The module $\operatorname{Ext}^d(M,N)$ corresponds to equivalence classes of extensions of $N$ by $M$. In particular, an element of this module is represented by an exact sequence of the form \[ 0 \leftarrow M \leftarrow C_1 \leftarrow C_2 \leftarrow \dots \leftarrow C_{d-1} \leftarrow N \leftarrow 0 \] In particular, we have $M = C_0$ and $N = C_d$. For any such exact sequence, this method returns the map $f \colon R^1 \to \operatorname{Ext}^d(M,N)$ corresponding to the element in the Ext module.
In our first example, the module $\operatorname{Ext}^1(M,R^1)$ has one generator, in degree 0. The middle term in the corresponding short exact sequence determines an irreducible rank 2 vector bundle on the elliptic curve.
i1 : R = ZZ/101[x,y,z]/(y^2*z-x*(x-z)*(x-2*z)); |
i2 : M = truncate(1,R^1)
o2 = image | z y x |
1
o2 : R-module, submodule of R
|
i3 : N = R^1; |
i4 : E = coker map(R^{3:-1} ++ R^1,,{
{y, x, 0, 0},
{-z, 0, x, -y*z},
{0, -z, -y, x^2-3*x*z+2*z^2},
{x^2-3*x*z+2*z^2, y*z, 0, 0}
})
o4 = cokernel {1} | y x 0 0 |
{1} | -z 0 x -yz |
{1} | 0 -z -y x2-3xz+2z2 |
{0} | x2-3xz+2z2 yz 0 0 |
4
o4 : R-module, quotient of R
|
i5 : d1 = map(M, E, (cover E)^[0])
o5 = {1} | 1 0 0 0 |
{1} | 0 1 0 0 |
{1} | 0 0 1 0 |
o5 : Matrix
|
i6 : d2 = map(E, N, (cover E)_[1])
o6 = {1} | 0 |
{1} | 0 |
{1} | 0 |
{0} | 1 |
o6 : Matrix
|
i7 : C = complex{d1,d2}
1
o7 = image | z y x | <-- cokernel {1} | y x 0 0 | <-- R
{1} | -z 0 x -yz |
0 {1} | 0 -z -y x2-3xz+2z2 | 2
{0} | x2-3xz+2z2 yz 0 0 |
1
o7 : Complex
|
i8 : assert isWellDefined C |
i9 : assert isHomogeneous C |
i10 : assert(HH C == 0) |
i11 : f = yonedaExtension' C
o11 = {-1} | 0 |
{-1} | 0 |
{-1} | 0 |
{0} | 1 |
o11 : Matrix
|
Although the complex representing $f$ is only defined up to equivalence of extensions, this method returns the same complex in this example.
i12 : assert(C == yonedaExtension f) |
i13 : assert(basis(0, Ext^1(M,N)) == f) |
The trivial extension corresponds to the zero element in the Ext module.
i14 : R = ZZ/101[a,b,c,d,e]; |
i15 : M = coker matrix"a,b,d,e"
o15 = cokernel | a b d e |
1
o15 : R-module, quotient of R
|
i16 : N = coker matrix"c,d,e"
o16 = cokernel | c d e |
1
o16 : R-module, quotient of R
|
i17 : Ext^3(M,N)
o17 = subquotient ({-3} | 1 0 0 0 0 0 0 0 0 |, {-3} | 0 -b a 0 0 0 e d c 0 0 0 0 0 0 0 0 0 |)
{-3} | 0 1 0 0 0 0 0 0 0 | {-3} | 0 0 0 -b a 0 0 0 0 e d c 0 0 0 0 0 0 |
{-3} | 0 0 e d c 0 0 0 a | {-3} | 0 0 0 0 0 a 0 0 0 0 0 0 e d c 0 0 0 |
{-3} | 0 0 0 0 0 e d c b | {-3} | 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 e d c |
4
o17 : R-module, subquotient of R
|
i18 : C = complex{id_M, map(M, R^0, 0), map(R^0, N, 0), id_N}
o18 = cokernel | a b d e | <-- cokernel | a b d e | <-- 0 <-- cokernel | c d e | <-- cokernel | c d e |
0 1 2 3 4
o18 : Complex
|
i19 : assert isWellDefined C |
i20 : assert(HH C == 0) |
i21 : f = yonedaExtension' C o21 = 0 o21 : Matrix |
i22 : assert(target f === Ext^3(M,N)) |
i23 : assert(f == 0) |
i24 : D = yonedaExtension f
1 4
o24 = cokernel | a b d e | <-- R <-- R <-- cokernel {2} | d e 0 0 0 0 0 | <-- cokernel | c d e |
{2} | -b 0 e 0 0 0 0 |
0 1 2 {2} | a 0 0 e 0 0 0 | 4
{2} | 0 -b -d 0 0 0 0 |
{2} | 0 a 0 -d 0 0 0 |
{2} | 0 0 a b 0 0 0 |
{0} | 0 0 0 0 c d e |
3
o24 : Complex
|
i25 : assert(C != D) |