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<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>
<div class="ChapSects"><a href="chap5.html#X7A489A5D79DA9E5C">5 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8426A658837B4911">5.1 <span class="Heading">An Easy Polynomial Example</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7820475F7C884EA5">5.2 <span class="Heading">Hom(Hom(-,Z128),Z16)</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CC8EA507E7AABA4">5.3 <span class="Heading">ResidueClass</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7958E7417BB312F0">5.4 <span class="Heading">Testing the Intersection Formula</span></a>
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<h3>5 <span class="Heading">Examples</span></h3>
<p><a id="X8426A658837B4911" name="X8426A658837B4911"></a></p>
<h4>5.1 <span class="Heading">An Easy Polynomial Example</span></h4>
<p>The ground ring used in this example is <span class="SimpleMath">F_3[x,y]</span>. We want to see, how the different rings in this package can be used to localize at different points and how the results differ.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("RingsForHomalg");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F3xy := HomalgRingOfIntegersInSingular(3) * "x,y";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x1 := HomalgRingElement( "x+2", F3xy );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">y0 := HomalgRingElement( "y", F3xy );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("LocalizeRingForHomalg");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R00 := LocalizeAtZero( F3xy );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R10 := LocalizeAt( F3xy, [ x1, y0 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RMora := LocalizePolynomialRingAtZeroWithMora( F3xy );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[\</span>
<span class="GAPprompt">></span> <span class="GAPinput"> y^3+2*y^2+x+x^2+2*x*y+y^4+x*y^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x*y^3+2*x^2*y+y^3+y^2+x+2*y+x^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x^2*y^2+2*x^3+x^2*y+y^3+2*x^2+2*x*y+y^2+2*y\</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ]", 1, 3, F3xy );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "Modules" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I := RightPresentation( M );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M00 := HomalgLocalMatrix( M, R00 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M10 := HomalgLocalMatrix( M, R10 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MMora := HomalgLocalMatrix( M, RMora );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I00 := RightPresentation( M00 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">I10 := RightPresentation( M10 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IMora := RightPresentation( MMora );;</span>
</pre></div>
<p>This ring is able to compute a standard basis of the module.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( IMora );</span>
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 >
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( IMora );</span>
<A cyclic torsion right module on a cyclic generator satisfying 2 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( IMora );</span>
GF(3)[x,y]_< x, y >/< x/1, y/1 >
</pre></div>
<p>This ring recognizes, that the module is not zero, but is not able to find better generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I00 );</span>
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 >
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( I00 );</span>
<A cyclic right module on a cyclic generator satisfying 3 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I00 );</span>
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>
<p>We are able to change the ring, to compute a nicer basis.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">I00ToMora := RMora * I00;</span>
<A cyclic right module on a cyclic generator satisfying 3 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I00ToMora );</span>
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 >
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( I00ToMora );</span>
<A cyclic torsion right module on a cyclic generator satisfying 2 relations>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I00ToMora );</span>
GF(3)[x,y]_< x, y >/< x/1, y/1 >
</pre></div>
<p>We are able to find out, that this module is actually zero.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I10 );</span>
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 >
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( I10 );</span>
<A zero right module>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( I10 );</span>
0
</pre></div>
<p><a id="X7820475F7C884EA5" name="X7820475F7C884EA5"></a></p>
<h4>5.2 <span class="Heading">Hom(Hom(-,Z128),Z16)</span></h4>
<p>The following example is taken from Section 2 of <a href="chapBib.html#biBBREACA">[BR06]</a>. <br /> <br /> The computation takes place over the local ring <span class="SimpleMath">R=ℤ_⟨ 2⟩</span> (i.e. ℤ localized at the maximal ideal generated by <span class="SimpleMath">2</span>).</p>
<p>Here we compute the (infinite) long exact homology sequence of the covariant functor <span class="SimpleMath">Hom(Hom(-,R/2^7R),R/2^4R)</span> (and its left derived functors) applied to the short exact sequence<br /> <br /> <span class="SimpleMath">0 -> M_=R/2^2R --alpha_1--> M=R/2^5R --alpha_2--> _M=R/2^3R -> 0</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "LocalizeRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GlobalR := HomalgRingOfIntegersInExternalGAP( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( GlobalR );</span>
<An external ring residing in the CAS GAP>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "RingsForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := LocalizeAt( GlobalR , [ 2 ] );</span>
Z_< 2 >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R );</span>
<A local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( HomalgMatrix( [ 2^5 ], R ) );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">_M := LeftPresentation( HomalgMatrix( [ 2^3 ], R ) );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := HomalgMap( HomalgMatrix( [ 1 ], R ), M, _M );</span>
<A "homomorphism" of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">M_ := Kernel( alpha2 );</span>
<A cyclic left module presented by yet unknown relations for a cyclic generato\
r>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha1 := KernelEmb( alpha2 );</span>
<A monomorphism of left modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">seq := HomalgComplex( alpha2 );</span>
<A "complex" containing a single morphism of left modules at degrees
[ 0 .. 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Add( seq, alpha1 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsShortExactSequence( seq );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">K := LeftPresentation( HomalgMatrix( [ 2^7 ], R ) );</span>
<A cyclic left module presented by 1 relation for a cyclic generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := RightPresentation( HomalgMatrix( [ 2^4 ], R ) );</span>
<A cyclic right module on a cyclic generator satisfying 1 relation>
<span class="GAPprompt">gap></span> <span class="GAPinput">triangle := LHomHom( 4, seq, K, L, "t" );</span>
<An exact triangle containing 3 morphisms of left complexes at degrees
[ 1, 2, 3, 1 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">lehs := LongSequence( triangle );</span>
<A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">ByASmallerPresentation( lehs );</span>
<A non-zero sequence containing 14 morphisms of left modules at degrees
[ 0 .. 14 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsExactSequence( lehs );</span>
true
</pre></div>
<p><a id="X7CC8EA507E7AABA4" name="X7CC8EA507E7AABA4"></a></p>
<h4>5.3 <span class="Heading">ResidueClass</span></h4>
<p>We want to show, how localization can work together with residue class rings.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "RingsForHomalg", ">= 2020.04.17" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Qxy := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y";</span>
Q[x,y]
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat := HomalgMatrix(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "[ y^3-y^2 , x^3-x^2 , y^3+y^2 , x^3+x^2 ]",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> 2, 2, Qxy );</span>
<A 2 x 2 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">ec := HomalgRingElement( "-x^3-x^2+2*y^2", Qxy );</span>
-x^3-x^2+2*y^2
</pre></div>
<p>Compute globally:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "Modules" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W := LeftPresentation( wmat );</span>
<A left module presented by 2 relations for 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Res := Resolution( 2 , W );</span>
<A right acyclic complex containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Res );</span>
-------------------------
at homology degree: 2
0
-------------------------
(an empty 0 x 2 matrix)
the map is currently represented by the above 0 x 2 matrix
------------v------------
at homology degree: 1
Q[x,y]^(1 x 2)
-------------------------
y^2, x^2,
x*y^2-y^3,0
the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Q[x,y]^(1 x 2)
-------------------------
</pre></div>
<p>Try a localization of a residue class ring:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R1 := Qxy / ec;</span>
Q[x,y]/( -x^3-x^2+2*y^2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R1 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat1 := R1 * wmat;</span>
<A 2 x 2 matrix over a residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "LocalizeRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R10 := LocalizeAt( R1 ,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ HomalgRingElement( "x", R1 ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> HomalgRingElement( "y", R1 ) ]</span>
<span class="GAPprompt">></span> <span class="GAPinput"> );</span>
Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R10 );</span>
<A local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat10 := HomalgLocalMatrix( wmat, R10 );</span>
<A 2 x 2 matrix over a local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W10 := LeftPresentation( wmat10 );</span>
<A left module presented by 2 relations for 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Res10 := Resolution( 2 , W10 );</span>
<A right acyclic complex containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Res10 );</span>
-------------------------
at homology degree: 2
0
-------------------------
(an empty 0 x 2 matrix)
the map is currently represented by the above 0 x 2 matrix
------------v------------
at homology degree: 1
Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >^(1 x 2)
-------------------------
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^2 ]
/ |[ 1 ]|
the map is currently represented by the above 2 x 2 matrix
------------v------------
at homology degree: 0
Q[x,y]/( x^3+x^2-2*y^2 )_< |[ x ]|, |[ y ]| >^(1 x 2)
-------------------------
</pre></div>
<p>Try a residue class ring of a localization:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R0 := LocalizeAtZero( Qxy );</span>
Q[x,y]_< x, y >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R0 );</span>
<A local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat0 := R0 * wmat;</span>
<A 2 x 2 matrix over a local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">R01 := R0 / ( ec / R0 );</span>
Q[x,y]_< x, y >/( (-x^3-x^2+2*y^2)/1 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( R01 );</span>
<A residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat01 := R01 * wmat0;</span>
<A 2 x 2 matrix over a residue class ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">W01 := LeftPresentation( wmat01 );</span>
<A left module presented by 2 relations for 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Res01 := Resolution( 2 , W01 );</span>
<A right acyclic complex containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Res01 );</span>
-------------------------
at homology degree: 2
0
-------------------------
(an empty 0 x 2 matrix)
the map is currently represented by the above 0 x 2 matrix
------------v------------
at homology degree: 1
Q[x,y]_< x, y >/( (x^3+x^2-2*y^2)/1 )^(1 x 2)
-------------------------
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)
-------------------------
</pre></div>
<p><a id="X7958E7417BB312F0" name="X7958E7417BB312F0"></a></p>
<h4>5.4 <span class="Heading">Testing the Intersection Formula</span></h4>
<p>We want to check Serre's intersection formula <span class="SimpleMath">i(I_1, I_2; 0)=∑_i(-1)^i length(Tor^R_0_i(R_0/I_1,R_0/I_2))</span> on an easy affine example.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "RingsForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInSingular() * "w,x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "LocalizeRingForHomalg" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R0 := LocalizePolynomialRingAtZeroWithMora( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M1 := HomalgMatrix( "[\</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (w-x^2)*y, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (w-x^2)*z, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (x-w^2)*y, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (x-w^2)*z \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ]", 4, 1, R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M2 := HomalgMatrix( "[\</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (w-x^2)-y, \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (x-w^2)-z \</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ]", 2, 1, R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "Modules" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RmodI1 := LeftPresentation( M1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RmodI2 := LeftPresentation( M2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=Tor( RmodI1, RmodI2 );</span>
<A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( ObjectsOfComplex( T ), AffineDegree );</span>
[ 12, 4, 0, 0 ]
</pre></div>
<p>We read, that the intersection multiplicity is 12-4=8 globally.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M10 := R0 * M1;</span>
<A 4 x 1 matrix over a local (Mora) ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M20 := R0 * M2;</span>
<A 2 x 1 matrix over a local (Mora) ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">R0modI10 := LeftPresentation( M10 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R0modI20 := LeftPresentation( M20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T0 := Tor( R0modI10, R0modI20 );</span>
<A graded homology object consisting of 4 left modules at degrees [ 0 .. 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( ObjectsOfComplex( T0 ), AffineDegree );</span>
[ 3, 1, 0, 0 ]
</pre></div>
<p>The intersection multiplicity at zero is 3-1=2.</p>
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