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<title>GAP (LocalizeRingForHomalg) - Chapter 5: Examples</title>
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<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap4.html">[Previous Chapter]</a> Here compute the() long homology sequence the functor< R,24R)<spanand leftderived) appliedto exactsequence /> < classSimpleMath0-gt=R/2^2-alpha_1--&; M=R25 -alpha_2--&; _M=R/^3 -> <spanpjava.lang.StringIndexOutOfBoundsException: Index 340 out of bounds for length 340

>span> span="">R : ( GlobalR [ 2])
<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>

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

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

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

</div>
</div>

<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 <<_< 2 >

The="GAPprompt">gap>Display R)Here we compute the (infinite) long exact homology sequence of the covariant functor Hom(Hom(-,R/2^7R),R/2^4R)</span<;A cyclicleft presented 1 for cyclic>

<div class="example"><pre>
an=""gap;</span<spanclass">LoadPackage( LocalizeRingForHomalg" );/span
<span=GAPpromptgap/><spanGAPinput> =HomalgRingOfIntegersInExternalGAP
Z
<spanclassGAPprompt>>/pan < classGAPinput( GlobalR
>/span> span class="GAPinputalpha1KernelEmb ;gapltA left&;
Z_< 2 >
gap> Display( R );
<A local ring>
gap> ap><> < class : ( alpha2gap> gap> alpha2 class"gapgtGAPinput( seq</>
<A "homomorphism" ofspan="">> <span="GAPinput">K:LeftPresentation ( [27 , ;/>
gap></> <span="GAPinput">: Kernel/>
<A cyclic left module presented by yet unknown relations for a cyclic generato\
r>
<span="g>s> < class="GAPinputalpha1KernelEmb;/>
<A monomorphism of left modules>
gap> 
&t;/ < class"> :=LongSequence( triangle )
true
gap> K := LeftPresentation
<A cyclic left module presented by 1 relation for a cyclic generator>
<spanclass=GAPpromptg>> <span=""> : RightPresentation( HomalgMatrix( [24] R ) );
gap> triangle := LHomHom( 4, seq, K, [0. 14]gt
&;Anexact triangle 3morphisms left
[ 1, 2, 3, 1 ]>
<span="gap; lehs := LongSequence( triangle );
<Asequence 14morphismsof modules degrees 0.1 ]>
("","&;=000.7 );<s>
[ 0n classGAPprompt&;<> span=GAPinputQxy HomalgFieldOfRationalsInDefaultCAS,";
gap> IsExactSequence( lehs );
true
</pre></div>

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

<h4>5.< class=GAPpromptgap/> <spanclass=GAPinputwmat: HomalgMatrixspan

We classGAPpromptgt< classGAPinput [^3y2 ^-^2,y3y^ ,x^+^2]"

<div class="example">< over external>
< =GAPpromptgap</>< classGAPinput">(RingsForHomalg",&;= 000.7 );/span>
<spanclass"APprompt">gap><span< class"GAPinput"> :=HomalgFieldOfRationalsInDefaultCASy"
Q[x,y]
<span^-^+*^
&<pre/div>
< classGAPprompt> 2, 2, Qxy)pre
< ="GAPprompt">gap> ec: HomalgRingElement"x3x22y2,Qxy ;/>
-x^-^+*y2
</pre></div>

<p; left presented 2relations 2 generatorsgt

<div class="example"><pre>
<span=GAPprompt>&;/> <=GAPinput Modules);/pan
gap> < 0 .2]&;
&;A left presented
<java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<rightacycliccomplex of modules degrees
[ 0 .. 2 ]>
<span class="GAPpromptthe map iscurrently representedby the 0 x matrix
--------- homology: 1
athomologydegree2
0xy^1x)
------------------------
(an empty 0 x 2 matrix)

the map is currently represented the 0 x2matrix
------------v------------
at homology degree: 1
Q[x,y]^(1 x 2)
-----------------^,x2
y^2, x^*^-^,java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
x*y^2-y^3,0

----v------------
------------v------------
at homology degree: 0
Q[x,y]^(1 x 2)
---------------------Qx,]( x2java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
</pre></div>

Try a localization of a residue class--------------

<div class="example"><pre>
</iv>
Q[x,y]/( -x^3-<> a localization a residue ringpjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
gap> Display( R1 );
&t; residue ring
gap>[,]( x^-^+*^ )
<A 2 x2 matrixover residue ringgt;
gap;<span<spanclass">Display R1 );s>
< classGAPprompt>/> <pan="GAPinput">R10 := LocalizeAt R1/span
<pan="">&; HomalgRingElement( x, R1 ),
> HomalgRingElement( "y", R1 ) ]
<span=GAPprompt"> >gapgt/span> <spanclass"">( "");/pan
<span=""gap;<spanspanGAPinputDisplay ;span
<A local ring>
<span">> < class=>wmat10 : HomalgLocalMatrix( wmat R10 );
<A 2 x 2 matrix over a local ringgt
gap> W10span="GAPprompt">> HomalgRingElement( "y", R1 ]/span>
<A left module presented by 2 relations for 2 generators>
<span="GAPprompt">gapg;/> =GAPinput :Resolution( ,W10 ); s classGAPinputDisplayR10;</>
[ 0 .. 2 ]>
gap<A local ring>
-------------------------
at homology degree: 2
0
-------------------------
(an 0 x 2matrix

the map is <A 2 x 2 matrix o a local ring>
------------v------------
at degree 1
Q[x,y]/( x^3+x^2-2*lt forgenerators&t;
-------------------------
x*y^2+y^2,2*y^2,
y^2, y^4-2*y^3+2*y^2

modulo classGAPpromptgapgt < classGAPinputRes10 = ( );/>
/ |[ 1 ]|

the map islt; acyclic containing of modules degrees
--------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
at homology degree: 0
Q[x,y]/( -----------------------
-------------------------
</pre</div

<pat degree 2

<div classexamplepre
gap> Display( R0 );
<Alocal>
<c=GAPinputR01= R0 /(ec R0
Q[x,y]_ |1 ]|
gap> Display( R01 );
<A residue class ring>
gap map is represented above x matrix
<A 2 x 2 matrixhomology: 0
^(1 x2
<A left module/pre<div>

<A right acyclic complex containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
-------------------------
at class=>&t;/pan class">R0 : LocalizeAtZero( Qxy ;/span>
0
--------------
(an empty 0 x 2 matrix)

the map is currently represented by the above 0 x 2 matrix
-------
at homology&t;A local>
Q[x,y]_< x, y >/( >spanspan=GAPinputwmat0=R0 ;</span
-------------
y^3+y^2,2*y^2,
0, x*y^2-y^3
/ 1

modulo [ (x^3+x^2-2*y^2)/1 ]

the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Q[x,y]_< x, y >/( (x^3+x^2-2*y^2)/1 )^(1 x 2)
-spanclass="">gap;<spanspan="GAPinput"> := / ec );/span
</pre[,y]&; ,y gt;( -x^3-^2+2y^)/ )

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

<h4>5.4 Testing gap> <sclass="GAPprompt"gapgt/span>< class=""GAPinput>Display( );We"GAPinput>mat01 := R01 * ;

<div class="example"><pre>

gap>spanclass=""></> <spanclassGAPinputW01= LeftPresentationwmat01 )</>
<spanltA left presentedby2relations for 2 generators>
<spanclass"">gap>/span <pan classGAPinput>oadPackage "" );;
gap> R0 := LocalizePolynomialRingAtZeroWithMora&;A rightacycliccomplexcontaining2 morphisms of leftmodulesat
gap>Display( Res01 );
> (w-x^2)*y, \
> (w-x^2)*z, \
<spanclass="GAPprompt">>></><spanclass"">(^2y, \
> (x-w^2)*z \
> ]", 4, 1, R );;
<spanclass=GAPpromptgapgt<span< class="GAPinput"">2 = HomalgMatrix( "\
> (w-x^2)-y, \
> (x-w^2)-z \
<span="">>
gap> LoadPackage( "Modules" );;
gap> RmodI1 homology degree 1
<gap> span class"GAPinput>RmodI2 :=LeftPresentation ));;span>
gap> T:=Tor( RmodI1, RmodI2 );
d homologyobject of left modules at degrees .. 3 ]&tjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
<span="GAPprompt">gap>List ObjectsOfComplex( T) AffineDegree);
[ 2, 4 , 0 ]
</pre></div>

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

<div class="example"><pre>
gap[xy]< x, y >/( (x^3+x^2-2*y^2)/1 )^(1 x 2)
<A 4 x 1 matrix over a local (Mora) ring>
gap> M20-------------------
<A 2 x 1 matrix over a local (Mora) ring>
gap> R0modI10
gap> R0modI20
Pinput := Tor,R0modI20<spanclass="GAPinput">List( ObjectsOfComplex T0,AffineDegree ;
[ 3< classexample<>
</pre></div>

Theintersectionmultiplicity at zero 3-1=2</>

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