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<p><a id="X7EDB390B82D9E644" name="X7EDB390B82D9E644"></a></p>
<div class="ChapSects"><a href="chap11_mj.html#X7EDB390B82D9E644">11 <span class="Heading">Projective resolutions and the bounded derived category</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X7C2FFD3E7D0D5D7F">11.1 <span class="Heading">Projective and injective complexes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X854F304E7A7F6FF2">11.1-1 InjectiveResolution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X8581C1F37EA425DE">11.1-2 IsProjectiveComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X7BDFFF987E99A5EA">11.1-3 IsInjectiveComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X8273999C7B352F22">11.1-4 ProjectiveResolution</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11_mj.html#X83D4593C80C2C4F6">11.2 <span class="Heading">The bounded derived category</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X7BC3EE977FA24151">11.2-1 ProjectiveResolutionOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X84F050D07B19ABC4">11.2-2 ProjectiveToInjectiveComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X82CB35E07DD2D96D">11.2-3 TauOfComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X85861B017AEEC50B">11.2-4 <span class="Heading">Example</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11_mj.html#X84D4D700876395AF">11.2-5 StarOfMapBetweenProjectives</a></span>
</div></div>
</div>

<h3>11 <span class="Heading">Projective resolutions and the bounded derived category</span></h3>

<p>What is implemented so far for working with the bounded derived category <span class="SimpleMath">\(\mathcal{D}^{b}( \modc A )\)</span>. We use the isomorphism <span class="SimpleMath">\(\mathcal{D}^{b}( \modc A ) \cong \mathcal{K}^{-,b}(\proj A)\)</span>, and will hence need a way to describe complexes where all objectives are projective (or, dually, injective).</p>

<p><a id="X7C2FFD3E7D0D5D7F" name="X7C2FFD3E7D0D5D7F"></a></p>

<h4>11.1 <span class="Heading">Projective and injective complexes</span></h4>

<p><a id="X854F304E7A7F6FF2" name="X854F304E7A7F6FF2"></a></p>

<h5>11.1-1 InjectiveResolution</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; InjectiveResolution</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">M</var> -- a module.<br /></p>

<p>Returns: The injective resolution of <var class="Arg">M</var> with <var class="Arg">M</var> in degree <span class="SimpleMath">\(-1\)</span>.</p>

<p><a id="X8581C1F37EA425DE" name="X8581C1F37EA425DE"></a></p>

<h5>11.1-2 IsProjectiveComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsProjectiveComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Arguments: <var class="Arg">C</var> -- a complex.<br /></p>

<p>Returns: true if <var class="Arg">C</var> is either a finite complex of projectives or an infinite complex of projectives constructed as a projective resolution (<code class="func">ProjectiveResolutionOfComplex</code> (<a href="chap11_mj.html#X7BC3EE977FA24151"><span class="RefLink">11.2-1</span></a>)), false otherwise.</p>

<p>A complex for which this property is true, will be printed in a different manner than ordinary complexes. Instead of writing the dimension vector of the objects in each degree, the indecomposable direct summands are listed (for instance <code class="code">P1</code>, <code class="code">P2</code> … , where <span class="SimpleMath">\(P_i\)</span> is the indecomposable projective module corresponding to vertex <span class="SimpleMath">\(i\)</span> of the quiver). Note that if a complex is both projective and injective, it is printed as a projective complex.</p>

<p><a id="X7BDFFF987E99A5EA" name="X7BDFFF987E99A5EA"></a></p>

<h5>11.1-3 IsInjectiveComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInjectiveComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Arguments: <var class="Arg">C</var> -- a complex.<br /></p>

<p>Returns: true if <var class="Arg">C</var> is either a finite complex of injectives or an infinite complex of injectives constructed as <span class="SimpleMath">\(D\mathrm{Hom}_{A}(-,A)\)</span> of a projective complex (<code class="func">ProjectiveToInjectiveComplex</code> (<a href="chap11_mj.html#X84F050D07B19ABC4"><span class="RefLink">11.2-2</span></a>)), false otherwise.</p>

<p>A complex for which this property is true, will be printed in a different manner than ordinary complexes. Instead of writing the dimension vector of the objects in each degree, the indecomposable direct summands are listed (for instance <code class="code">I1</code>, <code class="code">I2</code> … , where <span class="SimpleMath">\(I_i\)</span> is the indecomposable injective module corresponding to vertex <span class="SimpleMath">\(i\)</span> of the quiver). Note that if a complex is both projective and injective, it is printed as a projective complex.</p>

<p><a id="X8273999C7B352F22" name="X8273999C7B352F22"></a></p>

<h5>11.1-4 ProjectiveResolution</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveResolution</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">M</var> -- a module.<br /></p>

<p>Returns: The projective resolution of <var class="Arg">M</var> with <var class="Arg">M</var> in degree <span class="SimpleMath">\(-1\)</span>.</p>

<p><a id="X83D4593C80C2C4F6" name="X83D4593C80C2C4F6"></a></p>

<h4>11.2 <span class="Heading">The bounded derived category</span></h4>

<p>Let <span class="SimpleMath">\(\mathcal{D}^{b}( \modc A )\)</span> denote the bounded derived category. If <span class="SimpleMath">\(C\)</span> is an element of <span class="SimpleMath">\(\mathcal{D}^{b}( \modc A )\)</span>, that is, a bounded complex of <span class="SimpleMath">\(A\)</span>-modules, there exists a projective resolution <span class="SimpleMath">\(P\)</span> of <span class="SimpleMath">\(C\)</span> which is a complex of projective <span class="SimpleMath">\(A\)</span>-modules quasi-isomorphic to <span class="SimpleMath">\(C\)</span>. Moreover, there exists such a <span class="SimpleMath">\(P\)</span> with the following properties:</p>


<ul>
<li><p><span class="SimpleMath">\(P\)</span> is minimal (in the homotopy category).</p>

</li>
<li><p><span class="SimpleMath">\(C\)</span> is bounded, so <span class="SimpleMath">\(C_i = 0\)</span> for <span class="SimpleMath">\(i < k\)</span> for a lower bound <span class="SimpleMath">\(k\)</span> and <span class="SimpleMath">\(C_i = 0\)</span> for <span class="SimpleMath">\(i > j\)</span> for an upper bound <span class="SimpleMath">\(j\)</span>. Then <span class="SimpleMath">\(P_i = 0\)</span> for <span class="SimpleMath">\(i < k\)</span>, and <span class="SimpleMath">\(P\)</span> is exact in degree <span class="SimpleMath">\(i\)</span> for <span class="SimpleMath">\(i > j\)</span>.</p>

</li>
</ul>
<p>The function <code class="code">ProjectiveResolutionOfComplex</code> computes such a projective resolution of any bounded complex. If <span class="SimpleMath">\(A\)</span> has finite global dimension, then <span class="SimpleMath">\(\mathcal{D}^{b}( \modc A )\)</span> has AR-triangles, and there exists an algorithm for computing the AR-translation of a complex <span class="SimpleMath">\(C \in \mathcal{D}^{b}( \modc A )\)</span>:</p>


<ul>
<li><p>Compute a projective resolution <span class="SimpleMath">\(P'\)</span> of <span class="SimpleMath">\(C\)</span>.</p>

</li>
<li><p>Shift <span class="SimpleMath">\(P'\)</span> one degree to the right.</p>

</li>
<li><p>Compute <span class="SimpleMath">\(I = D\mathrm{Hom}_{A}(P',A)\)</span> to get a complex of injectives.</p>

</li>
<li><p>Compute a projective resolution <span class="SimpleMath">\(P\)</span> of <span class="SimpleMath">\(I\)</span>.</p>

</li>
</ul>
<p>Then <span class="SimpleMath">\(P\)</span> is the AR-translation of <span class="SimpleMath">\(C\)</span>, sometimes written <span class="SimpleMath">\(\tau(C)\)</span>. The following documents the <strong class="pkg">QPA</strong> functions for working with complexes in the derived category.</p>

<p><a id="X7BC3EE977FA24151" name="X7BC3EE977FA24151"></a></p>

<h5>11.2-1 ProjectiveResolutionOfComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveResolutionOfComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">C</var> -- a finite complex.<br /></p>

<p>Returns: A projective complex <span class="SimpleMath">\(P\)</span> which is the projective resolution of <span class="SimpleMath">\(C\)</span>, as described in the introduction to this section.</p>

<p>If the algebra has infinite global dimension, the projective resolution of <span class="SimpleMath">\(C\)</span> could possibly be infinite.</p>

<p><a id="X84F050D07B19ABC4" name="X84F050D07B19ABC4"></a></p>

<h5>11.2-2 ProjectiveToInjectiveComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveToInjectiveComplex</code>( <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProjectiveToInjectiveFiniteComplex</code>( <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">P</var> -- a bounded below projective complex.<br /></p>

<p>Returns: An injective complex <span class="SimpleMath">\(I = D\mathrm{Hom}_{A}(P,A)\)</span>.</p>

<p><span class="SimpleMath">\(P\)</span> and <span class="SimpleMath">\(I\)</span> will always have the same length. Especially, if <span class="SimpleMath">\(P\)</span> is unbounded above, then so is <span class="SimpleMath">\(I\)</span>. If <span class="SimpleMath">\(P\)</span> is a finite complex (that is; <code class="code">LengthOfComplex(P)</code> is an integer) then the simpler method <code class="code">ProjectiveToInjectiveFiniteComplex</code> is used.</p>

<p><a id="X82CB35E07DD2D96D" name="X82CB35E07DD2D96D"></a></p>

<h5>11.2-3 TauOfComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TauOfComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">C</var> -- a finite complex over an algebra of finite global dimension.<br /></p>

<p>Returns: A projective complex <span class="SimpleMath">\(P\)</span> which is the AR-translation of <var class="Arg">C</var>.</p>

<p>This function only works when the algebra has finite global dimension. It will always assume that both the projective resolutions computed are finite.</p>

<p><a id="X85861B017AEEC50B" name="X85861B017AEEC50B"></a></p>

<h5>11.2-4 <span class="Heading">Example</span></h5>

<p>The following example illustrates the above mentioned functions and properties. Note that both <code class="code">ProjectiveResolutionOfComplex</code> and <code class="code">ProjectiveToInjectiveComplex</code> return complexes with a nonzero <em>positive</em> part, whereas <code class="code">TauOfComplex</code> always returns a complex for which <code class="code">IsFiniteComplex</code> returns true. Also note that after the complex <code class="code">C</code> in the example is found to have the <code class="code">IsInjectiveComplex</code> property, the printing of the complex changes.</p>

<p>The algebra in the example is <span class="SimpleMath">\(kQ/I\)</span>, where <span class="SimpleMath">\(Q\)</span> is the quiver <span class="SimpleMath">\(1 \longrightarrow 2 \longrightarrow 3\)</span> and <span class="SimpleMath">\(I\)</span> is generated by the composition of the arrows. We construct <span class="SimpleMath">\(C\)</span> as the stalk complex with the injective <span class="SimpleMath">\(I_1\)</span> in degree 0.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">alg;</span>
<Rationals[<quiver with 3 vertices and 2 arrows>]/
<two-sided ideal in <Rationals[<quiver with 3 vertices and 2 arrows>]>, 
  (1 generators)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">cat := CatOfRightAlgebraModules(alg);</span>
<cat: right modules over algebra>
<span class="GAPprompt">gap></span> <span class="GAPinput">C := StalkComplex(cat, IndecInjectiveModules(alg)[1], 0);</span>
0 -> 0:(1,0,0) -> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjC := ProjectiveResolutionOfComplex(C);</span>
--- -> 0: P1 -> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">InjC := ProjectiveToInjectiveComplex(ProjC);</span>
--- -> 1: I2 -> 0: I1 -> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">TauC := TauOfComplex(C);</span>
0 -> 1: P3 -> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsProjectiveComplex(C);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjectiveComplex(C);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">C;</span>
0 -> 0: I1 -> 0 
</pre></div>

<p><a id="X84D4D700876395AF" name="X84D4D700876395AF"></a></p>

<h5>11.2-5 StarOfMapBetweenProjectives</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; StarOfMapBetweenProjectives</code>( <var class="Arg">f</var>, <var class="Arg">list_i</var>, <var class="Arg">list_j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; StarOfMapBetweenIndecProjectives</code>( <var class="Arg">f</var>, <var class="Arg">i</var>, <var class="Arg">list_j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; StarOfMapBetweenDecompProjectives</code>( <var class="Arg">f</var>, <var class="Arg">list_i</var>, <var class="Arg">list_j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Arguments: <var class="Arg">f</var> -- a map between to projective modules <span class="SimpleMath">\(P = \bigoplus P_i\)</span> and <span class="SimpleMath">\(Q = \bigoplus Q_j\)</span>, each of which were constructed as direct sums of indecomposable projective modules; <var class="Arg">list_i</var> -- describes the summands of <span class="SimpleMath">\(P\)</span>; <var class="Arg">list_j</var> -- describes the summands of <span class="SimpleMath">\(Q\)</span>. If <span class="SimpleMath">\(P = P_1 \oplus P_3 \oplus P_3\)</span> (where <span class="SimpleMath">\(P_i\)</span> is the indecomposable projective representation in vertex <span class="SimpleMath">\(i\)</span>), then <var class="Arg">list_i</var> is [1,3,3].<br /></p>

<p>Returns: The map <span class="SimpleMath">\(f^* = \Hom_A(f,A): \Hom_A(Q,A) \rightarrow \Hom_A(P,A)\)</span> in <span class="SimpleMath">\(A^{\mathrm{op}}\)</span> (where <span class="SimpleMath">\(A\)</span> is the original algebra).</p>

<p>The function <code class="code">StarOfMapBetweenProjectives</code> is supposed to be called from within the <code class="code">ProjectiveToInjectiveComplex</code> method, and might not do as expected when called from somewhere else.</p>

<p>The other similarly named functions are called from within the first.</p>


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