From 8b54d2103fee3dab66de7fb233aa28a5316474e1 Mon Sep 17 00:00:00 2001
From: carlos alquezar <alquezar@unizar.es>
Date: Fri, 1 Apr 2022 13:58:23 +0200
Subject: [PATCH] working on injetivity

---
 notebook/tutorial AplK.ipynb | 382 ++++++++++++++++++++---------------
 1 file changed, 216 insertions(+), 166 deletions(-)

diff --git a/notebook/tutorial AplK.ipynb b/notebook/tutorial AplK.ipynb
index 9d42879..ee0bbd9 100644
--- a/notebook/tutorial AplK.ipynb	
+++ b/notebook/tutorial AplK.ipynb	
@@ -1566,7 +1566,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 128,
+   "execution_count": 33,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -1754,7 +1754,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 129,
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
    "metadata": {},
    "outputs": [
     {
@@ -1763,7 +1770,7 @@
        "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
       ]
      },
-     "execution_count": 129,
+     "execution_count": 34,
      "metadata": {},
      "output_type": "execute_result"
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@@ -1776,7 +1783,7 @@
   },
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-   "execution_count": 130,
+   "execution_count": 35,
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    "outputs": [
     {
@@ -1785,7 +1792,7 @@
        "{(0,): 0, (1,): 0, (2,): 0, (0, 1): 0, (0, 2): 0, (1, 2): 1, (0, 1, 2): 0}"
       ]
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-     "execution_count": 130,
+     "execution_count": 35,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1869,7 +1876,7 @@
   },
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-   "execution_count": 134,
+   "execution_count": 40,
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    "outputs": [
     {
@@ -1878,7 +1885,7 @@
        "[(0, 1), (0, 2), (1, 2)]"
       ]
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-     "execution_count": 134,
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@@ -1899,7 +1906,7 @@
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@@ -1945,16 +1952,16 @@
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      "data": {
       "text/plain": [
-       "The graded commutative differential cochain algebra Apl(K) over Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
+       "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
       ]
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+     "execution_count": 42,
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@@ -1967,7 +1974,7 @@
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@@ -1989,7 +1996,7 @@
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@@ -2003,7 +2010,7 @@
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@@ -2038,7 +2045,7 @@
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@@ -2087,16 +2094,16 @@
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-       "The graded commutative differential cochain algebra Apl(K) over Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
+       "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
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        "{(0,): 2, (1,): 1, (2,): 3, (0, 1): -t1 + 2, (0, 2): t1 + 2, (1, 2): 2*t1 + 1, (0, 1, 2): -t1 + t2 + 2}"
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@@ -2174,20 +2181,9 @@
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-       "[(0, 1), (0, 2), (1, 2)]"
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-     "execution_count": 127,
-     "metadata": {},
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-       "The graded commutative differential cochain algebra Apl(K) over Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
+       "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
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@@ -2270,7 +2266,7 @@
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@@ -2288,7 +2284,7 @@
        "{(0,): 2, (1,): 1, (2,): 3, (0, 1): -t1 + 2, (0, 2): t1 + 2, (1, 2): 2*t1 + 1, (0, 1, 2): -t1 + t2 + 2}"
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@@ -2323,7 +2319,7 @@
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@@ -2363,7 +2359,7 @@
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@@ -2459,16 +2455,16 @@
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-       "The graded commutative differential cochain algebra Apl(K) over Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
+       "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
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        "{(0,): 2, (1,): 1, (2,): 3, (0, 1): -t1 + 2, (0, 2): t1 + 2, (1, 2): 2*t1 + 1, (0, 1, 2): -t1 + t2 + 2}"
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@@ -2525,7 +2521,7 @@
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@@ -2554,7 +2550,7 @@
        "{0: {(0,): 2, (1,): 1, (2,): 3, (0, 1): -t1 + 2, (0, 2): t1 + 2, (1, 2): 2*t1 + 1, (0, 1, 2): -t1 + t2 + 2}}"
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@@ -2574,7 +2570,7 @@
        "{(0,): 2, (1,): 1, (2,): 3, (0, 1): -t1 + 2, (0, 2): t1 + 2, (1, 2): 2*t1 + 1, (0, 1, 2): -t1*y1*y2 - t2*y1*y2 - t1^2*y2 - t1*t2*y2 + y1*y2 + t1*y2 - t1 + t2 + 2}"
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-       "The graded commutative differential cochain algebra Apl(K) over Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
+       "The graded commutative differential cochain algebra Apl(K) of Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}"
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        "{(0, 1, 2, 3): -6*y1*y2*y3 - 2*t3*y1*y2 + 2*t2*y1*y3 - 2*t1*y2*y3 - t3*y1 + t1*y3, (0, 1, 2): 0, (0, 1, 3): -t2*y1 + t1*y2, (0, 2, 3): 0, (1, 2, 3): -2*y1*y2 + t2*y1 - t1*y2 + y2, (0, 1): 0, (0, 2): 0, (0, 3): 0, (1, 2): 0, (1, 3): y1, (2, 3): 0, (0,): 0, (1,): 0, (2,): 0, (3,): 0}"
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        "{(1, 0): h^{1,0}}"
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-       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,0,\\,-1,\\,0,\\,0\\right)</script></html>"
+       "<html>\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,0,\\,-1,\\,0,\\,0\\right)\\]</html>"
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        "  0): {(0, 1, 2): 0, (0, 1): 0, (0, 2): 0, (1, 2): 0, (1, 3): 0, (2, 3): y1, (0,): 0, (1,): 0, (2,): 0, (3,): 0}}"
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@@ -3338,7 +3332,7 @@
        "{(2, 0): h^{2,0}, (2, 1): h^{2,1}}"
       ]
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+     "execution_count": 80,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3353,7 +3347,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 79,
+   "execution_count": 81,
    "metadata": {},
    "outputs": [
     {
@@ -3388,7 +3382,7 @@
        " (4, 5, 6): 0}"
       ]
      },
-     "execution_count": 79,
+     "execution_count": 81,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3402,7 +3396,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 80,
+   "execution_count": 82,
    "metadata": {},
    "outputs": [
     {
@@ -3414,7 +3408,7 @@
        "  1): {(0, 1, 2): 0, (0, 1, 3): 0, (0, 2, 3): 0, (1, 2, 3): 0, (3, 4, 5): 0, (3, 4, 6): 0, (3, 5, 6): 0, (4, 5, 6): -2*y1*y2, (0, 1): 0, (0, 2): 0, (0, 3): 0, (1, 2): 0, (1, 3): 0, (2, 3): 0, (3, 4): 0, (3, 5): 0, (3, 6): 0, (4, 5): 0, (4, 6): 0, (5, 6): 0, (0,): 0, (1,): 0, (2,): 0, (3,): 0, (4,): 0, (5,): 0, (6,): 0}}"
       ]
      },
-     "execution_count": 80,
+     "execution_count": 82,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3447,7 +3441,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 81,
+   "execution_count": 83,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -3570,7 +3564,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 82,
+   "execution_count": 84,
    "metadata": {},
    "outputs": [
     {
@@ -3586,7 +3580,7 @@
        "[1, y4, y3, y2, t1^2, t2*t3*y4, t2*t4*y3, t3*t4*y2]"
       ]
      },
-     "execution_count": 82,
+     "execution_count": 84,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3603,7 +3597,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 83,
+   "execution_count": 85,
    "metadata": {},
    "outputs": [
     {
@@ -3612,7 +3606,7 @@
        "[t1*y1, t2*y3*y4, t3*y2*y4, t4*y2*y3]"
       ]
      },
-     "execution_count": 83,
+     "execution_count": 85,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3624,7 +3618,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 84,
+   "execution_count": 86,
    "metadata": {},
    "outputs": [
     {
@@ -3633,7 +3627,7 @@
        "(-6, 0, -10, -10)"
       ]
      },
-     "execution_count": 84,
+     "execution_count": 86,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3645,7 +3639,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 85,
+   "execution_count": 87,
    "metadata": {},
    "outputs": [
     {
@@ -3657,7 +3651,7 @@
        "[ 0  0  0  0  0  0  1 -1]"
       ]
      },
-     "execution_count": 85,
+     "execution_count": 87,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3669,7 +3663,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 86,
+   "execution_count": 88,
    "metadata": {},
    "outputs": [
     {
@@ -3678,7 +3672,7 @@
        "-10*t2*t4*y3 - 10*t2*t3*y4 - 3*t1^2"
       ]
      },
-     "execution_count": 86,
+     "execution_count": 88,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3689,7 +3683,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 87,
+   "execution_count": 89,
    "metadata": {},
    "outputs": [
     {
@@ -3698,7 +3692,7 @@
        "True"
       ]
      },
-     "execution_count": 87,
+     "execution_count": 89,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3723,7 +3717,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 88,
+   "execution_count": 90,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -3963,7 +3957,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 89,
+   "execution_count": 91,
    "metadata": {},
    "outputs": [
     {
@@ -3979,7 +3973,7 @@
        "{(0,): 0, (1,): 0, (2,): 0, (3,): 0, (4,): 0, (0, 1): 0, (0, 2): 0, (0, 3): 0, (0, 4): 0, (1, 2): 0, (1, 3): 0, (1, 4): 0, (2, 3): 0, (2, 4): 0, (3, 4): 0, (0, 1, 2): 0, (0, 1, 3): 0, (0, 1, 4): 0, (0, 2, 3): 0, (0, 2, 4): 0, (0, 3, 4): 0, (1, 2, 3): 0, (1, 2, 4): 0, (1, 3, 4): 0, (2, 3, 4): 0, (0, 1, 2, 3): 3*y1*y2*y3, (0, 1, 2, 4): -3*y1*y2*y3, (0, 1, 3, 4): -3*y1*y2*y3, (0, 2, 3, 4): 0, (1, 2, 3, 4): -3*y1*y2*y3, (0, 1, 2, 3, 4): 3*y1*y2*y3 - 3*y1*y2*y4 - 3*y1*y3*y4}"
       ]
      },
-     "execution_count": 89,
+     "execution_count": 91,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4028,7 +4022,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 90,
+   "execution_count": 92,
    "metadata": {},
    "outputs": [
     {
@@ -4037,7 +4031,7 @@
        "{(0, 1, 2, 3, 4): t3*y1*y2 - t4*y1*y2 - t2*y1*y3 - t4*y1*y3 + t1*y2*y3 + t2*y1*y4 + t3*y1*y4 - t1*y2*y4 - t1*y3*y4 + y1*y2 + y1*y3, (0, 1, 2, 3): t3*y1*y2 - t2*y1*y3 + t1*y2*y3 + y1*y2 + y1*y3, (0, 1, 2, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 1, 3, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 2, 3, 4): 0, (1, 2, 3, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 1, 2): y1*y2, (0, 1, 3): y1*y2, (0, 1, 4): 0, (0, 2, 3): 0, (0, 2, 4): 0, (0, 3, 4): 0, (1, 2, 3): y1*y2, (1, 2, 4): 0, (1, 3, 4): 0, (2, 3, 4): 0, (0, 1): 0, (0, 2): 0, (0, 3): 0, (0, 4): 0, (1, 2): 0, (1, 3): 0, (1, 4): 0, (2, 3): 0, (2, 4): 0, (3, 4): 0, (0,): 0, (1,): 0, (2,): 0, (3,): 0, (4,): 0}"
       ]
      },
-     "execution_count": 90,
+     "execution_count": 92,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4049,7 +4043,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 91,
+   "execution_count": 93,
    "metadata": {},
    "outputs": [
     {
@@ -4058,7 +4052,7 @@
        "True"
       ]
      },
-     "execution_count": 91,
+     "execution_count": 93,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4069,7 +4063,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 92,
+   "execution_count": 94,
    "metadata": {},
    "outputs": [
     {
@@ -4078,7 +4072,7 @@
        "True"
       ]
      },
-     "execution_count": 92,
+     "execution_count": 94,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4110,7 +4104,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 93,
+   "execution_count": 95,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4147,7 +4141,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 94,
+   "execution_count": 96,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4186,16 +4180,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 95,
+   "execution_count": 97,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(0, 4, 5), (1, 2, 5), (0, 1, 2, 3)}"
+       "Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and facets {(0,), (2, 3), (4, 5), (1, 2, 5)}"
       ]
      },
-     "execution_count": 95,
+     "execution_count": 97,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4209,7 +4203,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 96,
+   "execution_count": 98,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4220,7 +4214,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 97,
+   "execution_count": 99,
    "metadata": {},
    "outputs": [
     {
@@ -4229,7 +4223,6 @@
      "text": [
       "deg:  1\n",
       "deg:  2\n",
-      "deg:  3\n",
       "dif(recover) == dif(e):  True\n"
      ]
     }
@@ -4260,7 +4253,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 98,
+   "execution_count": 100,
    "metadata": {},
    "outputs": [
     {
@@ -4326,7 +4319,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
+   "execution_count": 101,
    "metadata": {},
    "outputs": [
     {
@@ -4342,7 +4335,7 @@
        "True"
       ]
      },
-     "execution_count": 99,
+     "execution_count": 101,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4378,7 +4371,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 100,
+   "execution_count": 102,
    "metadata": {},
    "outputs": [
     {
@@ -4394,7 +4387,7 @@
        "True"
       ]
      },
-     "execution_count": 100,
+     "execution_count": 102,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4432,7 +4425,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 101,
+   "execution_count": 103,
    "metadata": {},
    "outputs": [
     {
@@ -4448,7 +4441,7 @@
        "{(0,): 0, (1,): 0, (2,): 0, (3,): 0, (4,): 0, (0, 1): 0, (0, 2): 0, (0, 3): 0, (0, 4): 0, (1, 2): 0, (1, 3): 0, (1, 4): 0, (2, 3): 0, (2, 4): 0, (3, 4): 0, (0, 1, 2): y1*y2, (0, 1, 3): y1*y2, (0, 1, 4): 0, (0, 2, 3): 0, (0, 2, 4): 0, (0, 3, 4): 0, (1, 2, 3): y1*y2, (1, 2, 4): 0, (1, 3, 4): 0, (2, 3, 4): 0, (0, 1, 2, 3): t3*y1*y2 - t2*y1*y3 + t1*y2*y3 + y1*y2 + y1*y3, (0, 1, 2, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 1, 3, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 2, 3, 4): 0, (1, 2, 3, 4): -t3*y1*y2 + t2*y1*y3 - t1*y2*y3 + y1*y2, (0, 1, 2, 3, 4): t3*y1*y2 - t4*y1*y2 - t2*y1*y3 - t4*y1*y3 + t1*y2*y3 + t2*y1*y4 + t3*y1*y4 - t1*y2*y4 - t1*y3*y4 + y1*y2 + y1*y3}"
       ]
      },
-     "execution_count": 101,
+     "execution_count": 103,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4496,7 +4489,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 102,
+   "execution_count": 104,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4506,7 +4499,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 103,
+   "execution_count": 105,
    "metadata": {},
    "outputs": [
     {
@@ -4539,7 +4532,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 104,
+   "execution_count": 106,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4805,7 +4798,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 105,
+   "execution_count": 107,
    "metadata": {},
    "outputs": [
     {
@@ -4870,7 +4863,7 @@
        "   y3_2 --> x2_1^2"
       ]
      },
-     "execution_count": 105,
+     "execution_count": 107,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4906,7 +4899,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 106,
+   "execution_count": 108,
    "metadata": {},
    "outputs": [
     {
@@ -4915,7 +4908,7 @@
        "[x2_0^3, x2_0^2*x2_1, x2_0*x2_1^2, x2_1^3, y3_0*y3_1, y3_0*y3_2, y3_1*y3_2]"
       ]
      },
-     "execution_count": 106,
+     "execution_count": 108,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4926,7 +4919,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 107,
+   "execution_count": 109,
    "metadata": {},
    "outputs": [
     {
@@ -4935,7 +4928,7 @@
        "{2: [x2_0, x2_1], 5: [x2_1*y3_0 - x2_0*y3_1, x2_1*y3_1 - x2_0*y3_2]}"
       ]
      },
-     "execution_count": 107,
+     "execution_count": 109,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4946,7 +4939,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 108,
+   "execution_count": 110,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4956,7 +4949,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 109,
+   "execution_count": 111,
    "metadata": {},
    "outputs": [
     {
@@ -4966,9 +4959,9 @@
      "traceback": [
       "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
       "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
-      "\u001b[0;32m<ipython-input-109-21949880a51d>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
-      "\u001b[0;32m<ipython-input-88-ffaf432dae2a>\u001b[0m in \u001b[0;36msolve_primitive\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    214\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    215\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0msolve_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 216\u001b[0;31m             \u001b[0mp_basis\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_primitive_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    217\u001b[0m             \u001b[0md_basis\u001b[0m \u001b[0;34m=\u001b[0m  \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_differential_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_basis\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    218\u001b[0m             \u001b[0mdiff_matrix\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_differential_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_basis\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md_basis\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
-      "\u001b[0;32m<ipython-input-88-ffaf432dae2a>\u001b[0m in \u001b[0;36m_get_primitive_basis\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m     35\u001b[0m                         \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     36\u001b[0m                     \u001b[0;31m# add primitive elements related to the assigned polynomial\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 37\u001b[0;31m                     \u001b[0mpol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melements\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     38\u001b[0m                     \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mpol\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_primitive_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     39\u001b[0m                     \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/tmp/ipykernel_7283/764669733.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m/tmp/ipykernel_7283/334124851.py\u001b[0m in \u001b[0;36msolve_primitive\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    214\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    215\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0msolve_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 216\u001b[0;31m             \u001b[0mp_basis\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_primitive_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    217\u001b[0m             \u001b[0md_basis\u001b[0m \u001b[0;34m=\u001b[0m  \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_differential_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_basis\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    218\u001b[0m             \u001b[0mdiff_matrix\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_differential_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_basis\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md_basis\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m/tmp/ipykernel_7283/334124851.py\u001b[0m in \u001b[0;36m_get_primitive_basis\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m     35\u001b[0m                         \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     36\u001b[0m                     \u001b[0;31m# add primitive elements related to the assigned polynomial\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 37\u001b[0;31m                     \u001b[0mpol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melements\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     38\u001b[0m                     \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mpol\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_get_primitive_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     39\u001b[0m                     \u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimitive_basis\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
       "\u001b[0;31mKeyError\u001b[0m: ()"
      ]
     }
@@ -4986,7 +4979,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 112,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -4998,9 +4991,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 113,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Finite family {(0, 0): h^{0,0}, (2, 0): h^{2,0}, (2, 1): h^{2,1}}"
+      ]
+     },
+     "execution_count": 113,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "S.cohomology_ring().basis()"
    ]
@@ -5014,10 +5018,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 137,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{[x2_1*y3_0 - x2_0*y3_1]: (), [x2_1*y3_1 - x2_0*y3_2]: ()}"
+      ]
+     },
+     "execution_count": 137,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
+    "#def get_ker_elements(sullivan_model, model_morphism, A, dim):\n",
     "dim = 5\n",
     "# Get the basis of Hm\n",
     "hm_basis = sullivan_model.cohomology(dim).basis()\n",
@@ -5027,7 +5043,8 @@
     "Hc_correspondence = {}\n",
     "M = zero_matrix(QQ,len(Hc.basis()),len(hm_basis))\n",
     "i = 0\n",
-    "for gen in list(hm_basis.keys()):\n",
+    "hm_basis_keys = list(hm_basis.keys())\n",
+    "for gen in hm_basis_keys:\n",
     "    # get an element of the basis of Hm\n",
     "    # get a representative of a cohomology class in M\n",
     "    gen_repr = gen.representative()\n",
@@ -5045,6 +5062,7 @@
     "    Hc_correspondence[gen] = Hc(vcochain)\n",
     "    M[:,i] = Hc_correspondence[gen]\n",
     "    i+=1\n",
+    "    \n",
     "Hc_correspondence\n",
     "\n",
     "# falta crear la matriz"
@@ -5052,11 +5070,43 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 138,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((1, 0), (0, 1))"
+      ]
+     },
+     "execution_count": 138,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "show(M)"
+    "Kgens = M.right_kernel().gens()\n",
+    "Kgens"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 148,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[x2_1*y3_0 - x2_0*y3_1, x2_1*y3_1 - x2_0*y3_2]"
+      ]
+     },
+     "execution_count": 148,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "[e.representative() for e in hm_basis_keys]"
    ]
   },
   {
@@ -5437,7 +5487,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 9.2",
+   "display_name": "SageMath 9.5.beta2",
    "language": "sage",
    "name": "sagemath"
   },
@@ -5451,7 +5501,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.5"
+   "version": "3.8.10"
   },
   "toc-autonumbering": false,
   "toc-showmarkdowntxt": false