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<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap5.html#X7A489A5D79DA9E5C">5 Examples</a>
<div class="ContSect"> <a href="chap5.html#X8426A658837B4911">5.1 An Easy Polynomial Example</a>

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<div class="ContSect"> <a href="chap5.html#X7820475F7C884EA5">5.2 Hom(Hom(-,Z128),Z16)</a>

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<div class="ContSect"> <a href="chap5.html#X7CC8EA507E7AABA4">5.3 ResidueClass</a>

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<div class="ContSect"> <a href="chap5.html#X7958E7417BB312F0">5.4 Testing the Intersection Formula</a>

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<h3>5 Examples</h3>

<a id="X8426A658837B4911" name="X8426A658837B4911"></a>

<h4>5.1 An Easy Polynomial Example</h4>

The ground ring used in this example is F_3[x,y]. We want to see, how the different rings in this package can be used to localize at different points and how the results differ.

<div class="example"><pre>
gap> LoadPackage("RingsForHomalg");;
gap> F3xy := HomalgRingOfIntegersInSingular(3) * "x,y";;
gap> x1 := HomalgRingElement( "x+2", F3xy );;
gap> y0 := HomalgRingElement( "y", F3xy );;
gap> LoadPackage("LocalizeRingForHomalg");;
gap> R00 := LocalizeAtZero( F3xy );;
gap> R10 := LocalizeAt( F3xy, [ x1, y0 ] );;
gap> RMora := LocalizePolynomialRingAtZeroWithMora( F3xy );;
gap> M := HomalgMatrix( "[\
> y^3+2*y^2+x+x^2+2*x*y+y^4+x*y^2, \
> x*y^3+2*x^2*y+y^3+y^2+x+2*y+x^2, \
> x^2*y^2+2*x^3+x^2*y+y^3+2*x^2+2*x*y+y^2+2*y\
> ]", 1, 3, F3xy );;
gap> LoadPackage( "Modules" );;
gap> I := RightPresentation( M );;
gap> M00 := HomalgLocalMatrix( M, R00 );;
gap> M10 := HomalgLocalMatrix( M, R10 );;
gap> MMora := HomalgLocalMatrix( M, RMora );;
gap> I00 := RightPresentation( M00 );;
gap> I10 := RightPresentation( M10 );;
gap> IMora := RightPresentation( MMora );;
</pre></div>

This ring is able to compute a standard basis of the module.

<div class="example"><pre>
gap> Display( IMora );
GF(3)[x,y]_< x, y >/< (x+x^2-x*y-y^2+x*y^2+y^3+y^4)/1, (x-y+x^2+y^2-x^2*y+y^3+\
x*y^3)/1, (-y-x^2-x*y+y^2-x^3+x^2*y+y^3+x^2*y^2)/1 >
gap> ByASmallerPresentation( IMora );
<A cyclic torsion right module on a cyclic generator satisfying 2 relations>
gap> Display( IMora );
GF(3)[x,y]_< x, y >/< x/1, y/1 >
</pre></div>

This ring recognizes, that the module is not zero, but is not able to find better generators.

<div class="example"><pre>
gap> Display( I00 );
GF(3)[x,y]_< x, y >/< (y^4+x*y^2+y^3+x^2-x*y-y^2+x)/1, (x*y^3-x^2*y+y^3+x^2+y^\
2+x-y)/1, (x^2*y^2-x^3+x^2*y+y^3-x^2-x*y+y^2-y)/1 >
gap> ByASmallerPresentation( I00 );
<A cyclic right module on a cyclic generator satisfying 3 relations>
gap> Display( I00 );
GF(3)[x,y]_< x, y >/< (y^4+x*y^2+y^3+x^2-x*y-y^2+x)/1, (x*y^3-x^2*y+y^3+x^2+y^\
2+x-y)/1, (x^2*y^2-x^3+x^2*y+y^3-x^2-x*y+y^2-y)/1 >
</pre></div>

We are able to change the ring, to compute a nicer basis.

<div class="example"><pre>
gap> I00ToMora := RMora * I00;
<A cyclic right module on a cyclic generator satisfying 3 relations>
gap> Display( I00ToMora );
GF(3)[x,y]_< x, y >/< (x+x^2-x*y-y^2+x*y^2+y^3+y^4)/1, (x-y+x^2+y^2-x^2*y+y^3+\
x*y^3)/1, (-y-x^2-x*y+y^2-x^3+x^2*y+y^3+x^2*y^2)/1 >
gap> ByASmallerPresentation( I00ToMora );
<A cyclic torsion right module on a cyclic generator satisfying 2 relations>
gap> Display( I00ToMora );
GF(3)[x,y]_< x, y >/< x/1, y/1 >
</pre></div>

We are able to find out, that this module is actually zero.

<div class="example"><pre>
gap> Display( I10 );
GF(3)[x,y]_< x-1, y >/< (y^4+x*y^2+y^3+x^2-x*y-y^2+x)/1, (x*y^3-x^2*y+y^3+x^2+\
y^2+x-y)/1, (x^2*y^2-x^3+x^2*y+y^3-x^2-x*y+y^2-y)/1 >
gap> ByASmallerPresentation( I10 );
<A zero right module>
gap> Display( I10 );
0
</pre></div>

<a id="X7820475F7C884EA5" name="X7820475F7C884EA5"></a>

<h4>5.2 Hom(Hom(-,Z128),Z16)</h4>

The following example is taken from Section 2 of <a href="chapBib.html#biBBREACA">[BR06]</a>. The computation takes place over the local ring R=ℤ_⟨ 2⟩ (i.e. ℤ localized at the maximal ideal generated by 2).

H we infinite exact of covariant spanclass="SimpleMath">Hom(Hom(-,R/2^7)R/^4R)<> (andits functors the short <brspan="SimpleMath"> -; M_2 -alpha_1--gt/25-alpha_2--gtR/R>0/>.</>

<div class="example"><pre>
gap> LoadPackage( "LocalizeRingForHomalg" );;
gap> GlobalR := HomalgRingOfIntegersInExternalGAP( );
Z
gap> Display( GlobalR );
<An external ring residing in the CAS GAP>
gap> LoadPackage( "RingsForHomalg" );;
<span="GAPprompt"gap;< classGAPinput:LocalizeAt ,[ ;/spanjava.lang.StringIndexOutOfBoundsException: Range [104, 105) out of bounds for length 104
;
class>span(R )/>
<A local ring>
gap> M := LeftPresentation( HomalgMatrix( [ 2^5 ], R ) );
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> _M := LeftPresentation( HomalgMatrix( [ 2^3 ], R ) );
lt module byrelation a generator;
gap> alpha2 := HomalgMap( HomalgMatrix( [
< class"GAPprompt>>/span> < class="GAPinputLoadPackage"LocalizeRingForHomalg;<>
>;
< ="">apgt;</pan>span="">Display );/>
r;
<span=GAPpromptgap<< classGAPinput"> := KernelEmb( alpha2)
<Amonomorphismof modulesgtjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
< class=GAPprompt>;<spanspan="GAPinput">seq=HomalgComplex );/>
<A "complex" containing a single morphism of; cyclicmodule by relation a cyclic>
[ 0 .1]gt
gap></span&tA cyclic modulepresented 1 relationcyclic>
<spanclass="GAPprompt>&;/pan >IsShortExactSequence );gap classGAPinputK = LeftPresentation(HomalgMatrix[^ ],R))_ =Kernel( alpha2 );
ap;/panspanclassGAPinput"> := KernelEmb( alpha2 );
&;cyclicm on generator 1relationgt
gap> of modules
<An exact triangle containing 3 morphisms of left complexes at degrees 0 .1];
1,,3 1
<spanclass"GAPprompt>gap&t;span>GAPinputlehs LongSequencetriangle;span
java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 81
ap;
<A non-zero;A rightmodule a generatorsatisfying1relation&;
 .14 &;
gap> 
<lt; containing morphisms left atdegrees[ . 4]

<a id="X7CC8EA507E7AABA4" name="X7CC8EA507E7AABA4"></a>

<h4>5.3 ResidueClass</h4>

We want to show, how localization can work together with residue class rings.

<div class="example"><pre>
<span=">>LoadPackage RingsForHomalg "gt; 22.41");;/pan
<nclass="">gap><span< class"GAPinput"> :=( ) * "x,yjava.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
Q[x,y]
<pan="">><spanspan class="">wmat = (</>
<span="">>/span span="GAPinput"> "[y^3y^ ,x3x ^+2, x3x^ ,/>
>", "gt22.41"); ="APpromptgt;/> span="">Qxy= ( ) * "x,y;/>
-x3x22y2
/></>

Compute globally:<span="GAPprompt">gt )/>

< class"><>
gap> LoadPackage("" ;
[0. ]gt
gap></lt module by 2 relations for 2 generators>
-------------------------
at homology degree: 2
0
-------------------------
(an;A right acyclic complex containing2morphisms left atdegrees

the currently byabove 2matrix
------------v------------
at degree homology degree: 2
[,]( 2
-----currently by above java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
y2 ^,
xy2y30

the map is currently represented by the above 2 x 2 matrix
---------
at homology degree)
[,^1 )
-----------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<prediv

pTry of class:</>

<div class="example"><pre>

Qxy/ -x^x22y2 a class>
<=GAPprompt>/> (R1;</pan
<A span="GAPprompt">gap;</pan< class= LocalizeAt( ,</>
span> classGAPinput [HomalgRingElement ""java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
<A 2 x 2 matrix over a residue class ring>< class"">spanclass"> );
< classGAPpromptgap&;</ class=GAPinputLoadPackage"LocalizeRingForHomalg );
gap>>/> < class="">( R10 )
> );
Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, class>gap&t</span<spanclass"">Res10= Resolution 2 W10/>
<span="GAPprompt">gap><span<pan="">( R10 )</span
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
<spanan emptyx 2 )
ver
> span="">Res10:=Resolution 2,W10;------v------------
gap> Display( Res10 );
---></>
homology:
0
------------div="example"><>
(an )

the is represented by the
------------v------------
atlt; ring;
Qxy/ 3+^-*^ _lt |[x]|,| |&;^( 2
--------------&; 2 x 2matrix a local >
x*y^2+y^2,2*y^2,
y^2, y^4-2*y^3+2*y^2

modulo [ x^3+x^2-2*y">gap&;/span span lass="> : / );/span
/| 1]java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

the currently bythe 2 2
------------v------------
at degree
Q[x,](x3x22y2 )&; | x]|, |[]^1 )
-------------------------
</></div

Try class oflocalization:<pjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

<div class="example"><pre>
<spanclass=GAPprompt"gap&t >gap&-------v------------
&t; ring
gap;</> < class"">wmat0 : R0 *wmat>
<A 2 x 2 matrix over a local ring-------------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
< classGAPprompt>/> < classGAPinputR01 R0 ( / R0;</>
Q[x_ltx &;/(3-x2+*y21java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
&; span=""Display R01/>
<A residue
gap> mat01 *wmat0span
<A 2 x 2 matrix over a residue
< classGAPpromptgap>span =""> :=( wmat01 ;span
< module 2generators;
<pan="">(LocalizeRingForHomalg/>
lt 2morphismsof degrees
[ 0 .. 2 ]>
<class;> spanjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 86
-------------------------
at homology degree ="gt;/span span =GAPinput x-w2*,<>
0
-------------------------
(an empty 0 x 2 matrix ="">&;/> span=GAPinput>:HomalgMatrix[<spanjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92

the map is classGAPprompt;/><spanclass>" 2);;spanjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
------------v------------
at:
Q[x,y]_< x, y >/([x,y]_< x, y >/( gt<span<="RmodI2 (M2 ;;/java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
-------------------------
y^3+y^2,2*<A grade objectconsisting 4leftmodules at [0. &t;
0, x*y^2-y^3
/ 1

modulo [ (x^3+x^2-2*y classGAPpromptgap;/>< =GAPinput(ObjectsOfComplex ,

the1,,00java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
------------v------------
at
Q,_java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
------------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
</pre></div>

<a id="X7958E7417BB312F0" name="X7958E7417BB312F0"></a>

<h4>5.4 Testing the Intersection">T0: Tor( R0modI10 );/>

<p< class"GAPprompt">/> classList( ) )span

div=""><re

 ahref=chap4[ </a&;&<h=.htmlChapternbsp/>
gap>gapgtf bounds for length 118
< =">>span> <="> =HomalgMatrix\>
> (w-x^2)*y, \
> (w-x^2)*z, \
> (x-w^2)*y, \
> (x-w^2)*z \
> ]", 4, 1, R );;
gap> M2 := HomalgMatrix( "[\
> (w-x^2)-y, \
> (x-w^2)-z \
> ]", 2, 1, R );;
gap> LoadPackage( "Modules" );;
gap> RmodI1 := LeftPresentation( M1 );;
gap> RmodI2 := LeftPresentation( M2 );;
gap> T:=Tor( RmodI1, RmodI2 );
<A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
gap> List( ObjectsOfComplex( T ), AffineDegree );
[ 12, 4, 0, 0 ]
</pre></div>

We read, that the intersection multiplicity is 12-4=8 globally.

<div class="example"><pre>
gap> M10 := R0 * M1;
<A 4 x 1 matrix over a local (Mora) ring>
gap> M20 := R0 * M2;
<A 2 x 1 matrix over a local (Mora) ring>
gap> R0modI10 := LeftPresentation( M10 );;
gap> R0modI20 := LeftPresentation( M20 );;
gap> T0 := Tor( R0modI10, R0modI20 );
<A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
gap> List( ObjectsOfComplex( T0 ), AffineDegree );
[ 3, 1, 0, 0 ]
</pre></div>

The intersection multiplicity at zero is 3-1=2.

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