<p>The argument is a homalg ring <span class="Math">R</span>. The output is a functor which takes a left presentation as input and computes its standard presentation.</p>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is a functor which takes a right presentation as input and computes its standard presentation.</p>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is functor which takes a left presentation as input and computes a presentation having less generators.</p>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is functor which takes a right presentation as input and computes a presentation having less generators.</p>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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 class="Arg">Hom( Hom(M, R), R )</var> as a right presentation.</p>
<p>The arguments are an 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 <spanclass="Math">M</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsMorphismBetweenFreeLeftPresentations</code>( <var class="Arg">m</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(F^r,F^c)</span></p>
<p>The argument is a homalg matrix <span class="Math">m</span>. The output is a morphism <span class="Math">F^r \rightarrow F^c</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsMorphismBetweenFreeRightPresentations</code>( <var class="Arg">m</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(F^c,F^r)</span></p>
<p>The argument is a homalg matrix <span class="Math">m</span>. The output is a morphism <span class="Math">F^c \rightarrow F^r</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 <spanclass="Math">m</span>.</p>
<p>The argument is a homalg matrix <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>
<p>The argument is a homalg matrix <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>
<p>The arguments are a non-negative integer <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>
<p>The arguments are a non-negative integer <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">R</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>
<p>The argument is an 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>
<p>The argument is an 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Annihilator</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(I, F)</span></p>
<p>The argument is an 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>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
<p>The argument is a morphism <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>
<p>The argument is a morphism <span class="Math">\alpha</span> in the category of left or right presentations. The output is its underlying homalg matrix.</p>
<p>The argument is an 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoverByFreeModule</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(F,A)</span></p>
<p>The argument is an 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>, which maps the standard generators of the free module to the generators of <span class="Math">A</span>.</p>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is the natural isomorphism from the identity functor to the left standard module functor.</p>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is the natural isomorphism from the identity functor to the right standard module functor.</p>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is the natural morphism from the identity functor to the left less generators functor.</p>
<p>The argument is a homalg ring <span class="Math">R</span>. The output is the natural morphism from the identity functor to the right less generator functor.</p>
<p>The argument is a homalg ring <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>
<p>The argument is a homalg ring <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>
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