------------------------------------------------------------------------+

For the programmer
==================

The object roosTable, is a hash table (Macaulay2Doc).

higherDepthTable -- Creates hashtable of Jan-Erik Roos' examples of quadratic ideals with positive depth
********************************************************************************************************

Synopsis
========

  * Usage:
        H = higherDepthTable ()
  * Outputs:
      * H, a hash table (Macaulay2Doc),

Description
===========

This outputs the examples in Tables 3-7 of positive depth. These are those in
the tables with non-bold row index.

+-------------------------------------------------------+
|i1 : higherDepthTable                                  |
|                                                       |
|o1 = HashTable{1 => ideal 0                           }|
|                           2                           |
|               2 => ideal x                            |
|                            2   2                      |
|               3 => ideal (x , y )                     |
|                            2                          |
|               4 => ideal (x , x*y)                    |
|                            2   2   2                  |
|               5 => ideal (x , y , z )                 |
|                            2   2                      |
|               6 => ideal (x , y  + x*z, y*z)          |
|                            2   2                      |
|               8 => ideal (x , y , x*z)                |
|                            2        2                 |
|               9 => ideal (x , x*y, y )                |
|                             2                         |
|               10 => ideal (x , x*y, x*z)              |
|                                       2   2           |
|               23 => ideal (x*z, x*u, y , z )          |
|                                  2                    |
|               26 => ideal (x*z, y , y*u, z*u)         |
|                                       2               |
|               27 => ideal (x*y, x*z, y , y*z)         |
|                             2             2   2       |
|               50 => ideal (x , x*y, x*z, y , z )      |
|                             2             2           |
|               52 => ideal (x , x*y, x*z, y , y*z)     |
|                             2             2        2  |
|               68 => ideal (x , x*y, x*z, y , y*z, z ) |
|                                                       |
|o1 : HashTable                                         |
+-------------------------------------------------------+

For the programmer
==================

The object higherDepthTable, is a hash table (Macaulay2Doc).

depthZeroTable -- Creates hashtable of Jan-Erik Roos' examples of quadratic ideals with depth zero
**************************************************************************************************

Synopsis
========

  * Usage:
        H = depthZeroTable ()
  * Outputs:
      * H, a hash table (Macaulay2Doc),

Description
===========

This outputs the examples in Tables 3-7 of depth zero. These are those in the
tables with bold row index.

+-----------------------------------------------------------------------------+
|i1 : depthZeroTable                                                          |
|                                                                             |
|                            2    2   2    2                                  |
|o1 = HashTable{7 => ideal (x  + y , z  + u , x*z + y*u)                      |
|                             2   2   2   2                                   |
|               11 => ideal (x , y , z , u )                                  |
|                             2         2         2               2           |
|               12 => ideal (x  + x*y, y  + x*u, z  + x*u, z*u + u )          |
|                             2    2    2   2                                 |
|               13 => ideal (x  + z  + u , y , x*z, y*u + z*u)                |
|                                  2   2    2                                 |
|               14 => ideal (x*z, y , z  + u , y*u + z*u)                     |
|                                  2         2                                |
|               15 => ideal (x*z, y , y*z + u , y*u + z*u)                    |
|                                   2         2                               |
|               16 => ideal (x*y + z  + y*u, y , y*u + z*u, x*z)              |
|                                             2                               |
|               17 => ideal (x*z, y*z + x*u, y , y*u + z*u)                   |
|                             2   2   2                                       |
|               18 => ideal (x , y , z , y*u)                                 |
|                                  2              2                           |
|               19 => ideal (x*z, y , y*u + z*u, u )                          |
|                                  2   2                                      |
|               20 => ideal (x*z, y , z  + y*u, y*u + z*u)                    |
|                                  2   2                                      |
|               21 => ideal (x*z, y , z , y*u + z*u)                          |
|                             2                         2                     |
|               22 => ideal (x  + x*y, x*u, x*z + y*u, y )                    |
|                                  2         2                                |
|               24 => ideal (x*z, y , y*z + z , y*u + z*u)                    |
|                             2             2                                 |
|               25 => ideal (x , x*y, x*z, u )                                |
|                             2                                               |
|               28 => ideal (x , x*y, x*z, x*u)                               |
|                             2         2         2               2           |
|               29 => ideal (x  + x*y, y  + x*u, z  + x*u, z*u + u , y*z)     |
|                                   2        2    2    2   2                  |
|               30 => ideal (x*y + u , x*z, x  + z  + u , y , y*u + z*u)      |
|                             2    2   2    2   2    2                2       |
|               31 => ideal (x  - y , y  - z , z  - u , x*z + y*u, - x  + x*y |
|                             2    2        2              2                  |
|               32 => ideal (x  + z , x*z, y , y*u + z*u, u )                 |
|                             2         2         2         2               2 |
|               33 => ideal (x  + x*y, y  + y*z, y  + x*u, z  + x*u, z*u + u )|
|                             2                2   2        2    2    2       |
|               34 => ideal (x  + x*y + y*u + u , y , x*z, x  + z  + u , y*u +|
|                             2    2    2   2                                 |
|               35 => ideal (x  + z  + u , y , x*z, x*y + y*z + y*u, y*u + z*u|
|                             2    2   2   2                                  |
|               36 => ideal (x  + y , z , u , y*z - y*u, x*z + z*u)           |
|                             2   2                                           |
|               37 => ideal (x , y , x*y - z*u, y*z - x*u, x*z - y*z - x*u + y|
|                             2   2   2        2                              |
|               38 => ideal (x , y , z , z*u, u )                             |
|                             2          2         2                          |
|               39 => ideal (x  + y*z + u , x*z + z  + y*u, x*y, x*u, z*u)    |
|                             2           2         2    2   2    2           |
|               40 => ideal (x  - x*u, - y  + x*u, y  - z , z  - u , x*z + y*u|
|                                  2   2        2                             |
|               41 => ideal (x*y, y , z , z*u, u )                            |
|                             2              2                                |
|               42 => ideal (x  + x*y, z*u, y , x*u, x*z + y*u)               |
|                             2   2             2                             |
|               43 => ideal (x , y , y*z, z*u, u )                            |
|                                       2              2    2                 |
|               44 => ideal (x*z, y*z, y , y*u + z*u, z  + u )                |
|                                              2                    2         |
|               45 => ideal (x*y + y*z, x*y + z  + y*u, y*u + z*u, y , x*z)   |
|                             2                  2                            |
|               46 => ideal (x , x*y, y*z, z*u, u )                           |
|                             2         2                     2               |
|               47 => ideal (x  + x*y, y , x*u, x*z + y*u, - x  + x*z - y*z)  |
|                                  2                    2                     |
|               48 => ideal (x*y, z  + y*u, y*u + z*u, y , x*z)               |
|                                  2   2                                      |
|               49 => ideal (x*z, y , z , y*u, z*u)                           |
|                                                  2                          |
|               51 => ideal (x*y, x*z, y*z + x*u, z , z*u)                    |
|                             2    2             2                            |
|               53 => ideal (y  - u , x*z, y*z, z , z*u)                      |
|                             2        2   2              2                   |
|               54 => ideal (x , x*z, y , z , y*u + z*u, u )                  |
|                             2                         2   2         2       |
|               55 => ideal (x  + x*y, x*z + y*u, x*u, y , z , z*u + u )      |
|                             2          2             2    2   2             |
|               56 => ideal (x  + x*z + u , x*y, x*u, x  - y , z , z*u)       |
|                             2          2        2                          2|
|               57 => ideal (x  + y*z + u , x*u, x  + x*y, x*z + y*u, z*u + u |
|                             2         2         2   2                       |
|               58 => ideal (x  + x*y, x  + z*u, y , z , x*z + y*u, x*u)      |
|                             2    2             2                            |
|               59 => ideal (x  - y , x*y, x*u, z , z*u, x*z + y*u)           |
|                             2          2                        2           |
|               60 => ideal (x  + y*z + u , x*z + y*u, z*u, x*y, z , x*u)     |
|                             2    2        2             2                   |
|               61 => ideal (x  - y , x*y, z , x*u, z*u, u )                  |
|                             2    2                        2                 |
|               62 => ideal (x  - y , x*y, x*u, y*z + y*u, z , z*u)           |
|                             2             2   2                             |
|               63 => ideal (x , x*y, x*u, y , z , z*u)                       |
|                             2    2        2                                 |
|               64 => ideal (x  - y , x*y, z , x*u, y*u, z*u)                 |
|                             2             2   2                             |
|               65 => ideal (x , x*y, x*z, y , z  + y*u, y*u + z*u)           |
|                                  2        2        2                        |
|               66 => ideal (x*z, y , y*u, z , z*u, u )                       |
|                                       2        2                            |
|               67 => ideal (x*y, x*z, y , y*u, z , z*u)                      |
|                             2                             2                 |
|               69 => ideal (x , x*z, x*u, x*y - z*u, y*z, z )                |
|                             2        2      2                               |
|               70 => ideal (x , x*y, x z*u, y , y*z)                         |
|                             2   2   2   2                                   |
|               71 => ideal (x , y , z , u , x*y, z*u, y*z + x*u)             |
|                             2    2                  2                       |
|               72 => ideal (x  - y , x*y, y*z, z*u, z , x*z + y*u, x*u)      |
|                             2   2   2   2                                   |
|               73 => ideal (x , y , z , u , z*u, y*u, x*u)                   |
|                             2         2                       2             |
|               74 => ideal (x , x*y + z , y*z, x*u, y*u, z*u, u )            |
|                             2                  2        2                   |
|               75 => ideal (x , x*y, x*z, x*u, y , y*z, u )                  |
|                             2                  2                            |
|               76 => ideal (x , x*y, x*z, x*u, z , z*u, y*u)                 |
|                             2                  2                            |
|               77 => ideal (x , x*y, x*z, x*u, y , y*z, y*u)                 |
|                             2        2   2        2                         |
|               78 => ideal (x , x*y, y , z , z*u, u , x*z + y*u, y*z - x*u)  |
|                             2                  2        2                   |
|               79 => ideal (x , x*y, x*z, x*u, y , y*u, z , z*u)             |
|                             2             2             2                   |
|               80 => ideal (x , x*y, x*z, y , y*z, y*u, z , z*u)             |
|                             2   2   2   2                                   |
|               81 => ideal (x , y , z , u , x*y, x*z, y*z - x*u, y*u, z*u)   |
|                             2                  2        2                   |
|               82 => ideal (x , x*y, x*z, x*u, y , z*u, u , y*z, y*u)        |
|                             2   2   2   2                                   |
|               83 => ideal (x , y , z , u , x*y, x*z, x*u, y*z, y*u, z*u)    |
|                                                                             |
|o1 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|            }                                                                |
|                                                                             |
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|                                                                             |
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|                                                                             |
|- y*z + x*u)                                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| z*u)                                                                        |
|                                                                             |
|)                                                                            |
|                                                                             |
|                                                                             |
|                                                                             |
|*u)                                                                          |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|)                                                                            |
|                                                                             |
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|                                                                             |
|   2    2                                                                    |
|, y  + z )                                                                   |
+-----------------------------------------------------------------------------+

For the programmer
==================

The object depthZeroTable, is a hash table (Macaulay2Doc).

roosIsotopes -- Creates hashtable of Jan-Erik Roos' quadratic "isotopes"
************************************************************************

Synopsis
========

  * Usage:
        H = depthZeroTable ()
  * Outputs:
      * H, a hash table (Macaulay2Doc),

Description
===========

This outputs the examples in Tables B,C,D of Roos' quadratic "isotopes" (see
Main Theorem in  "Homological properties of the homology algebra of the Koszul
complex of a local ring: Examples and questions", Journal of Algebra 465 (2016)
399-436). The key is the alphanumeric name provided by Roos.  While these have
the same bigraded Poincar\'e series as their corresponding example in Tables
3-7, they have different homology algebra (called "HKR" in Roos). These all
have depth zero. The coefficient field is QQ.

+-----------------------------------------------------------------------------+
|i1 : roosIsotopes                                                            |
|                                                                             |
|                                       2             2   2    2              |
|o1 = HashTable{"46va" => ideal (x*z + u , x*y, x*u, x , y  + z  + z*u)       |
|                                 2    2    2                             2   |
|               "57v2" => ideal (x  + y  + z , x*y, x*u, y*z, z*u, x*z + u )  |
|                                 2    2                        2             |
|               "59va" => ideal (x  - y , x*y, y*z, z*u, x*z + u , x*u)       |
|                                 2          2                  2             |
|               "62va" => ideal (x  + y*z + u , y*u, z*u, x*y, z , x*u)       |
|                                 2              2        2    2              |
|               "63ne" => ideal (x , x*y, x*z + u , x*u, y  + z , z*u)        |
|                                 2                        2                  |
|               "63v4" => ideal (y , x*z + y*u, z*u, x*y, z , x*u)            |
|                                 2                  2         2   2    2     |
|               "63v8" => ideal (x , x*y, x*u, y*u, z , x*z + u , y  + z  + z*|
|                                            2                  2             |
|               "66v5" => ideal (x*y, x*z + u , x*u, y*u, z*u, z )            |
|                                2                  2   2    2                |
|               "68v" => ideal (x , x*y, x*z, x*u, u , y  + z  + z*u)         |
|                                 2    2             2        2               |
|               "71v4" => ideal (x  + u , x*y, x*u, y , y*z, z , z*u)         |
|                                 2         2               2   2   2         |
|               "71v5" => ideal (x  + x*y, x  + y*z, x*y + y , z , z , x*u, z*|
|                                 2   2   2         2                         |
|               "71v7" => ideal (x , y , z , x*z + u , x*u, y*z, z*u)         |
|                                  2   2    2                        2        |
|               "71v16" => ideal (x , y  + z , x*y, y*z, z*u, x*z + u , x*u)  |
|                                       2   2         2         2         2   |
|               "72v1" => ideal (x*u + u , x  + x*y, y  + x*u, y  + y*z, y  + |
|                                       2                         2        2  |
|               "72v2e" => ideal (y*z, x  + x*y, x*z + y*u, x*u, z , z*u, x  +|
|                                 2                   2             2         |
|               "75v1" => ideal (y , x*z, y*z + x*u, z , y*u, z*u, u )        |
|                                            2              2             2   |
|               "75v2" => ideal (x*y, x*z + u , x*u, y*z + u , y*u, z*u, z )  |
|                                 2   2   2   2                               |
|               "78v1" => ideal (x , y , z , u , x*y, x*z, x*u, y*u)          |
|                                                       2   2         2   2   |
|               "78v2e" => ideal (x*u, x*z + y*u, y*z, x , y , x*y + z , u , z|
|                                                       2   2         2   2   |
|               "78v3v" => ideal (x*u, x*z + y*u, y*z, x , y , x*z + z , u , z|
|                                 2   2   2   2                               |
|               "81va" => ideal (x , y , z , u , x*y, x*z, x*u, y*u, z*u)     |
|                                                                             |
|o1 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|                                   }                                         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|u)                                                                           |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|          2                                                                  |
|u, x*z + u )                                                                 |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                 2               2                                           |
|y*z, y*u + z*u, z  + x*u, z*u + u )                                          |
|  2                                                                          |
| u )                                                                         |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|*u)                                                                          |
|                                                                             |
|*u)                                                                          |
+-----------------------------------------------------------------------------+

For the programmer
==================

The object roosIsotopes, is a hash table (Macaulay2Doc).

almostKoszul -- Examples discovered by Jan-Erik Roos
****************************************************

Synopsis
========

  * Usage:
        R = almostKoszul(kk,a)
  * Inputs:
      * kk, a ring (Macaulay2Doc), the field over which R will be
        defined
      * a, an integer (Macaulay2Doc), length of the linear part of
        the resolution of kk over R
  * Outputs:
      * R, a ring (Macaulay2Doc),

Description
===========

A standard graded ring R is Koszul if the minimal R-free resolution F of its
residue field kk is linear. Roos' examples, which are 2-dimensional rings of
depth 0 in 6 variables, show that it is not enough to require that F be linear
for a steps, no matter how large a is.

The examples are also remarkable in that (as far as we could check) the
(a+1)-st syzygy, and all subsequent syzygies of kk have socle summands, but
none before the (a+1)-st do. This shows that the the socle summands do NOT all
come from the Koszul complex, but leaves open the conjecture that (with the one
exception) the socle summand persist once they start.

It's also striking that (in this case) the first socle summands come from the
linear strand of the resolution, though they begin to appear exactly where the
resolution ceases to be linear.

+------------------------------------------------------------+
|i1 : R = almostKoszul(ZZ/32003, 4)                          |
|                                                            |
|o1 = R                                                      |
|                                                            |
|o1 : QuotientRing                                           |
+------------------------------------------------------------+
|i2 : F = res (coker vars R, LengthLimit =>6)                |
|                                                            |
|      1      6      26      104      403      1543      5880|
|o2 = R  <-- R  <-- R   <-- R    <-- R    <-- R     <-- R    |
|                                                            |
|     0      1      2       3        4        5         6    |
|                                                            |
|o2 : ChainComplex                                           |
+------------------------------------------------------------+
|i3 : betti F                                                |
|                                                            |
|            0 1  2   3   4    5    6                        |
|o3 = total: 1 6 26 104 403 1543 5880                        |
|         0: 1 6 26 104 403 1542 5870                        |
|         1: . .  .   .   .    1   10                        |
|                                                            |
|o3 : BettiTally                                             |
+------------------------------------------------------------+

References
==========

J.E.Roos, Commutative non Koszul algebras having a linear resolution of
arbitrarily high order. Applications to torsion in loop space homology, C. R.
Acad. Sci. Paris 316 (1993),1123-1128.

Ways to use almostKoszul:
=========================

  * "almostKoszul(Ring,ZZ)"

For the programmer
==================

The object almostKoszul, is a method function (Macaulay2Doc).

onedimToricIrrationalPoincare -- Produces the example of a one-dimensional toric ideal whose Poincar\'e series is irrational.
*****************************************************************************************************************************

Synopsis
========

  * Usage:
        I = onedimToricIrrationalPoincare ()
  * Outputs:
      * I, an ideal (Macaulay2Doc),

Description
===========

This returns the example from Fr\:oberg and Roos' "An affine monomial curve
with irrational with irrational Poincar\'e-Betti series" of a toric ideal whose
quotient ring has an irrational Poincar\'e series. It is the toric ideal of the
numerical subsemigroup of $\mathbb{N}$ generated by $\{18, 24, 25, 26, 28, 30,
33\}.$

+-----------------------------------------------------------------------------+
|i1 : I = onedimToricIrrationalPoincare                                       |
|                                                                             |
|             2          2                       2                       3    |
|o1 = ideal (w  - w w , w  - w w , w w  - w w , w  - w w , w w  - w w , w  -  |
|             2    1 6   3    2 4   3 4    1 7   4    2 5   4 5    2 6   1    |
|     ------------------------------------------------------------------------|
|            2                       2      2   2            2      2         |
|     w w , w  - w w , w w  - w w , w w  - w , w w  - w w , w w  - w , w w w  |
|      2 6   5    4 6   5 6    3 7   1 2    6   1 3    5 7   1 6    7   2 3 5 |
|     ------------------------------------------------------------------------|
|                                   2      2     2              2            3|
|     - w w w , w w w  - w w w , w w  - w w , w w  - w w w , w w  - w w w , w |
|        1 4 7   2 3 6    1 5 7   2 6    1 7   3 6    2 5 7   4 6    3 5 7   6|
|     ------------------------------------------------------------------------|
|          2                                                                  |
|     - w w )                                                                 |
|        2 7                                                                  |
|                                                                             |
|o1 : Ideal of QQ[w ..w ]                                                     |
|                  1   7                                                      |
+-----------------------------------------------------------------------------+

For the programmer
==================

The object onedimToricIrrationalPoincare,
is an ideal (Macaulay2Doc).

twodimToricIrrationalPoincare -- Produces the example of a two-dimensional toric ideal whose Poincar\'e series is irrational.
*****************************************************************************************************************************

Synopsis
========

  * Usage:
        I = twodimToricIrrationalPoincare ()
  * Outputs:
      * I, an ideal (Macaulay2Doc),

Description
===========

Returns the example from Roos and Sturmfels' "A toric ring with irrational
Poincar\'e-Betti series" (1998) of a toric ideal whose quotient ring has an
irrational Poincar\'e series. It is the toric ideal of the numerical
subsemigroup of $\mathbb{N}^2$ generated by ${(36,0), (33,3), (30,6), (28,8),
(26,10), (25,11), (24,12), (18,18), (0,36)}.$

+-----------------------------------------------------------------------------+
|i1 : I = onedimToricIrrationalPoincare                                       |
|                                                                             |
|             2          2                       2                       3    |
|o1 = ideal (w  - w w , w  - w w , w w  - w w , w  - w w , w w  - w w , w  -  |
|             2    1 6   3    2 4   3 4    1 7   4    2 5   4 5    2 6   1    |
|     ------------------------------------------------------------------------|
|            2                       2      2   2            2      2         |
|     w w , w  - w w , w w  - w w , w w  - w , w w  - w w , w w  - w , w w w  |
|      2 6   5    4 6   5 6    3 7   1 2    6   1 3    5 7   1 6    7   2 3 5 |
|     ------------------------------------------------------------------------|
|                                   2      2     2              2            3|
|     - w w w , w w w  - w w w , w w  - w w , w w  - w w w , w w  - w w w , w |
|        1 4 7   2 3 6    1 5 7   2 6    1 7   3 6    2 5 7   4 6    3 5 7   6|
|     ------------------------------------------------------------------------|
|          2                                                                  |
|     - w w )                                                                 |
|        2 7                                                                  |
|                                                                             |
|o1 : Ideal of QQ[w ..w ]                                                     |
|                  1   7                                                      |
+-----------------------------------------------------------------------------+

For the programmer
==================

The object twodimToricIrrationalPoincare,
is an ideal (Macaulay2Doc).