pkghomalg.gi

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##
##  simpcomp / pkghomalg.gi
##
##  Loaded when package `homalg' is available
##
##  $Id$
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##<#GAPDoc Label="SCHomalgBoundaryMatrices">
## <ManSection>
## <Meth Name="SCHomalgBoundaryMatrices" Arg="complex,modulus"/>
## <Returns>a list of <Package>homalg</Package> objects upon success, 
## <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the boundary operator matrices for the simplicial 
## complex <Arg>complex</Arg> with a ring of coefficients as specified by 
## <Arg>modulus</Arg>: a value of <C>0</C> yields <M>\mathbb{Q}</M>-matrices, 
## a value of <C>1</C> yields <M>\mathbb{Z}</M>-matrices and a value of 
## <C>q</C>, q a prime or a prime power, computes the 
## <M>\mathbb{F}_q</M>-matrices.<P/>
## <Example>
## gap&gt; SCLib.SearchByName("CP^2 (VT)");
## [ [ 16, "CP^2 (VT)" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgBoundaryMatrices(c,0);
## [ &lt;A 36 x 9 mutable matrix over an internal ring&gt;, 
##   &lt;A 84 x 36 mutable matrix over an internal ring&gt;, 
##   &lt;A 90 x 84 mutable matrix over an internal ring&gt;, 
##   &lt;A 36 x 90 mutable matrix over an internal ring&gt;, 
##   &lt;An unevaluated 0 x 36 zero matrix over an internal ring&gt; ]
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCHomalgCoboundaryMatrices">
## <ManSection>
## <Meth Name="SCHomalgCoboundaryMatrices" Arg="complex,modulus"/>
## <Returns>a list of <Package>homalg</Package> objects upon success, 
## <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the coboundary operator matrices for the simplicial 
## complex <Arg>complex</Arg> with a ring of coefficients as specified by 
## <Arg>modulus</Arg>: a value of <C>0</C> yields <M>\mathbb{Q}</M>-matrices, 
## a value of <C>1</C> yields <M>\mathbb{Z}</M>-matrices and a value of 
## <C>q</C>, q a prime or a prime power, computes the 
## <M>\mathbb{F}_q</M>-matrices.<P/>
## <Example>
## gap&gt; SCLib.SearchByName("CP^2 (VT)");
## [ [ 16, "CP^2 (VT)" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgCoboundaryMatrices(c,0);
## [ &lt;A 9 x 36 mutable matrix over an internal ring&gt;, 
##   &lt;A 36 x 84 mutable matrix over an internal ring&gt;, 
##   &lt;A 84 x 90 mutable matrix over an internal ring&gt;, 
##   &lt;A 90 x 36 mutable matrix over an internal ring&gt;, 
##   &lt;An unevaluated 36 x 0 zero matrix over an internal ring&gt; ]
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCHomalgHomology">
## <ManSection>
## <Meth Name="SCHomalgHomology" Arg="complex,modulus"/>
## <Returns>a list of integers upon success, <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the ranks of the homology groups of 
## <Arg>complex</Arg> with a ring of coefficients as specified by 
## <Arg>modulus</Arg>: a value of <C>0</C> computes the 
## <M>\mathbb{Q}</M>-homology, a value of <C>1</C> computes the 
## <M>\mathbb{Z}</M>-homology and a value of <C>q</C>, q a prime or a 
## prime power, computes the <M>\mathbb{F}_q</M>-homology ranks.<P/>
## Note that if you are interested not only in the ranks of the homology 
## groups, but rather their full structure, have a look at the function 
## <Ref Meth="SCHomalgHomologyBasis" />.
## <Example>
## gap&gt; SCLib.SearchByName("K3");
## [ [ 7648, "K3_16" ], [ 7649, "K3_17" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgHomology(c,0);
## #I  SCHomalgHomologyOp: Q-homology ranks: [ 1, 0, 22, 0, 1 ]
## [ 1, 0, 22, 0, 1 ]
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCHomalgHomologyBasis">
## <ManSection>
## <Meth Name="SCHomalgHomologyBasis" Arg="complex,modulus"/>
## <Returns>a <Package>homalg</Package> object upon success, <K>fail</K> 
## otherwise.</Returns>
## <Description>
## This function computes the homology groups (including explicit bases of 
## the modules involved) of <Arg>complex</Arg> with a ring of coefficients 
## as specified by <Arg>modulus</Arg>: a value of <C>0</C> computes the 
## <M>\mathbb{Q}</M>-homology, a value of <C>1</C> computes the 
## <M>\mathbb{Z}</M>-homology and a value of <C>q</C>, q a prime or a prime 
## power, computes the <M>\mathbb{F}_q</M>-homology groups.<P/>
## The <M>k</M>-th homology group <C>hk</C> can be obtained by calling 
## <C>hk:=CertainObject(homology,k);</C>, where <C>homology</C> is the 
## <Package>homalg</Package> object returned by this function. The 
## generators of <C>hk</C> can then be obtained via 
## <C>GeneratorsOfModule(hk);</C>.<P/>
## Note that if you are only interested in the ranks of the homology groups, 
## then it is better to use the funtion <Ref Meth="SCHomalgHomology" /> 
## which is way faster.
## <Example>
## gap&gt; SCLib.SearchByName("K3");
## [ [ 7648, "K3_16" ], [ 7649, "K3_17" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgHomologyBasis(c,0);
## #I  SCHomalgHomologyBasisOp: constructed Q-homology groups.
## &lt;A graded homology object consisting of 5 left modules at degrees [ 0 .. 4 ]&gt;
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCHomalgCohomology">
## <ManSection>
## <Meth Name="SCHomalgCohomology" Arg="complex,modulus"/>
## <Returns>a list of integers upon success, <K>fail</K> otherwise.</Returns>
## <Description>
## This function computes the ranks of the cohomology groups of 
## <Arg>complex</Arg> with a ring of coefficients as specified by 
## <Arg>modulus</Arg>: a value of <C>0</C> computes the 
## <M>\mathbb{Q}</M>-cohomology, a value of <C>1</C> computes the 
## <M>\mathbb{Z}</M>-cohomology and a value of <C>q</C>, q a prime or a 
## prime power, computes the <M>\mathbb{F}_q</M>-cohomology ranks.<P/>
## Note that if you are interested not only in the ranks of the cohomology 
## groups, but rather their full structure, have a look at the function 
## <Ref Meth="SCHomalgCohomologyBasis" />.
## <Example>
## gap&gt; SCLib.SearchByName("K3");
## [ [ 7648, "K3_16" ], [ 7649, "K3_17" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgCohomology(c,0);
## #I  SCHomalgCohomologyOp: Q-cohomology ranks: [ 1, 0, 22, 0, 1 ]
## [ 1, 0, 22, 0, 1 ]
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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##<#GAPDoc Label="SCHomalgCohomologyBasis">
## <ManSection>
## <Meth Name="SCHomalgCohomologyBasis" Arg="complex,modulus"/>
## <Returns>a <Package>homalg</Package> object upon success, <K>fail</K> 
## otherwise.</Returns>
## <Description>
## This function computes the cohomology groups (including explicit bases of 
## the modules involved) of <Arg>complex</Arg> with a ring of coefficients as 
## specified by <Arg>modulus</Arg>: a value of <C>0</C> computes the 
## <M>\mathbb{Q}</M>-cohomology, a value of <C>1</C> computes the 
## <M>\mathbb{Z}</M>-cohomology and a value of <C>q</C>, q a prime or a prime 
## power, computes the <M>\mathbb{F}_q</M>-homology groups.<P/>
## The <M>k</M>-th cohomology group <C>ck</C> can be obtained by calling 
## <C>ck:=CertainObject(cohomology,k);</C>, where <C>cohomology</C> is the 
## <Package>homalg</Package> object returned by this function. The 
## generators of <C>ck</C> can then be obtained via 
## <C>GeneratorsOfModule(ck);</C>.<P/>
## Note that if you are only interested in the ranks of the cohomology groups, 
## then it is better to use the funtion <Ref Meth="SCHomalgCohomology" /> 
## which is way faster.
## <Example>
## gap&gt; SCLib.SearchByName("K3");
## [ [ 7648, "K3_16" ], [ 7649, "K3_17" ] ]
## gap&gt; c:=SCLib.Load(last[1][1]);;
## gap&gt; SCHomalgCohomologyBasis(c,0);
## #I  SCHomalgCohomologyBasisOp: constructed Q-cohomology groups.
## &lt;A graded cohomology object consisting of 5 left modules at degrees 
## [ 1 .. 5 ]&gt;
## </Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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