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

## Synopsis

• Scripted functor: Ext
• Usage:
Ext^i(M,N)
• Inputs:
• Outputs:
• , The global Ext module $Ext^i_X(M,N)$

## 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}, 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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).