This is SCMAlgebras.info, produced by Macaulay2, version 1.25.05

SCMAlgebras -- sequentially Cohen-Macaulay modules or ideals
************************************************************

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

SCMAlgebras is a package to check whether a module or an ideal is sequentially
Cohen-Macaulay, by computing the modules of deficiency and the filter ideals.

Author
======

  * Ernesto Lax (see https://www.researchgate.net/profile/Ernesto-Lax )
    <erlax@unime.it>

Version
=======

This documentation describes version 1.0 of SCMAlgebras.

Citation
========

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

+-----------------------------------------------------------------------------+
|@misc{SCMAlgebrasSource,                                                     |
|  title = {{SCMAlgebras: sequentially                                        |
|  author = {Ernesto Lax},                                                    |
|  howpublished = {A \emph{Macaulay2}                                         |
|    \url{https://github.com/Macaulay2                                        |
|}                                                                            |
|-----------------------------------------------------------------------------|
| Cohen-Macaulay modules or ideals. Version~1.0}},                            |
|                                                                             |
|package available at                                                         |
|/M2/tree/master/M2/Macaulay2/packages}}                                      |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * canonicalModule -- computes the canonical
        module of a module $M$ or an ideal $I$.
      * deficiencyModule -- computes the $i$th module
        of deficiency of a module $M$ or an ideal $I$.
      * filterIdeal -- computes the $i$th filter ideal of
        $I$.
      * isSCM -- checks whether a module or an ideal is
        sequentially Cohen-Macaulay
      * minimumDimension -- computes the minimum
        dimension of $I$.
  * Methods
      * "canonicalModule(Ideal)" -- see canonicalModule,
        -- computes the canonical module of a module $M$ or an ideal $I$.
      * "canonicalModule(Module)" -- see canonicalModule,
        -- computes the canonical module of a module $M$ or an
        ideal $I$.
      * "deficiencyModule(Ideal,ZZ)" -- see deficiencyModule,
        -- computes the $i$th module of deficiency of a
        module $M$ or an ideal $I$.
      * "deficiencyModule(Module,ZZ)" -- see deficiencyModule,
        -- computes the $i$th module of deficiency of a
        module $M$ or an ideal $I$.
      * "filterIdeal(Ideal,ZZ)" -- see filterIdeal, --
        computes the $i$th filter ideal of $I$.
      * "isSCM(Ideal)" -- see isSCM, -- checks whether a module or
        an ideal is sequentially Cohen-Macaulay
      * "isSCM(Module)" -- see isSCM, -- checks whether a module
        or an ideal is sequentially Cohen-Macaulay
      * "minimumDimension(Ideal)" -- see minimumDimension,
        -- computes the minimum dimension of $I$.

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

The object SCMAlgebras, is a package (Macaulay2Doc),
defined in SCMAlgebras.m2.

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.

canonicalModule -- computes the canonical module of a module $M$ or an ideal $I$.
*********************************************************************************

  * Usage:
        canonicalModule M
        canonicalModule I
  * Inputs:
      * M, a module (Macaulay2Doc), module over the polynomial
        ring $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal $I\subset S$
  * Outputs:
      * K, a module (Macaulay2Doc), the canonical module

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

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring. Given a $S$-module $M$ (or an
ideal $I\subset S$), it returns the canonical module $\omega(M)$ (or
$\omega(S/I)$), as $\mathrm{Ext}_S^{n-d}(M,S(-n))$, where $d=\dim M$ (or
$d=\dim S/I$).

+-----------------------------------------------+
|i1 : S=QQ[x_1..x_5];                           |
+-----------------------------------------------+
|i2 : I=(x_1^2*x_3,x_2*x_3^2*x_4,x_1*x_3^3*x_5);|
+-----------------------------------------------+
|i3 : M=S^1/I;                                  |
+-----------------------------------------------+
|i4 : canonicalModule M                         |
|                                               |
|o4 = cokernel {4} | x_3 |                      |
|                                               |
|                            1                  |
|o4 : S-module, quotient of S                   |
+-----------------------------------------------+

See also
========

  * deficiencyModule -- computes the $i$th module of
    deficiency of a module $M$ or an ideal $I$.

Ways to use canonicalModule:
============================

  * "canonicalModule(Ideal)"
  * "canonicalModule(Module)"

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

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

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.

deficiencyModule -- computes the $i$th module of deficiency of a module $M$ or an ideal $I$.
********************************************************************************************

  * Usage:
        deficiencyModule(M,i)
        deficiencyModule(I,i)
  * Inputs:
      * M, a module (Macaulay2Doc), module over the polynomial
        ring $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal $I\subset S$
      * i, an integer (Macaulay2Doc), an integer
  * Outputs:
      * D, a module (Macaulay2Doc), the ith module of deficiency
        of $M$

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

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring. Given a $S$-module $M$ (or an
ideal $I\subset S$), it returns the $i$th module of deficiency $\omega^i(M)$
(or $\omega^i(S/I)$), defined as $\mathrm{Ext}_S^{n-i}(M,S(-n))$.

+-----------------------------------------------+
|i1 : S=QQ[x_1..x_5];                           |
+-----------------------------------------------+
|i2 : I=(x_1^2*x_3,x_2*x_3^2*x_4,x_1*x_3^3*x_5);|
+-----------------------------------------------+
|i3 : M=S^1/I;                                  |
+-----------------------------------------------+
|i4 : deficiencyModule(M,3)                     |
|                                               |
|o4 = cokernel {0} | x_1 0   x_2x_4 |           |
|              {1} | 0   x_3 x_1    |           |
|                                               |
|                            2                  |
|o4 : S-module, quotient of S                   |
+-----------------------------------------------+

See also
========

  * canonicalModule -- computes the canonical module of
    a module $M$ or an ideal $I$.
  * isSCM -- checks whether a module or an ideal is sequentially
    Cohen-Macaulay

Ways to use deficiencyModule:
=============================

  * "deficiencyModule(Ideal,ZZ)"
  * "deficiencyModule(Module,ZZ)"

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

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

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.

filterIdeal -- computes the $i$th filter ideal of $I$.
******************************************************

  * Usage:
        filterIdeal(I,i)
  * Inputs:
      * I, an ideal (Macaulay2Doc), a homogeneous ideal of the
        polynomial ring $S=K[x_1,\ldots,x_n]$, with $K$ a field
      * i, an integer (Macaulay2Doc), an integer greater or equal
        than -1
  * Outputs:
      * J, an ideal (Macaulay2Doc), the $i$th filter ideal of $I$

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

Let $I\subset S$ be a homogeneous ideal, with $d=\dim S/I$, and let
$I=\displaystyle\bigcap_{j=1}^r Q_j$ be the minimal primary decomposition of
$I$. For all $1\leq j\leq r$, let $P_j = \sqrt{Q_j}$ be the radical of $Q_j$.
For all $-1\leq i\leq d$, the $i$th filter ideal of $I$ is $$I^{<i>} =
\bigcap_{\dim S/{P_j}>i} Q_{j},$$ where $I^{<-1>}=I$ and $I^{<d>}=S$.

+-----------------------------------------------------------------------------+
|i1 : S = QQ[x_1..x_10,y_1..y_10];                                            |
+-----------------------------------------------------------------------------+
|i2 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9  |
|-----------------------------------------------------------------------------|
|},{8,10},{9,10}};                                                            |
+-----------------------------------------------------------------------------+
|i3 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));               |
|                                                                             |
|o3 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i4 : filterIdeal(J,5)                                                        |
|                                                                             |
|o4 = ideal (- x y  + x y , - x y  + x y , - x y  + x y , - x y  + x y , -    |
|               2 1    1 2     3 1    1 3     4 1    1 4     5 1    1 5       |
|     ------------------------------------------------------------------------|
|     x y  + x y , - x y  + x y , - x y  + x y , - x y  + x y , - x  y  +     |
|      6 1    1 6     7 1    1 7     8 1    1 8     9 1    1 9     10 1       |
|     ------------------------------------------------------------------------|
|     x y  , - x y  + x y , - x y  + x y , - x  y  + x y  , - x  y  + x y  )  |
|      1 10     7 6    6 7     9 8    8 9     10 8    8 10     10 9    9 10   |
|                                                                             |
|o4 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+

See also
========

  * minimumDimension -- computes the minimum dimension
    of $I$.
  * isSCM -- checks whether a module or an ideal is sequentially
    Cohen-Macaulay

Ways to use filterIdeal:
========================

  * "filterIdeal(Ideal,ZZ)"

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

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

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.

isSCM -- checks whether a module or an ideal is sequentially Cohen-Macaulay
***************************************************************************

  * Usage:
        isSCM M
        isSCM I
  * Inputs:
      * M, a module (Macaulay2Doc), a finitely generated graded
        $S$-module, $S=K[x_1,\ldots,x_n]$, with $K$ a field or an ideal
        $I\subset S$
  * Outputs:
      * b, a Boolean value (Macaulay2Doc), whether the module
        $M$ or the graded algebra $S/I$ is sequentially Cohen-Macaulay

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

Given a finitely generated graded $S$-module $M$, this method checks if the
$i$th module of deficiency of $M$, $\omega^{i}(M)$, if non-zero, is
Cohen-Macaulay of dimension $i$. For a homogeneous ideal $I\subset S$, the
function checks if $\mathrm{depth} S/{I^{<i>}} \geq i+1$, where $I^{<i>}$ is
the $i$th filter ideal.

+-----------------------------------------------------------------+
|i1 : S=QQ[x_1..x_5];                                             |
+-----------------------------------------------------------------+
|i2 : M=coker matrix{{x_1*x_2,x_3*x_4,0,0},{0,x_1*x_5,x_2*x_4,0}};|
+-----------------------------------------------------------------+
|i3 : isSCM M                                                     |
|                                                                 |
|o3 = false                                                       |
+-----------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i4 : S = QQ[x_1..x_10,y_1..y_10];                                            |
+-----------------------------------------------------------------------------+
|i5 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9  |
|-----------------------------------------------------------------------------|
|},{8,10},{9,10}};                                                            |
+-----------------------------------------------------------------------------+
|i6 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));               |
|                                                                             |
|o6 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i7 : isSCM J                                                                 |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+

Ways to use isSCM:
==================

  * "isSCM(Ideal)"
  * "isSCM(Module)"

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

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

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.

minimumDimension -- computes the minimum dimension of $I$.
**********************************************************

  * Usage:
        minimumDimension I
  * Inputs:
      * I, an ideal (Macaulay2Doc), a homogeneous ideal of the
        polynomial ring $S=K[x_1,\ldots,x_n]$, with $K$ a field
  * Outputs:
      * k, an integer (Macaulay2Doc), the minimum dimension of $I$

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

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring. Given a homogeneous ideal
$I\subset S$, it returns the minimum dimension, defined as the minimum integer
$k$ such that the filter ideal $I^{<k>}$ differs from $I$.

+-----------------------------------------------------------------------------+
|i1 : S = QQ[x_1..x_10,y_1..y_10];                                            |
+-----------------------------------------------------------------------------+
|i2 : E = {{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{1,10},{6,7},{8,9  |
|-----------------------------------------------------------------------------|
|},{8,10},{9,10}};                                                            |
+-----------------------------------------------------------------------------+
|i3 : J=ideal(for e in E list x_(e#0)*y_(e#1)-x_(e#1)*y_(e#0));               |
|                                                                             |
|o3 : Ideal of S                                                              |
+-----------------------------------------------------------------------------+
|i4 : minimumDimension J                                                      |
|                                                                             |
|o4 = 11                                                                      |
+-----------------------------------------------------------------------------+

See also
========

  * filterIdeal -- computes the $i$th filter ideal of $I$.
  * isSCM -- checks whether a module or an ideal is sequentially
    Cohen-Macaulay

Ways to use minimumDimension:
=============================

  * "minimumDimension(Ideal)"

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

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

-------------------------------------------------------------------------------

The source of this document is in SCMAlgebras.m2:315:0.