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Macaulay2Doc :: Ext^ZZ(CoherentSheaf,CoherentSheaf)

Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext

Synopsis

Description

If M or N is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.

M and N must be coherent sheaves on the same projective variety or scheme X.

As an example, we compute Hom_X(I_X,OO_X), and Ext^1_X(I_X,OO_X), for the rational quartic curve in $P^3$.

i1 : S = QQ[a..d];
i2 : I = monomialCurveIdeal(S,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of S
i3 : R = S/I

o3 = R

o3 : QuotientRing
i4 : X = Proj R

o4 = X

o4 : ProjectiveVariety
i5 : IX = sheaf (module I ** R)

o5 = cokernel {2} | c2 bd ac b2 |
              {3} | -b -a 0  0  |
              {3} | d  c  -b -a |
              {3} | 0  0  -d -c |

                                         1           3
o5 : coherent sheaf on X, quotient of OO  (-2) ++ OO  (-3)
                                        X           X
i6 : Ext^1(IX,OO_X)
-- ker (59) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (59) returned CacheFunction: -*a cache function*-
-- ker (59) called with Matrix: 0
--                            8
-- ker (59) returned Module: R
assert( ker(map(R^0,R^{{5}, {5}, {5}, {5}, {6}, {6}, {6}, {6}},0)) === (R^{{5}, {5}, {5}, {5}, {6}, {6}, {6}, {6}}))
-- ker (60) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (60) returned CacheFunction: -*a cache function*-
-- ker (60) called with Matrix: 0
--                            4
-- ker (60) returned Module: R
assert( ker(map(R^0,R^{{4}, {4}, {4}, {4}},0)) === (R^{{4}, {4}, {4}, {4}}))
-- ker (61) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (61) returned CacheFunction: -*a cache function*-
-- ker (61) called with Matrix: 0
--                            4
-- ker (61) returned Module: R
assert( ker(map(R^0,R^{{2}, {3}, {3}, {3}},0)) === (R^{{2}, {3}, {3}, {3}}))

o6 = 0

o6 : QQ-module
i7 : Hom(IX,OO_X)
-- ker (62) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (62) returned CacheFunction: -*a cache function*-
-- ker (62) called with Matrix: {-4} | c2 -b d  0  |
--                              {-4} | bd -a c  0  |
--                              {-4} | ac 0  -b -d |
--                              {-4} | b2 0  -a -c |
-- ker (62) returned Module: image {-2} | c   b   a   d   c  0   b  |
--                                 {-3} | 2d2 2c2 2bd 0   d2 -cd c2 |
--                                 {-3} | bd  ac  b2  -c2 0  -ad 0  |
--                                 {-3} | 0   0   0   2ac b2 ab  a2 |
assert( ker(map(R^{{4}, {4}, {4}, {4}},R^{{2}, {3}, {3}, {3}},{{c^2,-b,d,0}, {b*d,-a,c,0}, {a*c,0,-b,-d}, {b^2,0,-a,-c}})) === (image(map(R^{{2}, {3}, {3}, {3}},R^{{1}, {1}, {1}, {1}, {1}, {1}, {1}},{{c,b,a,d,c,0,b}, {2*d^2,2*c^2,2*b*d,0,d^2,-c*d,c^2}, {b*d,a*c,b^2,-c^2,0,-a*d,0}, {0,0,0,2*a*c,b^2,a*b,a^2}}))))
-- ker (63) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (63) returned CacheFunction: -*a cache function*-
-- ker (63) called with Matrix: 0
--                            1
-- ker (63) returned Module: S
assert( ker(map(S^0,S^{{-1}},0)) === (S^{{-1}}))
-- ker (64) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (64) returned CacheFunction: -*a cache function*-
-- ker (64) called with Matrix: 0
--                            9
-- ker (64) returned Module: S
assert( ker(map(S^0,S^{{-3}, {-3}, {-3}, {-3}, {-3}, {-2}, {-2}, {-2}, {-2}},0)) === (S^{{-3}, {-3}, {-3}, {-3}, {-3}, {-2}, {-2}, {-2}, {-2}}))
-- ker (65) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (65) returned CacheFunction: -*a cache function*-
-- ker (65) called with Matrix: {-3} | -d  0   0   0    0   0   0   c   0   0   -d   c   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   -d  0   0    0   0   0   0   0   0   0    0   -d  0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   -d  0    0   0   0   0   d   c   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   -d   0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    -d  0   0   0   0   0   2d   0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   -d  0   0   0   0   0    d   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   -d  0   0   0   0    0   d   c   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   -b  0   d    0   0   0   0   0   0   0    0   -b  0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | -d  0   -b  0    0   0   0   c   b   a   -d   c   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   -b  0   0   0   0   0    b   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   -2d  -c  0   -b  0   0   0   2c   0   b   a   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | -d  0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   -d  0   0    0   0   0   0   0   0   0    0   0   0   0  b  0  -d  0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   -d  0    0   0   0   0   0   0   0    0   0   0   d  0  b  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   -d   0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    -d  0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   -d  0   0   0   0   0    0   0   0   0  0  0  0   0  b   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   -d  0   0   0   0    0   0   0   0  0  0  2d  c  0   b  0 0 0 0  0 0  0 |
--                              {-3} | -c  0   0   d    -c  0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   d   0   0   0   0   0    0   0   0   0  0  0  0   0  -b  0  0 0 0 0  0 0  0 |
--                              {-3} | 0   -d  -c  0    0   0   0   0   0   0   0    0   0   0   c  b  a  -d  c  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   -2d  0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    -d  0   0   0   0   0   0    0   0   0   0  0  0  0   b  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   -d  -c  0   0   0   0    0   0   0   0  0  0  2c  0  b   a  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   -d 0  0  0   0  0   0  c 0 0 -d c 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  -d 0  0   0  0   0  0 0 0 0  0 -d 0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  -d 0   0  0   0  0 d c 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  -d  0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   -d 0   0  0 0 0 2d 0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  -d  0  0 0 0 0  d 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   -d 0 0 0 0  0 d  c |
--                              {-3} | 0   0   0   0    0   0   0   -d  0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   -d  0   0    0   0   0   0  0  0  0   0  0   0  0 b 0 -d 0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   -d  0    0   0   0   0  0  0  0   0  0   0  d 0 b 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   -d   0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    -d  0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   -d  0   0  0  0  0   0  0   0  0 0 0 0  0 b  0 |
--                              {-3} | 0   0   0   0    0   0   0   0   0   0   0    0   0   -d  0  0  0  0   0  0   0  0 0 0 2d c 0  b |
--                              {-3} | -d  0   0   0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   -d  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 0 |
--                              {-3} | 0   0   -d  0    0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  c b a -d c 0  0 |
--                              {-3} | 0   0   0   -d   0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    -d  0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-3} | 0   0   0   0    0   -d  0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  b 0  0 |
--                              {-3} | 0   0   0   0    0   0   -d  0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 2c 0 b  a |
--                              {-4} | 0   0   0   cd   0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | d2  0   bd  0    0   0   0   0   -bd 0   d2   0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   cd  0   0    0   -d2 cd  -d2 0   -bd 0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-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 |
--                              {-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 |
--                              {-4} | 0   0   0   2d2  0   0   bd  0   0   0   0    0   -bd 0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    d2  cd  0   0   0   0   -2d2 -cd 0   -bd 0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-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 |
--                              {-4} | -bd 0   0   0    0   0   0   0   0   0   -bd  0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   -ad -ac -cd  0   bd  0   bd  ac  b2  0    0   -ad 0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-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 |
--                              {-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 |
--                              {-4} | 0   0   0   -2bd 0   0   0   0   0   0   0    0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    -bd -ad -ac 0   0   0   2bd  ad  ac  b2  0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   -d2 0   0   0    0   0   0   0  0  0  cd  0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   -d2 0   0    0   0   0   d2 0  bd 0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   0   -d2 0    0   0   0   0  cd 0  0   0  -d2 cd 0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   0   0   -d2  0   0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   0   0   0    -d2 0   0   0  0  0  0   0  0   0  0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   0   0   0    0   -d2 0   0  0  0  2d2 0  0   bd 0 0 0 0  0 0  0 |
--                              {-4} | 0   0   0   0    0   0   0   0   0   0   0    0   0   -d2 0  0  0  0   d2 cd  0  0 0 0 0  0 0  0 |
-- ker (65) returned Module: subquotient ({-2} | b  0  a  0  d   -c 0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  c2  0   0   0   0  0  0   0   |, {-2} | b  0  a  0  d   -c 0  0   0  0   0  0  c2  0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |)
--                                        {-2} | 0  c  0  a  0   0  d   d  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 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  -d2 0   0   0   0  0  0   0   |  {-2} | 0  c  0  a  0   0  0  d   d  0   0  b  -d2 -bd b2  0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | -d -d -c -b 0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  c c  0  0 -b b 0  0  a 0  0  0  -d -d 0   -c 0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   c2 bd 0   0   |  {-2} | -d -d -c -b 0   0  -c 0   0  d   0  0  0   c2  -ac 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  c   0  b   0  0  0  0  0  0   0  0   0  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  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  c   0  0  b   0  a   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  -2d c  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  2b 0  0   a  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  -2d c  a  0   0  0   b  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   -d 0   c  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  a  0 0  0 0  0  0 0  0  0  0  0  0   0  b  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   -d -b 0   c  0   0  a  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  -2d -d 0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  -b 0 0  0 0  0  0 0  0  0  0  2c 0   0  0  0  b b 0  0   0  a 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  -2d -d -2c -c -b 0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  b  0  a  0  d   -c 0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   c2  0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   b  0  a  0  d   -c 0  0   0  0   0  0  c2  0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  c  0  a  0   0  d   d  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  b 0  0  0 0  0  -d -d -d 0   0  0  0  0 0 0  0   0  0 0   0  0  0   -d2 0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  c  0  a  0   0  0  d   d  0   0  b  -d2 -bd b2  0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  -d -d -c -b 0   0  0   0  d   0  0  0  0  0  0   0  0   0  0   0  0  d d  -c c 0  0 0  0  b 0  0  0  0  0  0   -d -d 0  0 c 0  0   0  0 0   0  0  0   0   0   0   cd c2 0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   -d -d -c -b 0   0  -c 0   0  d   0  0  0   c2  -ac 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  c   0  b   0  a   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  c   0  0  b   0  a   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  -2d c  0   0  0   b  0  0  0  0  0   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 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  -2d c  a  0   0  0   b  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   -d 0   c  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 a  0 0  0  0 0  0  0  0  0  0   0  0  0  0 b 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   -d -b 0   c  0   0  a  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  -2d -d -2c -c 0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 -b 0 0  0  0 0  0  2d 2d 2d 0   c  c  0  0 0 0  0   0  b 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  -2d -d -2c -c -b 0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  b  0  a  0  d   0  0   0  0   0  0  c 0  0  0 0  0 0  0  0 0  0  -d -d -d 0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  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  0  a  0  d   -c 0  0   0  0   0  0  c2  0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  c  0  a  0   0  d   d  0   0  b  0 0  -d d 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   -bd b2  d2 0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  c  0  a  0   0  0  d   d  0   0  b  -d2 -bd b2  0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  -d -d -c -b 0   -c 0   0  d   0  0  0 0  0  0 -d d 0  0  c 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   c2  -ac 0  d2 0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   -d -d -c -b 0   0  -c 0   0  d   0  0  0   c2  -ac 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  c   0  b   0  a   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  c   0  0  b   0  a   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  -2d a  0   0  0   b  0  0 c  0  0 0  0 0  0  0 0  0  2d 2d 2d 0   0  0  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  -2d c  a  0   0  0   b  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   -b 0   c  0   0  a  0 -d 0  0 0  0 0  0  0 0  0  0  0  0  0   d  d  0  0 0 0  0   0  0 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   -d -b 0   c  0   0  a  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  -2d -d -2c -c -b 0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  d d 0  0   0  c 0   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  -2d -d -2c -c -b 0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  d d  -c c 0  0 b  0  0 a  0  0  0  0  d   -d -d -c c c 0  0   0  0 0   0  0  0   0   0   0   cd 0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  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  0  a  0  d   -c 0  0   0  0   0  0  c2  0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 -d d 0  c  0 0  a  0  0  0  0   0  0  0  0 0 0  d   d  0 0   0  b  0   0   0   0   0  d2 -bd b2  |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  c  0  a  0   0  0  d   d  0   0  b  -d2 -bd b2  |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 -d -d d -c -b 0  0  0  0   0  0  0  0 0 -c 0   0  0 d   0  0  0   0   0   0   0  0  c2  -ac |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   -d -d -c -b 0   0  -c 0   0  d   0  0  0   c2  -ac |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  d  d  d  c   0  0  0  0 0 0  b   0  0 a   0  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  c   0  0  b   0  a   0  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  -2d d  d  c  0 0 a  0   0  0 0   b  0  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  -2d c  a  0   0  0   b  0  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  -d d d -b 0   c  0 0   0  a  0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   -d -b 0   c  0   0  a  0   0   0   |
--                                        {-2} | 0  0  0  0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0  0  0  0   0  0   0  0   0  0  0 0  0  0 0  0 0  0  0 0  0  0  0  0  0   0  0  0  0 0 0  -2d -d d -2c -c -b 0   0   0   0   0  0  0   0   |  {-2} | 0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  0   0  0   0  0  0   0   0   0  0  0  0  0   0  0  -2d -d -2c -c -b 0   0   0   |
assert( ker(map(cokernel(map(R^{{3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}},R^{{2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {1}, {1}, {1}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {2}, {2}, {2}, {3}, {3}, {3}, 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       16
o7 = QQ

o7 : QQ-module, free
The Ext^1 being zero says that the point corresponding to I on the Hilbert scheme is smooth (unobstructed), and vector space dimension of Hom tells us that the dimension of the component at the point I is 16.

The method used may be found in: Smith, G., Computing global extension modules, J. Symbolic Comp (2000) 29, 729-746

If the module $\oplus_{d\geq 0} Ext^i(M,N(d))$ is desired, see Ext^ZZ(CoherentSheaf,SumOfTwists).

See also