The module $\operatorname{Ext}^d_R(M,N)$ is constructed from a free resolution $F$ of $M$, \[ 0 \leftarrow M \leftarrow F_0 \leftarrow F_1 \leftarrow \dots \leftarrow F_d \leftarrow \ldots, \] by taking the homology of the complex $\operatorname{Hom}_R(F, N)$. An element of $\operatorname{Ext}^d_R(M,N)$ is represented by an element of $\operatorname{Hom}_R(F_d, N)$. Given a map $g$ extending a map of degree $-d$ from $F$ to the free resolution of $N$, this method returns the corresponding element in the Ext module.
We illustrate this method by choosing a random element in an Ext module, constructing the corresponding map $g$ between free resolutions.
i1 : S = ZZ/101[a..d] o1 = S o1 : PolynomialRing |
i2 : I = ideal"a2,ab,ac,b3"
2 3
o2 = ideal (a , a*b, a*c, b )
o2 : Ideal of S
|
i3 : E = Ext^1(I, S^1/I)
o3 = subquotient ({-3} | 0 a b 0 0 0 0 0 |, {-3} | -b a 0 0 0 0 0 ac ab a2 0 0 0 0 0 0 0 b3 0 0 |)
{-3} | 0 0 c a 0 0 0 b2 | {-3} | -c 0 a 0 0 0 0 0 0 0 ac ab a2 0 0 0 0 0 b3 0 |
{-3} | 0 0 0 0 c b a 0 | {-3} | 0 -c b 0 0 0 0 0 0 0 0 0 0 ac ab a2 0 0 0 b3 |
{-4} | 1 0 0 0 0 0 0 0 | {-4} | 0 b2 0 -a ac ab a2 0 0 0 0 0 0 0 0 0 b3 0 0 0 |
4
o3 : S-module, subquotient of S
|
i4 : B = basis(0, E)
o4 = {-4} | bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 0 0 0 0 0 0 0 |
{-2} | 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 |
{-2} | 0 0 0 0 0 0 0 0 0 0 d2 0 0 0 0 0 |
{-2} | 0 0 0 0 0 0 0 0 0 0 0 c2 cd d2 0 0 |
{-2} | 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 d2 0 |
{-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d |
o4 : Matrix
|
i5 : f0 = B * random(S^16, S^1)
o5 = {-4} | 24bc3+19c4-36bc2d+19c3d-30bcd2-10c2d2-29bd3-29cd3-8d4 |
{-2} | -22d2 |
{-2} | 0 |
{-2} | -29d2 |
{-2} | -24c2-38cd-16d2 |
{-2} | 0 |
{-2} | 39d2 |
{-1} | 21d |
o5 : Matrix
|
i6 : g = yonedaMap f0
1 4
o6 = 0 : S <-------------------------------------------------------------------------------------------------------- S : 1
| -22ad2 21b2d-29ad2 -24c3-38c2d+39ad2-16cd2 24bc3+19c4-36bc2d+19c3d-30bcd2-10c2d2-29bd3-29cd3-8d4 |
4 1
1 : S <-------------------------- S : 2
{2} | -39d2 |
{2} | -29d2 |
{2} | 24c2+38cd+38d2 |
{3} | 21d |
o6 : ComplexMap
|
i7 : assert(degree g === -1) |
i8 : f = yonedaMap' g
o8 = {-4} | 24bc3+19c4-36bc2d+19c3d-30bcd2-10c2d2-29bd3-29cd3-8d4 |
{-2} | -22d2 |
{-2} | 0 |
{-2} | -29d2 |
{-2} | -24c2-38cd-16d2 |
{-2} | 0 |
{-2} | 39d2 |
{-1} | 21d |
o8 : Matrix
|
i9 : assert isWellDefined f |
i10 : assert(degree f == {0})
|
i11 : assert isHomogeneous f |
i12 : source f === S^1 o12 = true |
i13 : target f === E o13 = true |
i14 : assert(f == f0) |
The method yonedaMap' is only a one-sided inverse to yonedaMap.
i15 : R = ZZ/101[x,y,z]/(y^2*z-x*(x-z)*(x-2*z)); |
i16 : M = truncate(1,R^1)
o16 = image | z y x |
1
o16 : R-module, submodule of R
|
i17 : B = basis(-4, Ext^3(M, M))
o17 = {-4} | 0 0 0 |
{-4} | 1 0 0 |
{-4} | 0 1 0 |
{-4} | 0 0 0 |
{-4} | 0 0 1 |
{-4} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
{-3} | 0 0 0 |
o17 : Matrix
|
i18 : f0 = B_{2}
o18 = {-4} | 0 |
{-4} | 0 |
{-4} | 0 |
{-4} | 0 |
{-4} | 1 |
{-4} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
o18 : Matrix
|
i19 : g = yonedaMap(f0, LengthLimit => 8)
3 4
o19 = 0 : R <------------------------- R : 3
{1} | -49 0 0 0 |
{1} | 0 -49 -50 0 |
{1} | 50 0 0 z |
4 4
1 : R <----------------------------------- R : 4
{2} | 49 0 -46x-3z 0 |
{2} | 0 -49x 0 -46x-3z |
{2} | 50 0 -49x 0 |
{3} | 0 50 0 49 |
4 4
2 : R <------------------------------- R : 5
{3} | 0 -46x-3z -49x yz |
{4} | -49 0 0 x |
{4} | 0 -49 -50 0 |
{4} | 50 0 0 0 |
4 4
3 : R <----------------------------- R : 6
{5} | 0 0 0 x |
{5} | -50 0 49x+z 0 |
{5} | 49 0 -47x-3z 0 |
{6} | 0 50 0 49 |
4 4
4 : R <-------------------------------- R : 7
{6} | 0 -47x-3z -49x-z 0 |
{7} | -49 0 0 x |
{7} | 0 -49 -50 0 |
{7} | 50 0 0 0 |
4 4
5 : R <----------------------------- R : 8
{8} | 0 0 0 x |
{8} | -50 0 49x+z 0 |
{8} | 49 0 -47x-3z 0 |
{9} | 0 50 0 49 |
o19 : ComplexMap
|
i20 : f = yonedaMap' g
o20 = {-4} | 0 |
{-4} | 0 |
{-4} | 0 |
{-4} | 0 |
{-4} | 1 |
{-4} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
{-3} | 0 |
o20 : Matrix
|
i21 : assert isWellDefined f |
i22 : assert isHomogeneous f |
i23 : assert(degree f === {-4})
|
i24 : assert(f != f0) |
i25 : assert(yonedaMap(f, LengthLimit => 8) == g) |