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Macaulay2Doc :: minimalPresentation(Module)

minimalPresentation(Module) -- minimal presentation of a module

Synopsis

Description

If the Module M is graded then the module N is a minimal presentation of M. If not, then an attempt is made to improve the presentation of M. An example follows.
i1 : R = ZZ/32003[a..d];
i2 : M = coker matrix {{a,1,b},{c,3,b+d}}

o2 = cokernel | a 1 b   |
              | c 3 b+d |

                            2
o2 : R-module, quotient of R
i3 : N = minimalPresentation M

o3 = cokernel | b+16001d a-10668c |

                            1
o3 : R-module, quotient of R
i4 : peek N.cache

o4 = CacheTable{cache => MutableHashTable{}}
                pruningMap => | -10668 |
                              | 0      |
i5 : g = N.cache.pruningMap

o5 = | -10668 |
     | 0      |

o5 : Matrix
i6 : g^-1

o6 = | -3 1 |

o6 : Matrix
This function also works when M is a graded module, a chain complex, or a coherent sheaf, by acting on the modules and maps within it.
i7 : I = ideal(a^2,b^3,c^4,d^7)

             2   3   4   7
o7 = ideal (a , b , c , d )

o7 : Ideal of R
i8 : X = Proj R

o8 = X

o8 : ProjectiveVariety
i9 : J = (module I)~

o9 = image | a2 b3 c4 d7 |

                                         1
o9 : coherent sheaf on X, subsheaf of OO
                                        X
i10 : minimalPresentation J
-- ker (36) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (36) returned CacheFunction: -*a cache function*-
-- ker (36) called with Matrix: 0
--                            1
-- ker (36) returned Module: R
assert( ker(map(R^0,R^{{12}},0)) === (R^{{12}}))
-- ker (37) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (37) returned CacheFunction: -*a cache function*-
-- ker (37) called with Matrix: 0
--                            4
-- ker (37) returned Module: R
assert( ker(map(R^0,R^{{5}, {8}, {9}, {10}},0)) === (R^{{5}, {8}, {9}, {10}}))
-- ker (38) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (38) returned CacheFunction: -*a cache function*-
-- ker (38) called with Matrix: {-24} | 0      0     0     0    a13   0     0     0    0     0     0     0    0     0    0   0   |
--                              {-23} | -a2b10 -b13  0     0    0     a13   0     0    0     0     0     0    0     0    0   0   |
--                              {-22} | 0      0     -b13  0    0     0     a13   0    0     0     0     0    0     0    0   0   |
--                              {-19} | 0      0     0     -b13 0     0     0     a13  0     0     0     0    0     0    0   0   |
--                              {-24} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    0   0   |
--                              {-23} | 0      0     0     0    0     0     0     0    a2b10 b13   0     0    0     0    0   0   |
--                              {-22} | 0      0     0     0    -a2c9 -b3c9 -c13  0    0     0     b13   0    0     0    0   0   |
--                              {-19} | 0      0     0     0    0     0     0     -c13 0     0     0     b13  0     0    0   0   |
--                              {-24} | 0      0     0     0    0     0     0     0    a13   0     0     0    0     0    0   0   |
--                              {-23} | 0      0     0     0    0     0     0     0    0     a13   0     0    0     0    0   0   |
--                              {-22} | -a2c9  -b3c9 -c13  0    0     0     0     0    0     0     a13   0    0     0    0   0   |
--                              {-19} | 0      0     0     -c13 0     0     0     0    0     0     0     a13  0     0    0   0   |
--                              {-24} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    0   0   |
--                              {-23} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    0   0   |
--                              {-22} | 0      0     0     0    0     0     0     0    0     0     0     0    a2c9  b3c9 c13 0   |
--                              {-19} | 0      0     0     0    0     0     0     0    -a2d6 -b3d6 -c4d6 -d13 0     0    0   c13 |
--                              {-24} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    0   0   |
--                              {-23} | 0      0     0     0    0     0     0     0    0     0     0     0    a2b10 b13  0   0   |
--                              {-22} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    b13 0   |
--                              {-19} | 0      0     0     0    -a2d6 -b3d6 -c4d6 -d13 0     0     0     0    0     0    0   b13 |
--                              {-24} | 0      0     0     0    0     0     0     0    0     0     0     0    a13   0    0   0   |
--                              {-23} | 0      0     0     0    0     0     0     0    0     0     0     0    0     a13  0   0   |
--                              {-22} | 0      0     0     0    0     0     0     0    0     0     0     0    0     0    a13 0   |
--                              {-19} | -a2d6  -b3d6 -c4d6 -d13 0     0     0     0    0     0     0     0    0     0    0   a13 |
-- ker (38) returned Module: subquotient ({-11} | b3  0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   0   0   0   0   0   a11 |, {-11} | b3  c4  0   d7  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |)
--                                        {-10} | -a2 0   0   0   0   0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   0   0   |  {-10} | -a2 0   c4  0   d7  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-9}  | 0   0   0   0   -a2 0   0   0   -b3 0   0   0   0   0   0   0   0   0   0   0   d7  0   0   0   0   |  {-9}  | 0   -a2 -b3 0   0   d7  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   -c4 0   0   0   0   |  {-6}  | 0   0   0   -a2 -b3 -c4 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-11} | 0   b3  0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   0   0   0   0   0   |  {-11} | 0   0   0   0   0   0   b3  c4  0   d7  0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-10} | 0   -a2 0   0   0   0   0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   b10 |  {-10} | 0   0   0   0   0   0   -a2 0   c4  0   d7  0   0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-9}  | 0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   0   0   0   0   0   0   0   0   d7  0   0   0   |  {-9}  | 0   0   0   0   0   0   0   -a2 -b3 0   0   d7  0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   -c4 0   0   0   |  {-6}  | 0   0   0   0   0   0   0   0   0   -a2 -b3 -c4 0   0   0   0   0   0   0   0   0   0   0   0   |
--                                        {-11} | 0   0   b3  0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   0   0   0   0   |  {-11} | 0   0   0   0   0   0   0   0   0   0   0   0   b3  c4  0   d7  0   0   0   0   0   0   0   0   |
--                                        {-10} | 0   0   -a2 0   0   0   0   0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   |  {-10} | 0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   c4  0   d7  0   0   0   0   0   0   0   |
--                                        {-9}  | 0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   0   0   0   0   0   0   0   0   d7  0   c9  |  {-9}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 -b3 0   0   d7  0   0   0   0   0   0   |
--                                        {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   -c4 0   0   |  {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 -b3 -c4 0   0   0   0   0   0   |
--                                        {-11} | 0   0   0   b3  0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   0   0   0   0   |  {-11} | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   b3  c4  0   d7  0   0   |
--                                        {-10} | 0   0   0   -a2 0   0   0   0   0   0   0   c4  0   0   0   0   0   0   0   d7  0   0   0   0   0   |  {-10} | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   c4  0   d7  0   |
--                                        {-9}  | 0   0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   0   0   0   0   0   0   0   0   d7  0   |  {-9}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 -b3 0   0   d7  |
--                                        {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 0   0   0   -b3 0   0   0   -c4 d6  |  {-6}  | 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -a2 -b3 -c4 |
assert( ker(map(cokernel(map(R^{{24}, {23}, {22}, {19}, {24}, {23}, {22}, {19}, {24}, {23}, {22}, {19}, {24}, {23}, {22}, {19}, {24}, {23}, {22}, {19}, {24}, {23}, {22}, {19}},R^{{21}, {20}, {19}, {17}, {16}, {15}, {21}, {20}, {19}, {17}, {16}, {15}, {21}, {20}, {19}, {17}, {16}, {15}, {21}, {20}, {19}, {17}, {16}, {15}, {21}, {20}, {19}, {17}, {16}, {15}, {21}, {20}, {19}, {17}, {16}, {15}},{{b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4}})),cokernel(map(R^{{11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}},R^{{8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}},{{b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4}})),{{0,0,0,0,a^13,0,0,0,0,0,0,0,0,0,0,0}, {-a^2*b^10,-b^13,0,0,0,a^13,0,0,0,0,0,0,0,0,0,0}, {0,0,-b^13,0,0,0,a^13,0,0,0,0,0,0,0,0,0}, {0,0,0,-b^13,0,0,0,a^13,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,a^2*b^10,b^13,0,0,0,0,0,0}, {0,0,0,0,-a^2*c^9,-b^3*c^9,-c^13,0,0,0,b^13,0,0,0,0,0}, {0,0,0,0,0,0,0,-c^13,0,0,0,b^13,0,0,0,0}, {0,0,0,0,0,0,0,0,a^13,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,a^13,0,0,0,0,0,0}, {-a^2*c^9,-b^3*c^9,-c^13,0,0,0,0,0,0,0,a^13,0,0,0,0,0}, {0,0,0,-c^13,0,0,0,0,0,0,0,a^13,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,a^2*c^9,b^3*c^9,c^13,0}, {0,0,0,0,0,0,0,0,-a^2*d^6,-b^3*d^6,-c^4*d^6,-d^13,0,0,0,c^13}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,a^2*b^10,b^13,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^13,0}, {0,0,0,0,-a^2*d^6,-b^3*d^6,-c^4*d^6,-d^13,0,0,0,0,0,0,0,b^13}, {0,0,0,0,0,0,0,0,0,0,0,0,a^13,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,a^13,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,a^13,0}, {-a^2*d^6,-b^3*d^6,-c^4*d^6,-d^13,0,0,0,0,0,0,0,0,0,0,0,a^13}})) === (subquotient(map(R^{{11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}},R^{{8}, {8}, {8}, {8}, {7}, {7}, {7}, {7}, {6}, {6}, {6}, {6}, {4}, {4}, {4}, {4}, {3}, {3}, {3}, {3}, {2}, {2}, {2}, {2}, {0}},{{b^3,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,a^11}, {-a^2,0,0,0,0,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,0,0}, {0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,0,0,0,0,0,0,0,0,d^7,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,-c^4,0,0,0,0}, {0,b^3,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0}, {0,-a^2,0,0,0,0,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,b^10}, {0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,0,0,0,0,0,0,0,0,d^7,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,-c^4,0,0,0}, {0,0,b^3,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,0,0,0,0}, {0,0,-a^2,0,0,0,0,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0}, {0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,0,0,0,0,0,0,0,0,d^7,0,c^9}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,-c^4,0,0}, {0,0,0,b^3,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0,0,0,0,0}, {0,0,0,-a^2,0,0,0,0,0,0,0,c^4,0,0,0,0,0,0,0,d^7,0,0,0,0,0}, {0,0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,0,0,0,0,0,0,0,0,d^7,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,0,0,-b^3,0,0,0,-c^4,d^6}}),map(R^{{11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}, {11}, {10}, {9}, {6}},R^{{8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}, {8}, {7}, {6}, {4}, {3}, {2}},{{b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,b^3,c^4,0,d^7,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,0,c^4,0,d^7,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,0,0,d^7}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-a^2,-b^3,-c^4}}))))

         1
o10 = OO
        X

o10 : coherent sheaf on X, free

See also