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    <title>aInfinity -- aInfinity algebra and module structures on free resolutions</title>
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o1 = S

o1 : PolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : R = S/(ideal(a)*ideal(a,b,c))

o2 = R

o2 : QuotientRing</code></pre>
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              <pre><code class="language-macaulay2">i3 : mR = aInfinity R;</code></pre>
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              <pre><code class="language-macaulay2">i4 : keys mR

o4 = {ring, {3, 2}, {2}, {3}, {2, 2}, resolution, {4}, {2, 3}}

o4 : List</code></pre>
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              <pre><code class="language-macaulay2">i5 : res coker presentation R

1      3      3      1
o5 = S  &lt;-- S  &lt;-- S  &lt;-- S
                           
     0      1      2      3

o5 : Complex</code></pre>
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              <pre><code class="language-macaulay2">i6 : mR#&quot;resolution&quot;

3      3      1
o6 = S  &lt;-- S  &lt;-- S
                    
     2      3      4

o6 : Complex</code></pre>
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              <pre><code class="language-macaulay2">i7 : mR#{2,2}

o7 = {3} | 0 -a 0  a 0 0  0 -c 0 |
     {3} | 0 0  -a 0 0 0  a b  0 |
     {3} | 0 0  0  0 0 -a 0 0  0 |

3      9
o7 : Matrix S  &lt;-- S</code></pre>
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              <pre><code class="language-macaulay2">i8 : X = coker map(R^2,R^{2:-1},matrix{{a,b},{b,c}})

o8 = cokernel | a b |
              | b c |

2
o8 : R-module, quotient of R</code></pre>
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              <pre><code class="language-macaulay2">i9 : mX = aInfinity(mR,X)

o9 = HashTable{{1} => | a b 0  0  0  0  |                                }
                      | b c a2 ab ac bc |
               {2, 0} => {1} | a  0 0  0 c  0 |
                         {1} | 0  0 a  0 0  0 |
                         {2} | 0  1 0  0 0  0 |
                         {2} | -1 0 0  1 0  0 |
                         {2} | 0  0 -1 0 0  1 |
                         {2} | 0  0 0  0 -1 0 |
               {2, 1} => {3} | 1 0  0 a 0 c 0 0 -a 0 0 0 0 0 0  c  0 0 |
                         {3} | 0 -1 0 0 0 0 1 0 0  0 0 0 0 0 0  0  0 0 |
                         {3} | 0 0  0 0 a 0 1 0 0  0 0 0 0 0 -a -b 0 0 |
                         {3} | 0 0  0 0 0 a 0 1 0  0 a b 0 0 0  0  0 0 |
                         {3} | 0 0  0 0 0 a 0 0 0  0 0 b 1 0 0  0  0 c |
                         {3} | 0 0  0 0 0 0 0 0 0  0 0 0 0 1 0  0  0 0 |
               {2, 2, 0} => {4} | 0 0 0 0 0 0 0 0 0 0 0  0 1 0 0  0 0 0 |
                            {4} | 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 |
               {2, 2, 1} => 0
               {2, 2} => {4} | 0 0 0 a -a 0 0 a -a 0 0 0 0 b 0 c 0 0 |
                         {4} | 0 0 0 0 0  a 0 0 0  0 0 b 0 a 0 0 0 0 |
               {2, 3, 0} => 0
               {2} => {1} | 0  ab  0  0  0  -bc |
                      {1} | 0  -a2 0  0  0  ac  |
                      {2} | -b c   -c 0  0  0   |
                      {2} | a  -b  0  -c 0  0   |
                      {2} | 0  0   a  b  -b -c  |
                      {2} | 0  0   0  0  a  b   |
               {3, 0} => {3} | 0  1 0  0 0  0 |
                         {3} | -1 0 0  0 0  0 |
                         {3} | -1 0 0  1 0  0 |
                         {3} | 0  0 -1 0 0  1 |
                         {3} | 0  0 -1 0 0  0 |
                         {3} | 0  0 0  0 -1 0 |
               {3, 1} => {4} | 0 1 0 0 a 0 -1 0 0 0 0 -c 0  0 a b 0 0 |
                         {4} | 0 0 0 0 0 0 0  1 0 0 0 0  -1 0 0 0 0 0 |
               {3, 2, 0} => 0
               {3} => {3} | c  0  |
                      {3} | 0  c  |
                      {3} | -b c  |
                      {3} | a  -b |
                      {3} | 0  -b |
                      {3} | 0  a  |
               {4, 0} => {4} | 0  1 |
                         {4} | -1 0 |
               &quot;module&quot; => cokernel | a b |
                                    | b c |
                                2      6      6      2
               &quot;resolution&quot; => S  &lt;-- S  &lt;-- S  &lt;-- S
                                                     
                               0      1      2      3

o9 : HashTable</code></pre>
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        <div>
          <p>Jesse Burke showed how to use mR,mX to make an R-free resolution</p>
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        <table class="examples">
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              <pre><code class="language-macaulay2">i10 : betti burkeResolution(X,8)

0 1  2  3  4   5   6   7    8
o10 = total: 2 6 12 26 56 120 258 554 1190
          0: 2 2  6 12 26  56 120 258  554
          1: . 4  6 14 30  64 138 296  636

o10 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i11 : betti res(X, LengthLimit =>8)

0 1 2 3  4  5  6   7   8
o11 = total: 2 2 2 6 12 26 56 120 258
          0: 2 2 2 6 12 26 56 120 258

o11 : BettiTally</code></pre>
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              <pre><code class="language-macaulay2">i12 : Y = image presentation X

o12 = image | a b |
            | b c |

2
o12 : R-module, submodule of R</code></pre>
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              <pre><code class="language-macaulay2">i13 : burkeResolution(Y,8)

2      2      6      12      26      56      120      258      554
o13 = R  &lt;-- R  &lt;-- R  &lt;-- R   &lt;-- R   &lt;-- R   &lt;-- R    &lt;-- R    &lt;-- R
                                                                      
      0      1      2      3       4       5       6        7        8

o13 : Complex</code></pre>
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        <h2>References</h2>
        <div>
          <p>Jesse Burke, Higher Homotopies and Golod Rings. arXiv:1508.03782v2, October 2015.</p>
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        <h2>Caveat</h2>
        <div>
          <p>Requires standard graded ring, module. Something to fix in a future version</p>
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        <h2>See also</h2>
        <ul>
          <li><span><a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> -- aInfinity algebra and module structures on free resolutions</span></li>
        </ul>
      </div>
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        <div class="waystouse">
          <h2>Ways to use <span class="tt">aInfinity</span>:</h2>
          <ul>
            <li><kbd>aInfinity(HashTable,Module)</kbd></li>
            <li><kbd>aInfinity(Module)</kbd></li>
            <li><kbd>aInfinity(Ring)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="aInfinity algebra and module structures on free resolutions" href="_a__Infinity.html">aInfinity</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">AInfinity.m2:1176:0</span>.</p>
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