<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>
<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)[y_lt;,y &;/lt; (+^-xy-2x*^2y^+^/, (-y+x2y^-x^*y+^3\
x*y^3)/1, ( class=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 109
<class="APprompt>&;/span>spanclass="GAPinput>ByASmallerPresentation( java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 101
&; cyclic right module a generatorsatisfying 2 relationsgt;
< =Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 86
GF(3)[x,y]_< />/>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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="GAPprompty^3)1 --x2-java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 55
GF(3)[x,y]_< x, y >/< (y^4+x*y^2+y^3+x^2s">ByASmallerPresentation(IMora );</span>
^3x2*y+^-^-*+^
<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>
<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 /Here computethe(nfinite)long ofcovariantfunctor< class"impleMath>(Hom-R2^R,2^R<span>andleftderived )appliedto exact <java.lang.StringIndexOutOfBoundsException: Range [217, 216) out of bounds for length 340
<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=<pan class"GAPprompt"gap&t;/span>> class=GAPinputGlobalR :HomalgRingOfIntegersInExternalGAP );spanjava.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 118
<lt; ring java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 48
<spanclass="gap&;/spanclassGAPinput>LoadPackage "" );;/>
<span class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <spanGAPinput> : LocalizeAt( [ 2 ;/pan
_; gt
<span class="GAPpromptgap&t<><pan =">java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 82
&;Alocal&;
<span lt;A cyclic module presentedby 1relation for a cyclic generator>
<A cyclic left module presented by 1 relation for a cyclic generator<pan =GAPprompt>>/> <span class"APinput"_ =LeftPresentation(HomalgMatrix([ 2^ , R));/pan>
<pan=""g;/ pan class=G">M LeftPresentation ( [3] R ))<span>
<A cyclic left module presented by 1 relation for a cyclic generator>
s class">><><span =>: HomalgMapHomalgMatrix[1, ),M M);</>
<A &t; ""of gt;
< class=GAPprompt>></span><span class=GAPinput> : Kernel );by yet unknown relations for a cyclic generato\
<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, ,3 ]gt;
<span class="GAPprompt">gap></span> <span class="GAPinput">lehs := java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 38
< sequence containing14 modules java.lang.StringIndexOutOfBoundsException: Range [67, 65) out of bounds for length 81
< =>&;</span><pan class=GAPinput"B( lehs ;/java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
<A span ""ggt</span><span ""A( ;/>
[ 0 .. 14 =""&;</ <pan class=GAPinput">IsShortExactSequence( seq );</span>
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
rue
</pre></div>
<p><a id="X7CC8EA507E7AABA4" name="X7CC8EA507E7AABA4; cyclic left module presented by 1 relation for a cyclic generator>
<<A cyclic java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 73
<t; containing3morphisms of left complexes at degrees
<div class="example">p>
<class=>apgt</pan < class"(" &t; 20200417";<span>
<span class="GAPprompt">gap<A sequence containing 14 morphisms degrees [ . 14 &;
Q[x,y]
<span class="spanclass=GAPprompt"g></java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 100
<spanclass=>g;/> s""> [3y2,x3-x^2,y^+java.lang.StringIndexOutOfBoundsException: Range [98, 97) out of bounds for length 120
<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>
the map is java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 58
---< ="xample"><pre>
at homology : 1
Q[,]^ 2)
------------
y =gapgt;span> <pan class"APinput =HomalgMatrix(/pan>
x*y^2-y^3,0
the map is "">&;><class=G""[ y3y2 ^3x2 , ^3y^2 x3+x2"<span>
------------v------------
at span class"GAPprompt">gt<span><panclass="GAPinput"> 2, 2, Qxy );</span>
1 x )
---------------<pan =GAPprompt"gap;<span span class=GAPinput">ec := HomalgRingElement( "-x^3-x^2+2*y^2", Qxy );</span>
</pre></div>
< ajava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 50
<div class="example"><pre>
<span <span class">g;<> class""LoadPackage("odules");<s>
Q[s class="gapgt<span <span class"">W : LeftPresentation wmat )</java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
<classG"gap&t<span> <span class="GAPinput">Display( R1 );</span>
<A residue class <A right acyclic complex morphisms of left modules at degrees
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat1 := R1 * wmat;</span>
<A 2 <pan class"GAPprompt&;/pan span class="GAPinput"> HomalgRingElement( "y", R1 ) ]</span>
<<pan class=GAPprompt"&t;</span> <span class="GAPinput"> );</span>
<span x,(x^+22y2 )lt;|[x | |[y ] ] gt;
<span class="GAPprompt">&<span class="GAPprompt">gap/pan<span class="APinput"Display R10)/pan
<span class="panclass="GAPprompt">gap>/ span ">: , ;>
<spanclass=GAPprompt">&t;/> s "> <s>
Q[< =GAPpromptgap>/pan> <panclass="GAPinput"W10 />
< class"java.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 84
<A local ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">wmat10 := HomalgLocalMatrix( wmat, R10 );</span>
;2 x 2over a ring&java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
<class">>span>< ="">W10= (java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 103
&;A java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 63
<span at degree java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
< empty x 2matrix
[ 0 .. 2 ]>
< class="GAPinput>( Res10 <spanjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
--------------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
degree java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 1
-------------------------
(an
the [ |
-----------the map is currently represented the above 2 x 2 matrix
at homology degree: 1--
Q[x,y]/( x^3+x^2-2*y^2 )_< |[
-------Q[x,/(x3x^22y2 )&t; [ x ],[ y | g;( x )
x*y^2+y^2,2*y^-------------
y^2, y^4-2*y^3+
modulo x^+x^2-2*^
/ |java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
the map is currently represented by the above 2 x 2 matrix
--------ss=G>;span class="GAPinput">Display( R0 );</span>
at homology degree: 0
&t localring&tjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
-------------------------
</pre></div>
<p;java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 40
<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>
<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="chlinkbot"><span class="chlink1"><pan class="APprompt">ap&t<span> <class=""R :=LeftPresentation( ;<java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
<hr />
<p class="lt;A graded homologyobject of 4 at [0 .3 ]gt;
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