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<h5>6.3-10 IsTriangle</h5>

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<h5>6

<class=func> class"width100"<body>
<p>Returns: a <strong class="pkg">homalg</strong> complex</p>

<p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkg">homalg</strong> (static) objects.</p>

<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
        "for homalg complexes",
        [ IsHomalgComplex ],

  function( C )

    List( ObjectsOfComplex( C ), ByASmallerPresentation );

    if Length( ObjectDegreesOfComplex( C ) ) > 1 then
        List( MorphismsOfComplex( C ), DecideZero );
    fi;

    IsZero( C );

    return C;

end );
</pre></div>

<p>This method performs side effects on its argument <var class="Arg">C</var> and returns it.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span class="GAPprompt">gap></span> <span class="GAPinput">M := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, zz );</span>
<A 2 x 3 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := LeftPresentation( M );</span>
<A non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ 2, 3, 4, 5,   6, 7, 8, 9 ]", 2, 4, zz );</span>
<A 2 x 4 matrix over an internal ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := LeftPresentation( N );</span>
<A non-torsion left module presented by 2 relations for 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := HomalgMatrix( "[ \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span>
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<h5>6p>Checkif the <strong class="pkg">homalg</strong> complex

<div class="func"><table span class="GAPprompt">gap>span><span class=GAPinput">N := HomalgMatrixAlt([ 1. 5 ])
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<h5>6.3-6 IsGradedObject</h5>

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<h5>6.3-IsExactSequence</h5>

<div class="func"><h5>6.3-7 IsExactSequence
<piv/pre></div

<p>Check if 76FD2367E64462D"name"X876FD2367E64462D">/>/p>

<p><a id="X87ADD4F685457000" name="

<h5>p>Returns: Thenumerical description of  outputof<code class="func">SimpleAlgebraByCharacter</code> (<a href="chap2.java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 0

<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>eturns <code class=code>true</ode>or<code class=code"false<</code></p>

<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<><

<6 nonzero graded cohomologyobjectconsisting asingle rightmodule at de\

<div class="lt;pc group of size with 7java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 10
<eclasscode">true</codeanclass=GAPinput">G:=GroupRing(Rationals,G);/span>

<p>Check <pan class="span class="GAPprompt>gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<&l; cocomplexcontaininga singlemorphism of[[ 4

<h5>6.3-10[ 4 NF(, 1,3])]]

<div class="func">/pre>/iv>>
<p>Returns: <code class="

<p>Settotrue if the <trong class="kg">homalg<strong>complex var class=ArgCphispan>

<p><a <A non>2-SimpleAlgebraByStrongSP</h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>.-11<pre>/v

< class="unc>able  java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 222
<Returns: <code java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0

<p>Check if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is an exact trianglejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>a id="<h4>6.3<span class="eading"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>. <panclass="eading">Complexes: Attributes</span></h4>

<p><a id="X7DE6E8D8875B515F" <p>Returns: A simple algebra.</p

<h5>6.4-1 BettiTable</h5>

<divhep>a=func>able =unc" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSequence</code>( <var java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: a <strong java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0

<p>The Betti diagram of the <

<p><a id="X80EDFDD281834882" name="X80EDFDD281834882"></a></p>

<56.

<div span class=h5>6.3- IsComplex<>
<p Betti diagram  of the<class="kg">omalg/strong complex<var class=Arg"C/var>  graded .<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The filtration<:code

<p

<h4>6.5 <span classrossedproduct

<p><a id="X7F10893B78FEDEB7

<h5>6.5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class=func"<table class=func width="
<div="100%"<tr"dleft"<codejava.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 239
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p>In

<p>

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<"GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Z
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
&;
<pan class=GAPprompt>gap&>/span><span class"GAPinput"phi
<A
<p> thethree-argument version the input must be formed by a <em>semisimple rational group algebra
<A non-zero acyclicp> versions ending in NC do notcheck if(
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 20
<span class="APprompt">gap></span> <span
<span class=GAPprompt"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
&tAck if the <strong class="pkg"><panclass="APprompt">gap></span> <span class="GAPinput">G:=F/[ a^16, b^2*a^8, b
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
-------------------------
at homology degree: 2
x2java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

[ [  0,  1 ],
  [  0,  0 ] ]

the  iscurrently java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 20
--------v------,NF(6,span class"APprompt"gap<span>><span class="GAPinput">Add( C, mat );</span>
at homology degree: 1
Z^(1 x 2)
-------------------------
[ [  p>Returns: <code class=<pansequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
  [0,  0 ]> 2)

the map is currently represented by the ------
---------v
at homology degree: 0
Z^(1 x 2)
-------------------------
<span class="GAPprompt">p>iv class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a hrefjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
< classGAPprompt"gap&<span><span class="GAPinput">IsAcyclic( C );</span>
true
<pan class=-----------
false
<spanclass=GAPprompt>gap&t<span> <span class="GAPinput">C  0, 0 ] ]
&;Anon-zero acyclic complex java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0
[0 . 2 ]gt;
/pre</div>

<p><a ids:<code class= class="code">true</Z^1x )

<ByASmallerPresentation/>

<class="unc><able class="func" width="100%"><tr><
<>Returns: a <strong class=pkg"homalg</strong> complex</p>

<p>It invokes <code class="code">ByASmallerPresentation</code> for <strong class="pkgtrue

<div class="example"><pre>
InstallMethod( ByASmallerPresentation,
        "for homalg complexes",
         IsHomalgComplex ],

  function( C )

    List( ObjectsOfComplex( C ), <A non-zero acyclic complex containing 2 of  modules at degrees

    if 2 ]gtjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
        Listp>heckif the <strong class="pkg">homalg</strong> complex <var class="Arg">java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
    fi;

    IsZero( C );

    return java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

end );
</pre></div>

<p>This method performs side effects on its argument <var class="Arg">C</java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 0

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
Z
<span functionC )
<A 2 x 3     List( ObjectsO ObjectsOfComplex),ByASmallerPresentation)java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
<span class==GAPprompt"gapgt;span <span class=GAPinput"> : LeftPresentation( M );</span>
<A non-torsion left module presented by 2 relations for 3 generators>
<span class="GAPprompt">gap></span> <List( MorphismsOfComplex( C ), DecideZero
<   4matrix
< ="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 4
<A non-torsion left module<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [1, 30) out of bounds for length 26
< class"APprompt"gap&;</pan> span class="GAPinput">zz := HomalgRingOfIntegers( );</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 2, 4, 6, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">0, 3, 6, 9  \</span>
<span class="GAPprompt">><pSet totrue if the <strong class="pkg">homalg</strong> complex <var class="Arg">C</var> is a triangle.</p>
&t; 3 x 4 matrix over an
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<pan class=GAPprompt>ap;</>span classG">M :=LeftPresentation( M );</spanjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
<rompt">&t;</an><span class="GAPinput>IsMorphism phi);<span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">phi;</span>
<Alt; 2 x 4 matrix over aninternalring&t
<span class"APprompt">gap></span> <span class="GAPinput">C := HomalgComplex( phi );</span>
<nonzeroacyclic complex containing a single morphism of left modules at de\
greesh4>.4 <span class="Heading">Complexes: Attributes</span></h4>
<spanp>aid=X7DE6E8D8875B515F name="7DE6E8D8875B515F"></a></p>
-------------------------
athomologydegree:1
[ [  2,  3,  4 ],
  [ 5, 6  7 ]]

Cokernel ofthe map

Z^(1java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

currently represented by the above matrix
-------------------------
[ [  0,  3,  6,  9 ],
  [  0,  2, 4,6 ,
  [0,  3,  6  9] ]

<spclass"APprompt>gt;<
-----------<5>.- class"">gt</pan <span  class"">,3 6, 9\/java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
at homologya <strong class="pkg">homalg</strong> diagram</p>
[ [  2,  3,  4,  5 ],
  [>< idX84E12E9C7A60D9BCname=X84E12E9C7A60D9BC>/>/p>

Cokernel>6. <pan class="Heading">Complexes: Operations and Functions</span></h4>

Z^x2) -gt

div class="func"><table class="func" width="100%"><tr&t;   >java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
----------------
</pre></div>

<p>And now:</p>

<div class="example"><pre>
<span class="GAPpromptspan class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <span=GAPinput">zz := HomalgRingOfIntegers( );</---------
&----------java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
grees  .1 ]&t
<span class="GAPprompt">GAPprompt">gap></span> <span class="GAPinput">Add( C, mat );</span>
-[,  ,6    java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
atdegree:1
java.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 25
----------
[ 0 , 0 ,
    ,0  0 ] ]

he is currently represented bythe
-----
at homology degree:
Z/< 4 &t +Z^(  2)
-----------  java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
</ [  0 homologydegree0

<="&

class=""<class="chlink1">oto Chapter </span<a

<hr />
<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</body>
</html>

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