minimalPresentation(Module) -- minimal presentation of a module

Synopsis

• Function: minimalPresentation
• Usage:
N = minimalPresentation M
• Inputs:
• M,
• Optional inputs:
• Exclude => ..., default value {}, compute a minimal presentation of a quotient ring
• Outputs:
• N, , isomorphic to M
• Consequences:
• The isomorphism from N to M as g = N.cache.pruningMap unless M.cache.pruningMap already exists, in which case N is the same as M. The inverse isomorphism can be obtained as g^-1

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 , , or , 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