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<div class="ChapSects"><a href="chap5.html#X7A489A5D79DA9E5C">5 Examples</a>
<div class="ContSect"> java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 6

</div>
<div class="ContSect"> <a
</pan>
</div "mathjaxlink ""<java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 80
<div class="ContSect"> <a href="chap5.html#X7CC8EA507E7AABA4">5.3 ResidueClass</a>
 href=.#">. s ="> the </>/ajava.lang.StringIndexOutOfBoundsException: Index 185 out of bounds for length 185

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<h3>5 

<>a id"" name=java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62

<h4>5.1 

<<s

<div classjava.lang.StringIndexOutOfBoundsException: Index 185 out of bounds for length 185
<pan class=G"&ap&;<span<pan "" :=java.lang.StringIndexOutOfBoundsException: Range [103, 100) out of bounds for length 120
<<span">apgt<s>< ="y0 : ( "" );<>
<spanclass=GAPpromptgap& <pan="" =HomalgRingElement"" );<s>
gap> LoadPackage("LocalizeRingForHomalg");;spanclass="g&;sclass="GAPinput>00 : LocalizeAtZero F3xy );/span
< class"GAPprompt>gap&;/pan < "APinputR :( ;</spanjava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
< 
 < ="">M =HomalgMatrix("[<>
gap> <<pan class"GAPprompt"gt;<spansclass=" ],1 ,F3xy);/java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
<spanclass">&<s java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 107
> x*y^3+2*x^2*y+y^3+y^2+x+2*y+x^2, \
gap&t;s><class"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 105
<pan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 89
<spanclass=GAPprompt>/><panclass">oadPackage");<>
gap> I := RightPresentation( M );;
gap&gap>"GAPprompt>& : java.lang.StringIndexOutOfBoundsException: Range [90, 88) out of bounds for length 106
gap> MMora := HomalgLocalMatrix( M, RMora );;
< class=""(IMora );
gap> gap> Display( IMora );
<pre>/div>

This ring is able to compute a standard basis java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0

<div class="example"><pre>

<=GAPinput" java.lang.StringIndexOutOfBoundsException: Range [93, 91) out of bounds for length 101
<A cyclic torsion+^+3x2xy+2-y)/1 >
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).

we i exacthomologysequence the span =S"Hom(-,/2^7)R/^4)<> ( its derived functors)applied totheshort sequencebr /> 0 -> M_=R/2^2R --alpha_1--> M=R/2^5R --alpha_2--> _M=R/2^3R -> 0.

<div class="example"><pre>
gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class=GAPprompt>g;/span : HomalgRingOfIntegersInExternalGAP( ;/>
Z
gap> Display( GlobalR );
&An external ring residing in the CAS GAP>
gap> < class="GAPinput"( RingsForHomalg);/span
< class="APinput"> :LocalizeAt GlobalR , ])</>
Z_< 2 >< 2&;
">gap&t;/span sclass"GAPinput"Display( R );
<A local ring>
gap> gap>
<A cyclic left modules class"APprompt"gap&t s ="APinput_ :=LeftPresentation(HomalgMatrix 2^3 ,R ;/java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 122
gap><pan class=GAPprompt">ap>/span span class=GAPinput"alpha2 =HomalgMap( HomalgMatrix( ],R , , _ )</pan
&A"omomorphism"leftmodules&;
<<spanclass""gap;/>span ""M_ =( alpha2)
<A cyclic left module presented java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
r>
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
<A monomorphism of left modules>
gap> seq := HomalgComplex( alpha2;A sequencecontaining 14 morphismsofleftmodulesat degrees [ 0 .. 14 ]>
&spanclass=GAPprompt"gap>< <pan "">yASmallerPresentation ))
[ 0 .. 1 ]>
<spanclass=GAPprompt">ap&;/span><class=GAPinput>dd(seq,alpha1 )gap>/pan><pan "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 97
true
gap> K := LeftPresentation( HomalgMatrixtjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<A java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 75
gap> 
ap>/pan> <> <pan class="GAPinput"LoadPackage "RingsForHomalg,"&; 2020..17 );/span
& of left modulesat[0.14]gtjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
apgt ByASmallerPresentation( lehs );
<A nonspan ="APprompt"&t " y^3-^ ^3x2 , 3y^2 , x^3+x^2 ]",
[ 0 .. 14 ]>
<classG"gapgt</pan>class=">java.lang.StringIndexOutOfBoundsException: Range [78, 77) out of bounds for length 93
true
<>div>

<7E7AABA4="7java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<h4>5.( x x 2matrix

We want the map is currentlycurrently represented by the above 0 x 2 matrix

<div class""><pre>
<atdegree
ap&t;/ sclass="">wmat : <sjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
="APprompt>&t;/span span =APinput" ^-y^,x-x^ y+^2, ^3+^ ],/java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 120
<=GAPprompt>;< s java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 88
<A 2 1 22)
<class=>>/>< class="java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 118
-x^3-x^2+2*y^2
</pre></div>

<<pTry a localization of a residue class ring:

<div class="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<="GAPprompt>gap&t(M );/pan>
<pan="APprompt"gap&;/span> <="GAPinput">W :(wmat)span>
<A left module presented by 2 relations for 2 generators>
gap><pan class="APprompt>apg;</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 83
&containing 2 java.lang.StringIndexOutOfBoundsException: Range [53, 50) out of bounds for length 77
[ 0 .. 2 ]>
gap> Display( Res );
-------------------------
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>

Try a localization of a residue class ring:

<div class="example"><pre>
gap> R1 := Qxy / ec;
Q[x,y]/( -x^3-x^2+2*y^2 )
gap> Display( R1 );
<A residue class ring>
gap> wmat1 := R1 * wmat;
<A 2 x 2 matrix over a residue class ring>
gap> LoadPackage( "LocalizeRingForHomalg" );;
gap> R10 := LocalizeAt( R1 ,
> &t<s><java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 107
<pan class">&java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 75
Q[,y/ ^+^-*y^ ))< |x ],|[y| &java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
<gt;</pan> ="">( R10 )<s>
<A local ring>
<pan =GAPprompt>gap<class=GAPinput"wmat10 =HomalgLocalMatrix(wmat,R10)&t;span> <panclass=GAPinput" );</pan>
<panclass"">&</pans ="GAPinput":=LeftPresentation(wmat10);
<A leftgap> Display( R10 );
<java.lang.StringIndexOutOfBoundsException: Range [51, 5) out of bounds for length 110
<A 2x2 matrix overa local >gt;
[ 0 .. 2 ]>
gap><pan class=GAPprompt"gap; := LeftPresentation wmat10 );
--------------------<A left module presented by 2 relations for 2 generators>
at homology degree:2
0
-------------------------
(an 0 22 )

the map is currently represented by the span class="GAPprompt">gap> <span">isplay( Res10 );</pan>
------------v-------------------
at homology degree: 1
Q[x,y]/( x^ :2
-------------------------
x*y^2
y^2, y^4-2*y^3+2*y^2

modulo
/| 1 ]

themap bythejava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 58
--java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
java.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 21
Q[y]/ ^+x-*^2 )&|[],| ] &t^12java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
--------------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 12

Try a[x^+x^*2]

<div class="example"><pre>
java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
Q[x,y]_< x, y >
<ss="APprompt">ap>/span><spanjava.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 83
&;A local ring&t;
gap> wmat0 := R0 * wmat;</java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 25
tA 2 x 2 matrix over a local ring>
gap> R01 := R0 / ( ec / R0 );
Q[x,y]_< x, y >/( (-x^3-x^2+2*y^2)/1 )
gap> Display( R01 );
<A residue class ring>
gap> wmat01 := R01 * wmat0;
<A 2 x 2 matrix over a residue class ring>
gap> W01 := LeftPresentation( wmat01 );
<A left module presented by 2 relations for 2 generators>
gap> Res01 := Resolution( 2 , W01 );
<A right acyclic complex containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
gap> Display( Res01 );
-------------------------
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>

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

<h4>5.4 Testing the Intersection Formula</h4>

We want to check Serre's intersection formula i(I_1, I_2; 0)=∑_i(-1)^i length(Tor^R_0_i(R_0/I_1,R_0/I_2)) on an easy affine example.

<div class="example"><pre>

gap> LoadPackage( "RingsForHomalg" );;
gap> R := HomalgFieldOfRationalsInSingular() * "w,x,y,z";;
gap> LoadPackage( "LocalizeRingForHomalg" );;
gap> R0 := LocalizePolynomialRingAtZeroWithMora( R );;
gap> M1 := 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, <panclassjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<">gap&t;s>< java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 95
<="APpromptgapg;/> span =GAPinput>modI1: (M1);/span>
gap> RmodI2 := LeftPresentation( M2 );;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
& objectconsisting 44left modulesat degrees [0. 3 ]java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
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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