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href="chap1.html" >1 </a> <a href="chap2.html" >2 </a> <a href="chapInd.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap0.html" >[Previous Chapter]</a> <a href="chap2.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap1_mj.html" >[MathJax on]</a></p>"X7B8C95CA7DA733B4" name="X7B8C95CA7DA733B4" ></a></p>div class="ChapSects" ><a href="chap1.html#X7B8C95CA7DA733B4" >1 <span class="Heading" >Module Presentations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X78D1062D78BE08C1" >1 .1 <span class="Heading" >Functors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F7AC44478418555" >1 .1 -1 FunctorStandardModuleLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X87AEC1177DB7F50D" >1 .1 -2 FunctorStandardModuleRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8427C0B17A445822" >1 .1 -3 FunctorGetRidOfZeroGeneratorsLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F1E779D8003146B" >1 .1 -4 FunctorGetRidOfZeroGeneratorsRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X819A04517B3601C0" >1 .1 -5 FunctorLessGeneratorsLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F80AE9B7EC07198" >1 .1 -6 FunctorLessGeneratorsRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X877B7ACE87E1BEC2" >1 .1 -7 FunctorDualLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D56611D7BF91B54" >1 .1 -8 FunctorDualRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7C9901F8851FD24A" >1 .1 -9 FunctorDoubleDualLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8016306881444DCA" >1 .1 -10 FunctorDoubleDualRight</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7D03633A7D98026B" >1 .2 <span class="Heading" >GAP Categories</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X79DBCB747E91FB70" >1 .2 -1 IsLeftOrRightPresentationMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X85E26CFF86855B6B" >1 .2 -2 IsLeftPresentationMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X873EFE29849F6998" >1 .2 -3 IsRightPresentationMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7BB95B7A7EB96854" >1 .2 -4 IsLeftOrRightPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7C71B8D17C60C6B5" >1 .2 -5 IsLeftPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7DBF478D7EE3FE63" >1 .2 -6 IsRightPresentation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X86EC0F0A78ECBC10" >1 .3 <span class="Heading" >Constructors</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X87010AB6819736C8" >1 .3 -1 PresentationMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7C9B36AD7B9CCC8D" >1 .3 -2 AsMorphismBetweenFreeLeftPresentations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F1AD55C852BE617" >1 .3 -3 AsMorphismBetweenFreeRightPresentations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7BE01A1381744627" >1 .3 -4 AsLeftPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X780443B07F43AA1C" >1 .3 -5 AsRightPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F345A2A87ABE417" >1 .3 -6 FreeLeftPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X85536E4E85D15252" >1 .3 -7 FreeRightPresentation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X86F926B27C579E66" >1 .3 -8 UnderlyingMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D8E30E486A08439" >1 .3 -9 UnderlyingHomalgRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7B5539AB8541F618" >1 .3 -10 Annihilator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X87946F997AD1005A" >1 .3 -11 LeftPresentations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7BDF988F7FFEAB8C" >1 .3 -12 RightPresentations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8145816A85BA2680" >1 .3 -13 LeftPresentations_as_FreydCategory_CategoryOfRows</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7ED9DFEB82DE9653" >1 .3 -14 RightPresentations_as_FreydCategory_CategoryOfColumns</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7C701DBF7BAE649A" >1 .4 <span class="Heading" >Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8157F3E8847B15E1" >1 .4 -1 UnderlyingHomalgRing</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X83CA6F06832162B7" >1 .4 -2 UnderlyingMatrix</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X81CDBC6E7DBB4EA0" >1 .5 <span class="Heading" >Non-Categorical Operations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8508310C7E908093" >1 .5 -1 StandardGeneratorMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7EF1493A7D341F5E" >1 .5 -2 CoverByFreeModule</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X836749D8814FEEE6" >1 .6 <span class="Heading" >Natural Transformations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X85D3DB1F856D05EF" >1 .6 -1 NaturalIsomorphismFromIdentityToStandardModuleLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7B64AF718133C945" >1 .6 -2 NaturalIsomorphismFromIdentityToStandardModuleRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7CA2B84E7F933125" >1 .6 -3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7A2DCBD6844093E3" >1 .6 -4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7B331B0A86010185" >1 .6 -5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X834AC0FD825FCD2F" >1 .6 -6 NaturalIsomorphismFromIdentityToLessGeneratorsRight</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7E37CB058378CBEE" >1 .6 -7 NaturalTransformationFromIdentityToDoubleDualLeft</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F92B4448041A68C" >1 .6 -8 NaturalTransformationFromIdentityToDoubleDualRight</a></span >div ></div >div >1 <span class="Heading" >Module Presentations</span ></h3>"X78D1062D78BE08C1" name="X78D1062D78BE08C1" ></a></p>1 .1 <span class="Heading" >Functors</span ></h4>"X7F7AC44478418555" name="X7F7AC44478418555" ></a></p>1 .1 -1 FunctorStandardModuleLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorStandardModuleLeft</code >( <var class="Arg" >R</var > )</td td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is a functor which takes a left presentation as input and computes its standard presentation.</p>"X87AEC1177DB7F50D" name="X87AEC1177DB7F50D" ></a></p>1 .1 -2 FunctorStandardModuleRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorStandardModuleRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is a functor which takes a right presentation as input and computes its standard presentation.</p>"X8427C0B17A445822" name="X8427C0B17A445822" ></a></p>1 .1 -3 FunctorGetRidOfZeroGeneratorsLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorGetRidOfZeroGeneratorsLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is a functor which takes a left presentation as input and gets rid of the zero generators.</p>"X7F1E779D8003146B" name="X7F1E779D8003146B" ></a></p>1 .1 -4 FunctorGetRidOfZeroGeneratorsRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorGetRidOfZeroGeneratorsRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is a functor which takes a right presentation as input and gets rid of the zero generators.</p>"X819A04517B3601C0" name="X819A04517B3601C0" ></a></p>1 .1 -5 FunctorLessGeneratorsLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorLessGeneratorsLeft</code >( <var class="Arg" >R</var > )</td td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is functor which takes a left presentation as input and computes a presentation having less generators.</p>"X7F80AE9B7EC07198" name="X7F80AE9B7EC07198" ></a></p>1 .1 -6 FunctorLessGeneratorsRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorLessGeneratorsRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is functor which takes a right presentation as input and computes a presentation having less generators.</p>"X877B7ACE87E1BEC2" name="X877B7ACE87E1BEC2" ></a></p>1 .1 -7 FunctorDualLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorDualLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" td ></tr ></table ></div >span class="Math" >R</span > that has an involution function. The output is functor which takes a left presentation <var class="Arg" >M</var > as input and computes its Hom(M, R) as a left presentation.</p>"X7D56611D7BF91B54" name="X7D56611D7BF91B54" ></a></p>1 .1 -8 FunctorDualRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorDualRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span > that has an involution function. The output is functor which takes a right presentation <var class="Arg" >M</var > as input and computes its Hom(M, R) as a right presentation.</p>"X7C9901F8851FD24A" name="X7C9901F8851FD24A" ></a></p>1 .1 -9 FunctorDoubleDualLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorDoubleDualLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span > that has an involution function. The output is functor which takes a left presentation <var class="Arg" >M</var > as input and computes its <var class="Arg" >Hom( Hom(M, R), R )</var > as a left presentation.</p>"X8016306881444DCA" name="X8016306881444DCA" ></a></p>1 .1 -10 FunctorDoubleDualRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FunctorDoubleDualRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span > that has an involution function. The output is functor which takes a right presentation <var class="Arg" >M</var > as input and computes its <var "Arg" >Hom( Hom(M, R), R )</var > as a right presentation.</p>"X7D03633A7D98026B" name="X7D03633A7D98026B" ></a></p>1 .2 <span class="Heading" >GAP Categories</span ></h4>"X79DBCB747E91FB70" name="X79DBCB747E91FB70" ></a></p>1 .2 -1 IsLeftOrRightPresentationMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsLeftOrRightPresentationMorphism</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X85E26CFF86855B6B" name="X85E26CFF86855B6B" ></a></p>1 .2 -2 IsLeftPresentationMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsLeftPresentationMorphism</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X873EFE29849F6998" name="X873EFE29849F6998" ></a></p>1 .2 -3 IsRightPresentationMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsRightPresentationMorphism</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X7BB95B7A7EB96854" name="X7BB95B7A7EB96854" ></a></p>1 .2 -4 IsLeftOrRightPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsLeftOrRightPresentation</code >( <var class="Arg" >object </var td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X7C71B8D17C60C6B5" name="X7C71B8D17C60C6B5" ></a></p>1 .2 -5 IsLeftPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsLeftPresentation</code >( <var class="Arg" >object </var > )</td ><td "tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X7DBF478D7EE3FE63" name="X7DBF478D7EE3FE63" ></a></p>1 .2 -6 IsRightPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsRightPresentation</code >( <var class="Arg" >object </var > )</td ><td class="tdright" >( filter )</td ></tr ></table ></div >code class="keyw" >true</code > or <code class="keyw" >false</code ></p>"X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10" ></a></p>1 .3 <span class="Heading" >Constructors</span ></h4>"X87010AB6819736C8" name="X87010AB6819736C8" ></a></p>1 .3 -1 PresentationMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; PresentationMorphism</code >( <var class="Arg" >A</var >, <var class="Arg" >M</var >, <var class="Arg" >B</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(A,B)</span ></p>object <span class="Math" >A</span >, a homalg matrix <span class="Math" >M</span >, and another object <span class="Math" >B</span >. <span class="Math" >A</span > and <span class="Math" >B</span > shall either both be objects in the category of left presentations or both be objects in the category of right presentations. The output is a morphism <span class="Math" >A \rightarrow B</span > in the the category of left or right presentations whose underlying matrix is given by <span class="Math" >M</span >.</p>"X7C9B36AD7B9CCC8D" name="X7C9B36AD7B9CCC8D" ></a></p>1 .3 -2 AsMorphismBetweenFreeLeftPresentations</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsMorphismBetweenFreeLeftPresentations</code >( <var class="Arg" >m</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(F^r,F^c)</span ></p>span class="Math" >m</span >. The output is a morphism <span class="Math" span > in the the category of left presentations whose underlying matrix is given by <span class="Math" >m</span >, where <span class="Math" >F^r</span > and <span class="Math" >F^c</span > are free left presentations of ranks given by the number of rows and columns of <span class="Math" >m</span >.</p>"X7F1AD55C852BE617" name="X7F1AD55C852BE617" ></a></p>1 .3 -3 AsMorphismBetweenFreeRightPresentations</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsMorphismBetweenFreeRightPresentations</code >( <var class="Arg" >m</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(F^c,F^r)</span ></p>span class="Math" >m</span >. The output is a morphism <span class="Math" span > in the the category of right presentations whose underlying matrix is given by <span class="Math" >m</span >, where <span class="Math" >F^r</span > and <span class="Math" >F^c</span > are free right presentations of ranks given by the number of rows and columns of <span class="Math" >m</span >.</p>"X7BE01A1381744627" name="X7BE01A1381744627" ></a></p>1 .3 -4 AsLeftPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsLeftPresentation</code >( <var class="Arg" >M</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >object </p>span class="Math" >M</span > over a ring <span class="Math" >R</span >. The output is an object in the category of left presentations over <span class="Math" >R</span >. This object has <span class="Math" >M</span > as its underlying matrix.</p>"X780443B07F43AA1C" name="X780443B07F43AA1C" ></a></p>1 .3 -5 AsRightPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; AsRightPresentation</code >( <var class="Arg" >M</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >object </p>span class="Math" >M</span > over a ring <span class="Math" >R</span >. The output is an object in the category of right presentations over <span class="Math" >R</span >. This object has <span class="Math" >M</span > as its underlying matrix.</p>"X7F345A2A87ABE417" name="X7F345A2A87ABE417" ></a></p>1 .3 -6 FreeLeftPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FreeLeftPresentation</code >( <var class="Arg" >r</var >, <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >object </p>span class="Math" >r</span > and a homalg ring <span class="Math" >R</span >. The output is an object in the category of left presentations over <span class="Math" >R</span >. It is represented by the <span class="Math" >0 \times r</span > matrix and thus it is free of rank <span class="Math" >r</span >.</p>"X85536E4E85D15252" name="X85536E4E85D15252" ></a></p>1 .3 -7 FreeRightPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; FreeRightPresentation</code >( <var class="Arg" >r</var >, <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >object </p>span class="Math" >r</span > and a homalg ring <span class="Math" >R</span >. The output is an object in the category of right presentations over <span class="Math" span >. It is represented by the <span class="Math" >r \times 0 </span > matrix and thus it is free of rank <span class="Math" >r</span >.</p>"X86F926B27C579E66" name="X86F926B27C579E66" ></a></p>1 .3 -8 UnderlyingMatrix</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; UnderlyingMatrix</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >object <span class="Math" >A</span > in the category of left or right presentations over a homalg ring <span class="Math" >R</span >. The output is the underlying matrix which presents <span class="Math" >A</span >.</p>"X7D8E30E486A08439" name="X7D8E30E486A08439" ></a></p>1 .3 -9 UnderlyingHomalgRing</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; UnderlyingHomalgRing</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >object <span class="Math" >A</span > in the category of left or right presentations over a homalg ring <span class="Math" >R</span >. The output is <span class="Math" >R</span >.</p>"X7B5539AB8541F618" name="X7B5539AB8541F618" ></a></p>1 .3 -10 Annihilator</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; Annihilator</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(I, F)</span ></p>object <span class="Math" >A</span > in the category of left or right presentations. The output is the embedding of the annihilator <span class="Math" >I</span > of <span class="Math" >A</span > into the free module <span class="Math" >F</span > of rank <span class="Math" >1 </span >. In particular, the annihilator itself is seen as a left or right presentation.</p>"X87946F997AD1005A" name="X87946F997AD1005A" ></a></p>1 .3 -11 LeftPresentations</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; LeftPresentations</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is the category of left presentations over <span class="Math" >R</span >.</p>"X7BDF988F7FFEAB8C" name="X7BDF988F7FFEAB8C" ></a></p>1 .3 -12 RightPresentations</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; RightPresentations</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >R</span >. The output is the category of right presentations over <span class="Math" >R</span >.</p>"X8145816A85BA2680" name="X8145816A85BA2680" ></a></p>1 .3 -13 LeftPresentations_as_FreydCategory_CategoryOfRows</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; LeftPresentations_as_FreydCategory_CategoryOfRows</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >R</span >. The output is the category of left presentations over <span class="Math" >R</span >, constructed internally as the <code class="code" >FreydCategory</code > of the <code class="code" >CategoryOfRows</code > of <var class="Arg" >R</var >. Only available if the package <code class="code" >FreydCategoriesForCAP</code > is available.</p>"X7ED9DFEB82DE9653" name="X7ED9DFEB82DE9653" ></a></p>1 .3 -14 RightPresentations_as_FreydCategory_CategoryOfColumns</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; RightPresentations_as_FreydCategory_CategoryOfColumns</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >R</span >. The output is the category of right presentations over <span class="Math" >R</span >, constructed internally as the <code class="code" >FreydCategory</code > of the <code class="code" >CategoryOfColumns</code > of <var class="Arg" >R</var >. Only available if the package <code class="code" >FreydCategoriesForCAP</code > is available.</p>"X7C701DBF7BAE649A" name="X7C701DBF7BAE649A" ></a></p>1 .4 <span class="Heading" >Attributes</span ></h4>"X8157F3E8847B15E1" name="X8157F3E8847B15E1" ></a></p>1 .4 -1 UnderlyingHomalgRing</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; UnderlyingHomalgRing</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\alpha</span > in the category of left or right presentations over a homalg ring <span class="Math" >R</span >. The output is <span class="Math" >R</span >.</p>"X83CA6F06832162B7" name="X83CA6F06832162B7" ></a></p>1 .4 -2 UnderlyingMatrix</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; UnderlyingMatrix</code >( <var class="Arg" >alpha</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\alpha</span > in the category of left or right presentations. The output is its underlying homalg matrix.</p>"X81CDBC6E7DBB4EA0" name="X81CDBC6E7DBB4EA0" ></a></p>1 .5 <span class="Heading" >Non-Categorical Operations</span ></h4>"X8508310C7E908093" name="X8508310C7E908093" ></a></p>1 .5 -1 StandardGeneratorMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; StandardGeneratorMorphism</code >( <var class="Arg" >A</var >, <var class="Arg" >i</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(F, A)</span ></p>object <span class="Math" >A</span > in the category of left or right presentations over a homalg ring <span class="Math" >R</span > with underlying matrix <span class="Math" >M</span > and an integer <span class="Math" >i</span >. The output is a morphism <span class="Math" >F \rightarrow A</span > given by the <span class="Math" >i</span >-th row or column of <span class="Math" >M</span >, where <span class="Math" >F</span > is a free left or right presentation of rank <span class="Math" >1 </span >.</p>"X7EF1493A7D341F5E" name="X7EF1493A7D341F5E" ></a></p>1 .5 -2 CoverByFreeModule</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; CoverByFreeModule</code >( <var class="Arg" >A</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Hom}(F,A)</span ></p>object <span class="Math" >A</span > in the category of left or right presentations. The output is a morphism from a free module <span class="Math" >F</span > to <span class="Math" >A</span span class="Math" >A</span >.</p>"X836749D8814FEEE6" name="X836749D8814FEEE6" ></a></p>1 .6 <span class="Heading" >Natural Transformations</span ></h4>"X85D3DB1F856D05EF" name="X85D3DB1F856D05EF" ></a></p>1 .6 -1 NaturalIsomorphismFromIdentityToStandardModuleLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToStandardModuleLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{StandardModuleLeft}</span ></p>span class="Math" >R</span >. The output is the natural isomorphism from the identity functor to the left standard module functor.</p>"X7B64AF718133C945" name="X7B64AF718133C945" ></a></p>1 .6 -2 NaturalIsomorphismFromIdentityToStandardModuleRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToStandardModuleRight</code var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{StandardModuleRight}</span ></p>span class="Math" >R</span >. The output is the natural isomorphism from the identity functor to the right standard module functor.</p>"X7CA2B84E7F933125" name="X7CA2B84E7F933125" ></a></p>1 .6 -3 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsLeft}</span ></p>span class="Math" >R</span >. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of left modules.</p>"X7A2DCBD6844093E3" name="X7A2DCBD6844093E3" ></a></p>1 .6 -4 NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsRight}</span ></p>span class="Math" >R</span >. The output is the natural isomorphism from the identity functor to the functor that gets rid of zero generators of right modules.</p>"X7B331B0A86010185" name="X7B331B0A86010185" ></a></p>1 .6 -5 NaturalIsomorphismFromIdentityToLessGeneratorsLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToLessGeneratorsLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{LessGeneratorsLeft}</span ></p>span class="Math" >R</span >. The output is the natural morphism from the identity functor to the left less generators functor.</p>"X834AC0FD825FCD2F" name="X834AC0FD825FCD2F" ></a></p>1 .6 -6 NaturalIsomorphismFromIdentityToLessGeneratorsRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalIsomorphismFromIdentityToLessGeneratorsRight</code var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{LessGeneratorsRight}</span ></p>span class="Math" >R</span >. The output is the natural morphism from the identity functor to the right less generator functor.</p>"X7E37CB058378CBEE" name="X7E37CB058378CBEE" ></a></p>1 .6 -7 NaturalTransformationFromIdentityToDoubleDualLeft</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalTransformationFromIdentityToDoubleDualLeft</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualLeft}</span ></p>span class="Math" >R</span >. The output is the natural morphism from the identity functor to the double dual functor in left Presentations category.</p>"X7F92B4448041A68C" name="X7F92B4448041A68C" ></a></p>1 .6 -8 NaturalTransformationFromIdentityToDoubleDualRight</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; NaturalTransformationFromIdentityToDoubleDualRight</code >( <var class="Arg" >R</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >span class="Math" >\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualRight}</span ></p>span class="Math" >R</span >. The output is the natural morphism from the identity functor to the double dual functor in right Presentations category.</p>div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap0.html" >[Previous Chapter]</a> <a href="chap2.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1 </a> <a href="chap2.html" >2 </a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html > Messung V0.5 in Prozent C=91 H=99 G=94
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