is an ideal or ring, it is regarded as a module in the evident way.
i1 : R = ZZ/32003[a..d];
|
i2 : I = monomialCurveIdeal(R,{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 R
|
i3 : M = R^1/I
o3 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |
1
o3 : R-module, quotient of R
|
i4 : Ext^1(M,R)
o4 = subquotient ({-2} | bc-ad |, {-2} | bc-ad |)
{-3} | b3-a2c | {-3} | b3-a2c |
{-3} | ac2-b2d | {-3} | ac2-b2d |
{-3} | c3-bd2 | {-3} | c3-bd2 |
4
o4 : R-module, subquotient of R
|
i5 : Ext^2(M,R)
o5 = subquotient ({-4} | c b 0 a 0 0 |, {-4} | -b2 c a 0 |)
{-4} | d 0 b 0 a 0 | {-4} | -ac d b 0 |
{-4} | 0 d -c 0 0 a | {-4} | -bd 0 -c a |
{-4} | 0 0 0 -d c b | {-4} | -c2 0 -d b |
4
o5 : R-module, subquotient of R
|
i6 : Ext^3(M,R)
o6 = cokernel {-5} | d -c -b a |
1
o6 : R-module, quotient of R
|
i7 : Ext^1(I,R)
o7 = subquotient ({-4} | c a 0 b 0 0 |, {-4} | b2 0 -a -c |)
{-4} | d 0 a 0 b 0 | {-4} | ac 0 -b -d |
{-4} | 0 -d c 0 0 b | {-4} | c2 -b d 0 |
{-4} | 0 0 0 d -c a | {-4} | bd -a c 0 |
4
o7 : R-module, subquotient of R
|
As an efficiency consideration, it is generally much more efficient to compute Ext^i(R^1/I,N) rather than Ext^(i-1)(I,N). The latter first computes a presentation of the ideal I, and then a free resolution of that. For many examples, the difference in time and space required can be very large.