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    <title>NCMatrix // NCMatrix -- Factor one map through another</title>
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o1 = B

o1 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : A = ambient B

o2 = A

o2 : NCPolynomialRing</code></pre>
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          <p>The element g below is central in B (the two is just for convenience).</p>
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              <pre><code class="language-macaulay2">i3 : g = 2*(-y^3-x*y*z+y*x*z+x^3)

3             3
o3 = -2y +2yxz-2xyz+2x

o3 : A</code></pre>
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              <pre><code class="language-macaulay2">i4 : J = (ideal B) + ncIdeal {g}

2      2       2           3             3
o4 = Two-sided ideal {zy+yz-x , zx-y +xz, -z +yx+xy, -2y +2yxz-2xyz+2x }

o4 : NCIdeal</code></pre>
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              <pre><code class="language-macaulay2">i5 : B' = A/J -- Factor of sklyanin
--Calling Bergman for NCGB calculation.
Complete!

o5 = B'

o5 : NCQuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i6 : k = ncMatrix {{x,y,z}}

o6 = | x y z |

o6 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i7 : BprimeToB = ncMap(B,B',gens B) -- way to lift back from B' to B

o7 = NCRingMap B &lt;--- B'

o7 : NCRingMap</code></pre>
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              <pre><code class="language-macaulay2">i8 : M = BprimeToB rightKernelBergman rightKernelBergman k  -- second syzygy of k over B
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!
--Calling Bergman for NCGB calculation.
Complete!

o8 : NCMatrix</code></pre>
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          <p>At this point, M is maximal Cohen-Macaulay B'-module, and hence the projective dimension of M as a B-module is 1.  Since M is a B' module, multiplication by g on the complex that gives the resolution over B is null homotopic.  This means we may factor the map f through f times the identity.  We do so below.</p>
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              <pre><code class="language-macaulay2">i9 : gId = g*(ncMatrix applyTable(entries id_(ZZ^4), i -> promote(i,B)))

o9 = | -2*y^3+2*y*x*z-2*x*y*z+2*x^3 0                            0                            0                            |
     | 0                            -2*y^3+2*y*x*z-2*x*y*z+2*x^3 0                            0                            |
     | 0                            0                            -2*y^3+2*y*x*z-2*x*y*z+2*x^3 0                            |
     | 0                            0                            0                            -2*y^3+2*y*x*z-2*x*y*z+2*x^3 |

o9 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i10 : assignDegrees(gId,{2,2,2,3},{5,5,5,6});</code></pre>
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              <pre><code class="language-macaulay2">i11 : M' = gId // M
--Calling Bergman for NCGB calculation.
Complete!

o11 : NCMatrix</code></pre>
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              <pre><code class="language-macaulay2">i12 : M*M' == gId

o12 = true</code></pre>
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        <div class="waystouse">
          <h2>Ways to use this method:</h2>
          <ul>
            <li><span><a title="Factor one map through another" href="___N__C__Matrix_sp_sl_sl_sp__N__C__Matrix.html">NCMatrix // NCMatrix</a> -- Factor one map through another</span></li>
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        <div class="waystouse">
          <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:1019:0</span>.</p>
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