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</span><a href="chap8_mj.html#X7C705F2A79F8E43 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X81A2D49D8592389 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7C8B72157BDE095 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X873C632A83DF48E <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8018953F7951F8F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X81BD4EEA831D4A5 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X78CA656D7D4F244 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X824D52847879AB9 </div></div> </div> <h3>8 <span class="Heading">Homological algebra</span></h3> <p>This chapter describes the homological algebra that is implemented in QPA.</p> <p>The example used throughout this chapter is the following.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Q := Quiver(3,[[1,2,"a"],[1,2,"b" <span class="GAPprompt">></span> <span class="GAPinput">[3,1,"e"]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">KQ := PathAlgebra(Rationals, Q);;</ <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(KQ);;</ <span class="GAPprompt">gap></span> <span class="GAPinput">rels := [d*e,c^2,a*c*d-b*d,e*a];;< <span class="GAPprompt">gap></span> <span class="GAPinput">A := KQ/rels;;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">mat := [["a",[[1,2],[0,3],[1,5]]] <span class="GAPprompt">></span> <span class="GAPinput">["c",[[0,0],[1,0]]],["d",[[1,2],[0 <span class="GAPprompt">gap></span> <span class="GAPinput">N := RightModuleOverPathAlgebra(A </pre></div> <p><a id="X7D310EDE847865C1" name="X7D310EDE847865C1"></a></p> <h4>8.1 <span class="Heading">Homological algebra</span></h4> <p><a id="X7EC01D588141BB96" name="X7EC01D588141BB96"></a></p> <h5>8.1-1 1stSyzygy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 1s <p>Arguments: <var class="Arg">M</var> -- a path algebra module (<code class="code">PathAlgebraMa <p>Returns: the first syzygy of the representation <var class="Arg">M</var> as a representation.</ <p><a id="X7902D1477B9FF116" name="X7902D1477B9FF116"></a></p> <h5>8.1-2 AllComplementsOfAlmostCompleteTiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <p>Arguments: <var class="Arg">M</var>, <var class="Arg">X</var> - two PathAlgebraMatModule's. <p>Returns: all the complements of the almost complete (co-)tilting module <var class="Arg">M</v <p><a id="X81479679781E04FA" name="X81479679781E04FA"></a></p> <h5>8.1-3 CotiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Arguments: <var class="Arg">M</var>, <var class="Arg">n</var> - a PathAlgebraMatModule and a posi <p>Returns: false if <var class="Arg">M</var> is not a cotilting module of injective dimension at mo <p><a id="X7D6AD2AB87865333" name="X7D6AD2AB87865333"></a></p> <h5>8.1-4 DominantDimensionOfAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Do <p>Arguments: <var class="Arg">A</var>, <var class="Arg">n</var> - a quiver algebra, a positive integ <p>Returns: the dominant dimension of the algebra <var class="Arg">A</var> if the dominant dimensi <p><a id="X8582841B79A11466" name="X8582841B79A11466"></a></p> <h5>8.1-5 DominantDimensionOfModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Do <p>Arguments: <var class="Arg">M</var>, <var class="Arg">n</var> - a PathAlgebraMatModule, a positi <p>Returns: the dominant dimension of the module <var class="Arg">M</var> if the dominant dimensio <p><a id="X7B9209CD7A46F544" name="X7B9209CD7A46F544"></a></p> <h5>8.1-6 ExtAlgebraGenerators</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ex <p>Arguments: <var class="Arg">M</var> - a module, <var class="Arg">n</var> - a positive integer.<br /></ <p>Returns: a list of three elements, where the first element is the dimensions of Ext^[0..n](M, <p>This function computes the generators of the Ext-algebra <span class="SimpleMath">\(Ext^*( <p><a id="X787217547958E9C3" name="X787217547958E9C3"></a></p> <h5>8.1-7 ExtOverAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ex <p>Arguments: <var class="Arg">M</var>, <var class="Arg">N</var> - two modules.<br /></p> <p>Returns: a list of three elements <code class="func">ExtOverAlgebra</code>, where the first el <p>The function checks if the arguments <var class="Arg">M</var> and <var class="Arg">N</var> are modu <p><a id="X7A7BF8178390DB7A" name="X7A7BF8178390DB7A"></a></p> <h5>8.1-8 FaithfulDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fa <p>Arguments: <var class="Arg">M</var> - a PathAlgebraMatModule.<br /></p> <p>Returns: the faithful dimension of the module <var class="Arg">M</var>.</p> <p><a id="X7B2001337C48B3C9" name="X7B2001337C48B3C9"></a></p> <h5>8.1-9 GlobalDimensionOfAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gl <p>Arguments: <var class="Arg">A</var>, <var class="Arg">n</var> - a quiver algebra, a positive integ <p>Returns: the global dimension of <var class="Arg">A</var> if the global dimension is less or equa <p><a id="X8380C47D80FAAED8" name="X8380C47D80FAAED8"></a></p> <h5>8.1-10 GorensteinDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Go <p>Arguments: <var class="Arg">A</var> - a quiver algebra.<br /></p> <p>Returns: the Gorenstein dimension of <var class="Arg">A</var>, if the Gorenstein dimension has <p><a id="X84BC33C07B4BBBC0" name="X84BC33C07B4BBBC0"></a></p> <h5>8.1-11 GorensteinDimensionOfAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Go <p>Arguments: <var class="Arg">A</var>, <var class="Arg">n</var> - a quiver algebra, a positive integ <p>Returns: the Gorenstein dimension of <var class="Arg">A</var> if the Gorenstein dimension is le <p><a id="X7B1A0308813BA30F" name="X7B1A0308813BA30F"></a></p> <h5>8.1-12 HaveFiniteCoresolutionInAddM</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ha <p>Arguments: <var class="Arg">N</var>, <var class="Arg">M</var>, <var class="Arg">n</var> - two PathAl <p>Returns: false if <var class="Arg">N</var> does not have a coresolution of length at most <var clas <p><a id="X83271D587CD605E1" name="X83271D587CD605E1"></a></p> <h5>8.1-13 HaveFiniteResolutionInAddM</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ha <p>Arguments: <var class="Arg">N</var>, <var class="Arg">M</var>, <var class="Arg">n</var> - two PathAl <p>Returns: false if <var class="Arg">N</var> does not have a resolution of length at most <var class= <p><a id="X80C6D02F7B543CC9" name="X80C6D02F7B543CC9"></a></p> <h5>8.1-14 InjDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>Arguments: <var class="Arg">M</var> - a PathAlgebraMatModule.<br /></p> <p>If the injective dimension of the module <var class="Arg">M</var> has been computed, then the pro <p><a id="X7C16FB658383C36F" name="X7C16FB658383C36F"></a></p> <h5>8.1-15 InjDimensionOfModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>Arguments: <var class="Arg">M, n</var> - a PathAlgebraMatModule, a positive integer.<br /></p> <p>Returns: Returns the injective dimension of the module <var class="Arg">M</var> if it is less or e <p><a id="X8434C5FC87773723" name="X8434C5FC87773723"></a></p> <h5>8.1-16 InjectiveEnvelope</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>Arguments: <var class="Arg">M</var> - a module.<br /></p> <p>Returns: the injective envelope of <var class="Arg">M</var>, that is, returns the map <span class <p>If the module <var class="Arg">M</var> is zero, then the zero map from <var class="Arg">M</var> is ret <p><a id="X81910CD67E69BBF4" name="X81910CD67E69BBF4"></a></p> <h5>8.1-17 IsCotiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Arguments: <var class="Arg">M</var> - a PathAlgebraMatModule.<br /></p> <p>Returns: true if the module <var class="Arg">M</var> has been checked to be a cotilting mdoule, ot <p><a id="X7AA1E69C7A1009DA" name="X7AA1E69C7A1009DA"></a></p> <h5>8.1-18 IsNthSyzygy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Arguments: <var class="Arg">M</var> -- a path algebra module (<code class="code">PathAlgebraMa <p>Returns: <code class="code">true</code>, if the representation <var class="Arg">M</var> is isomo <p><a id="X7BE6A5B586463067" name="X7BE6A5B586463067"></a></p> <h5>8.1-19 IsOmegaPeriodic</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Arguments: <var class="Arg">M</var> -- a path algebra module (<code class="code">PathAlgebraMa <p>Returns: <code class="code">i</code>, where <code class="code">i</code> is the smallest positive <p><a id="X7CA63DA785F30E99" name="X7CA63DA785F30E99"></a></p> <h5>8.1-20 IsTtiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Arguments: <var class="Arg">M</var> - a PathAlgebraMatModule.<br /></p> <p>Returns: true if the module <var class="Arg">M</var> has been checked to be a tilting mdoule, othe <p><a id="X7B433C86806B60A4" name="X7B433C86806B60A4"></a></p> <h5>8.1-21 IyamaGenerator</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Iy <p>Arguments: <var class="Arg">M</var> -- a path algebra module (<code class="code">PathAlgebraMa <p>Returns: a module <span class="SimpleMath">\(N\)</span> such that <var class="Arg">M</var> is a di <p><a id="X80800F4F80398356" name="X80800F4F80398356"></a></p> <h5>8.1-22 LeftApproximationByAddTHat</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Arguments: <var class="Arg">T</var>, <var class="Arg">M</var> -- two path algebra modules (<code cl <p>Returns: the minimal left <span class="SimpleMath">\(\widehat{\add T}\)</span>-approximat <p>The function checks if the first argument is a cotilting module, that is, checks if the attribu <p><a id="X806A7DF7864D9431" name="X806A7DF7864D9431"></a></p> <h5>8.1-23 LeftFacMApproximation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">C</var>, <var class="Arg">M</var> -- two path algebra modules (<code cl <p>Returns: a left <span class="SimpleMath">\(\operatorname{Fac} M\)</span>-approximation of <p><a id="X82720AAD7E80714B" name="X82720AAD7E80714B"></a></p> <h5>8.1-24 LeftMutationOfTiltingModuleComplement</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Arguments: <var class="Arg">M</var>, <var class="Arg">N</var> -- two path algebra modules (<code cl <p>Returns: a left mutation of the complement <var class="Arg">N</var> of the almost complete tilti <p><a id="X86EB50377F2E910D" name="X86EB50377F2E910D"></a></p> <h5>8.1-25 LeftSubMApproximation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">C</var>, <var class="Arg">M</var> -- two path algebra modules (<code cl <p>Returns: a minimal left <span class="SimpleMath">\(\operatorname{Sub} M\)</span>-approxim <p><a id="X848B51017E269802" name="X848B51017E269802"></a></p> <h5>8.1-26 LiftingInclusionMorphisms</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Li <p>Arguments: <var class="Arg">f</var>, <var class="Arg">g</var> - two homomorphisms with common ran <p>Returns: a factorization of <var class="Arg">g</var> in terms of <var class="Arg">f</var>, wheneve <p>Given two inclusions <span class="SimpleMath">\(f\colon B\to C\)</span> and <span class="Simp <p><a id="X7A10EF0587ADA535" name="X7A10EF0587ADA535"></a></p> <h5>8.1-27 LiftingMorphismFromProjective</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Li <p>Arguments: <var class="Arg">f</var>, <var class="Arg">g</var> - two homomorphisms with common ran <p>Returns: a factorization of <var class="Arg">g</var> in terms of <var class="Arg">f</var>, wheneve <p>Given two morphisms <span class="SimpleMath">\(f\colon B\to C\)</span> and <span class="Simpl <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">B := BasisVectors(Basis(N));</span> [ [ [ 1, 0, 0 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 0, 1, 0 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 0, 0, 1 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 1 ], [ 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0 ], [ 1, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0 ], [ 0, 1 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">g := SubRepresentationInclusion(N <<[ 1, 2, 2 ]> ---> <[ 3, 2, 2 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">f := SubRepresentationInclusion(N <<[ 2, 2, 2 ]> ---> <[ 3, 2, 2 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">LiftingInclusionMorphisms(f,g);< <<[ 1, 2, 2 ]> ---> <[ 2, 2, 2 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">S := SimpleModules(A); </span> [ <[ 1, 0, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 0, 1 ]> ] <span class="GAPprompt">gap></span> <span class="GAPinput">homNS := HomOverAlgebra(N,S[1]);</ [ <<[ 3, 2, 2 ]> ---> <[ 1, 0, 0 ]>> , <<[ 3, 2, 2 ]> ---> <[ 1, 0, 0 ]>> , <<[ 3, 2, 2 ]> ---> <[ 1, 0, 0 ]>> ] <span class="GAPprompt">gap></span> <span class="GAPinput">f := homNS[1];</span> <<[ 3, 2, 2 ]> ---> <[ 1, 0, 0 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">p := ProjectiveCover(S[1]);</span> <<[ 1, 4, 3 ]> ---> <[ 1, 0, 0 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">LiftingMorphismFromProjective(f <<[ 1, 4, 3 ]> ---> <[ 3, 2, 2 ]>> [ [(1)*v1], [(1)*v2], [(1)*v3], [(1)*a], [(1)*b], [(1)*c], [(1)*d], [(1)*e] ] )> > </pre></div> <p><a id="X853E593B83DDD4B8" name="X853E593B83DDD4B8"></a></p> <h5>8.1-28 LeftApproximationByAddM</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">C</var>, <var class="Arg">M</var> - two modules.<br /></p> <p>Returns: the minimal left <span class="SimpleMath">\(\add M\)</span>-approximation in the tw <p><a id="X7F689558823AC784" name="X7F689558823AC784"></a></p> <h5>8.1-29 RightApproximationByAddM</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">M</var>, <var class="Arg">C</var> - two modules.<br /></p> <p>Returns: the minimal right <span class="SimpleMath">\(\add M\)</span>-approximation in the t <p><a id="X7947619F7C8539E0" name="X7947619F7C8539E0"></a></p> <h5>8.1-30 RadicalRightApproximationByAddM</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <p>Arguments: <var class="Arg">modulelist</var>, <var class="Arg">t</var> - a list of modules and an i <p>Returns: a radical right approximation of <code class="code">moduleslist[ t ]</code> by the add <p><a id="X87AB8D2E8660869D" name="X87AB8D2E8660869D"></a></p> <h5>8.1-31 MorphismOnKernel</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mo <p>Arguments: <var class="Arg">f</var>, <var class="Arg">g</var>, <var class="Arg">alpha</var>, <var cl <p>Returns: the morphism induced on the kernels, the images or the cokernels of the morphisms <var <p>It is checked if <var class="Arg">f</var>, <var class="Arg">g</var>, <var class="Arg">alpha</var>, <va <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">hom := HomOverAlgebra(N,N);</span> [ <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>> , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>> , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>> , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>> , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>> ] <span class="GAPprompt">gap></span> <span class="GAPinput">g := MorphismOnKernel(hom[1],hom[ <<[ 2, 2, 2 ]> ---> <[ 2, 2, 2 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphicModules(Source(g),Ran true <span class="GAPprompt">gap></span> <span class="GAPinput">p := ProjectiveCover(N);</span> <<[ 3, 12, 9 ]> ---> <[ 3, 2, 2 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">N1 := Kernel(p);</span> <[ 0, 10, 7 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">pullback := PullBack(p,hom[1]);</s [ <<[ 3, 12, 9 ]> ---> <[ 3, 2, 2 ]>> , <<[ 3, 12, 9 ]> ---> <[ 3, 12, 9 ]>> ] <span class="GAPprompt">gap></span> <span class="GAPinput">Kernel(pullback[1]);</span> <[ 0, 10, 7 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphicModules(N1,Kernel(pul true <span class="GAPprompt">gap></span> <span class="GAPinput">t := LiftingMorphismFromProjectiv <<[ 3, 12, 9 ]> ---> <[ 3, 12, 9 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">s := MorphismOnKernel(p,p,t,hom[1 <<[ 0, 10, 7 ]> ---> <[ 0, 10, 7 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">Source(s) = N1;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">q := KernelInclusion(p);</span> <<[ 0, 10, 7 ]> ---> <[ 3, 12, 9 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">pushout := PushOut(q,s);</span> [ <<[ 0, 10, 7 ]> ---> <[ 3, 12, 9 ]>> , <<[ 3, 12, 9 ]> ---> <[ 3, 12, 9 ]>> ] <span class="GAPprompt">gap></span> <span class="GAPinput">U := CoKernel(pushout[1]);</span> <[ 3, 2, 2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphicModules(U,N);</span> true </pre></div> <p><a id="X7EB51A487B5B239D" name="X7EB51A487B5B239D"></a></p> <h5>8.1-32 NthSyzygy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nt <p>Arguments: <var class="Arg">M</var> -- a path algebra module (<code class="code">PathAlgebraMa <p>Returns: the <var class="Arg">n</var>-th syzygy of <var class="Arg">M</var>.</p> <p>This functions computes a projective resolution of <var class="Arg">M</var> and finds the <var cl <p><a id="X788CD7B282582D93" name="X788CD7B282582D93"></a></p> <h5>8.1-33 NumberOfComplementsOfAlmostCompleteTiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Nu <p>Arguments: <var class="Arg">M</var> -- a PathAlgebraMatModule.<br /></p> <p>Returns: the number complements of an almost complete tilting/cotilting module <var class=" <p><a id="X7D3A779D7EFF32D1" name="X7D3A779D7EFF32D1"></a></p> <h5>8.1-34 ProjDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Arguments: <var class="Arg">M</var> - a PathAlgebraMatModule.<br /></p> <p>Returns: the projective dimension of the module <var class="Arg">M</var>, if it has been compute <p><a id="X86DA085884BAF968" name="X86DA085884BAF968"></a></p> <h5>8.1-35 ProjDimensionOfModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Arguments: <var class="Arg">M, n</var> - a PathAlgebraMatModule, a positive integer.<br /></p> <p>Returns: Returns the projective dimension of the module <var class="Arg">M</var> if it is less or <p><a id="X83257E2F7F8E0068" name="X83257E2F7F8E0068"></a></p> <h5>8.1-36 ProjectiveCover</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Arguments: <var class="Arg">M</var> - a module.<br /></p> <p>Returns: the projective cover of <var class="Arg">M</var>, that is, returns the map <span class=" <p>If the module <var class="Arg">M</var> is zero, then the zero map to <var class="Arg">M</var> is retur <p><a id="X8346B3997D9AB706" name="X8346B3997D9AB706"></a></p> <h5>8.1-37 ProjectiveResolutionOfPathAlgebraModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Arguments: <var class="Arg">M</var> - a path algebra module (<code class="code">PathAlgebraMat <p>Returns: in terms of attributes <code class="code">RProjectives</code>, <code class="code">Pr <p>The algorithm for computing this projective resolution is based on the paper <a href="chapBib <p><a id="X80678790799AEC16" name="X80678790799AEC16"></a></p> <h5>8.1-38 ProjectiveResolutionOfSimpleModuleOverEndo</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>Arguments: <var class="Arg">modulelist</var> - a list of module, <var class="Arg">t</var> - an inde <p>Returns: information about the projective dimension and non-projective summands of the syz <p><a id="X7C705F2A79F8E43C" name="X7C705F2A79F8E43C"></a></p> <h5>8.1-39 PullBack</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pu <p>Arguments: <var class="Arg">f</var>, <var class="Arg">g</var> - two homomorphisms with a common ra <p>Returns: the pullback of the maps <var class="Arg">f</var> and <var class="Arg">g</var>.</p> <p>It is checked if <var class="Arg">f</var> and <var class="Arg">g</var> have the same range. Given the <p><a id="X81A2D49D85923894" name="X81A2D49D85923894"></a></p> <h5>8.1-40 PushOut</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pu <p>Arguments: <var class="Arg">f</var>, <var class="Arg">g</var> - two homomorphisms between module <p>Returns: the pushout of the maps <var class="Arg">f</var> and <var class="Arg">g</var>.</p> <p>It is checked if <var class="Arg">f</var> and <var class="Arg">g</var> have the same source. Given th <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">S := SimpleModules(A);</span> [ <[ 1, 0, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 0, 1 ]> ] <span class="GAPprompt">gap></span> <span class="GAPinput">Ext := ExtOverAlgebra(S[2],S[2]);< [ <<[ 0, 1, 2 ]> ---> <[ 0, 2, 2 ]>> , [ <<[ 0, 1, 2 ]> ---> <[ 0, 1, 0 ]>> ], function( map ) ... end ] <span class="GAPprompt">gap></span> <span class="GAPinput">Length(Ext[2]);</span> 1 <span class="GAPprompt">gap></span> <span class="GAPinput"># i.e. Ext^1(S[2],S[2]) is 1-dimens <span class="GAPprompt">gap></span> <span class="GAPinput">pushout := PushOut(Ext[2][1],Ext[ [ <<[ 0, 2, 2 ]> ---> <[ 0, 2, 0 ]>> , <<[ 0, 1, 0 ]> ---> <[ 0, 2, 0 ]>> ] <span class="GAPprompt">gap></span> <span class="GAPinput">f := CoKernelProjection(pushout[1 <<[ 0, 2, 0 ]> ---> <[ 0, 0, 0 ]>> <span class="GAPprompt">gap></span> <span class="GAPinput">U := Range(pushout[1]); </span> <[ 0, 2, 0 ]> </pre></div> <p><a id="X7C8B72157BDE0957" name="X7C8B72157BDE0957"></a></p> <h5>8.1-41 RightApproximationByPerpT</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Arguments: <var class="Arg">T</var>, <var class="Arg">M</var> -- two path algebra modules (<code cl <p>Returns: the minimal right <span class="SimpleMath">\(^\perp T\)</span>-approximation of <va <p>The function checks if the first argument is a cotilting module, that is, checks if the attribu <p><a id="X873C632A83DF48E2" name="X873C632A83DF48E2"></a></p> <h5>8.1-42 RightFacMApproximation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">M</var>, <var class="Arg">C</var> -- two path algebra modules (<code cl <p>Returns: a minimal right <span class="SimpleMath">\(\operatorname{Fac} M\)</span>-approxi <p><a id="X8018953F7951F8F2" name="X8018953F7951F8F2"></a></p> <h5>8.1-43 RightMutationOfTiltingModuleComplement</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Arguments: <var class="Arg">M</var>, <var class="Arg">N</var> -- two path algebra modules (<code cl <p>Returns: a right mutation of the complement <var class="Arg">N</var> of the almost complete tilt <p><a id="X81BD4EEA831D4A54" name="X81BD4EEA831D4A54"></a></p> <h5>8.1-44 RightSubMApproximation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Arguments: <var class="Arg">M</var>, <var class="Arg">C</var> -- two path algebra modules (<code cl <p>Returns: a right <span class="SimpleMath">\(\operatorname{Sub} M\)</span>-approximation o <p><a id="X78CA656D7D4F2446" name="X78CA656D7D4F2446"></a></p> <h5>8.1-45 N_RigidModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ N_ <p>Arguments: <var class="Arg">M</var>, <var class="Arg">n</var> - a PathAlgebraMatModule, an integ <p>Returns: true if <var class="Arg">M</var> is a <var class="Arg">n</var>-rigid module. Otherwise it <p><a id="X824D52847879AB91" name="X824D52847879AB91"></a></p> <h5>8.1-46 TiltingModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ti <p>Arguments: <var class="Arg">M</var>, <var class="Arg">n</var> - a PathAlgebraMatModule and a posi <p>Returns: false if <var class="Arg">M</var> is not a tilting module of projective dimension at mos <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAP </body> </html>
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