i1 : (err, output) = capture examples(resolution, Ideal)
o1 = (false,
i1 : -- examples for tag: freeResolution(Ideal)
-- /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaula
R = QQ[a..d];
i2 : I = ideal(c^2-b*d, b*c-a*d, b^2-a*c)
2 2
o2 = ideal (c - b*d, b*c - a*d, b - a*c)
o2 : Ideal of R
i3 : M = R^1/I
o3 = cokernel | c2-bd bc-ad b2-ac |
1
o3 : R-module, quotient of R
i4 : C = freeResolution M
1 3 2
o4 = R <-- R <-- R
0 1 2
o4 : Complex
i5 : length C
o5 = 2
i6 : betti C
0 1 2
o6 = total: 1 3 2
0: 1 . .
1: . 3 2
o6 : BettiTally
i7 : dd^C
1 3
o7 = 0 : R <------------------------- R : 1
| b2-ac bc-ad c2-bd |
3 2
1 : R <----------------- R : 2
{2} | -c d |
{2} | b -c |
{2} | -a b |
o7 : ComplexMap
i8 : assert isWellDefined C
i9 : assert(prune HH C == complex M)
i10 : assert(length HH C == 0)
i11 : f = augmentationMap C
1
o11 = 0 : cokernel | c2-bd bc-ad b2-ac | <--------- R : 0
| 1 |
o11 : ComplexMap
i12 : assert isWellDefined f
i13 : assert(source f === C)
i14 : assert(target f == complex M)
i15 : aC = cone f
1 3 2
o15 = M <-- R <-- R <-- R
0 1 2 3
o15 : Complex
i16 : assert(HH aC == 0)
i17 : assert(freeResolution I == C)
i18 : assert(resolution complex M == freeResolution M)
i19 : C' = freeResolution module I
3 2
o19 = R <-- R
0 1
o19 : Complex
i20 : assert isWellDefined C'
i21 : assert(C' != C)
i22 : assert(betti naiveTruncation(C, 1, infinity) == betti C'[-
i23 : S = ZZ/101[a,b];
i24 : R = S/(a^3+b^3);
i25 : C = freeResolution (coker vars R, LengthLimit => 7)
1 2 2 2 2 2 2 2
o25 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6 7
o25 : Complex
i26 : length C
o26 = 7
i27 : betti C
0 1 2 3 4 5 6 7
o27 = total: 1 2 2 2 2 2 2 2
0: 1 2 1 . . . . .
1: . . 1 2 1 . . .
2: . . . . 1 2 1 .
3: . . . . . . 1 2
o27 : BettiTally
i28 : dd^C
1 2
o28 = 0 : R <----------- R : 1
| a b |
2 2
1 : R <----------------- R : 2
{1} | -b a2 |
{1} | a b2 |
2 2
2 : R <------------------ R : 3
{2} | a2 -b2 |
{3} | b a |
2 2
3 : R <----------------- R : 4
{4} | -a b2 |
{4} | b a2 |
2 2
4 : R <------------------ R : 5
{5} | -a2 b2 |
{6} | b a |
2 2
5 : R <----------------- R : 6
{7} | -a b2 |
{7} | b a2 |
2 2
6 : R <------------------ R : 7
{8} | -a2 b2 |
{9} | b a |
o28 : ComplexMap
i29 : E = ZZ/101[e_1..e_6, SkewCommutative => true];
i30 : I = ideal(e_4*e_5-e_4*e_6+e_5*e_6,
e_2*e_3-e_2*e_6+e_3*e_6,
e_1*e_3-e_1*e_5+e_3*e_5,
e_1*e_2-e_1*e_4+e_2*e_4)
o30 = ideal (e e - e e + e e , e e - e e + e e , e e - e e
4 5 4 6 5 6 2 3 2 6 3 6 1 3 1 5
----------------------------------------------------------
- e e + e e )
1 4 2 4
o30 : Ideal of E
i31 : F = freeResolution(I, LengthLimit => 5)
1 4 10 21 45 91
o31 = E <-- E <-- E <-- E <-- E <-- E
0 1 2 3 4 5
o31 : Complex
i32 : assert isWellDefined F
i33 : assert isHomogeneous F
i34 : betti F
0 1 2 3 4 5
o34 = total: 1 4 10 21 45 91
0: 1 . . . . .
1: . 4 10 15 20 25
2: . . . 6 25 66
o34 : BettiTally
i35 : S = QQ[x,y,Dx,Dy, WeylAlgebra => {{x,Dx}, {y,Dy}}];
i36 : I = ideal(x*Dy, y*Dx)
o36 = ideal (x*Dy, y*Dx)
o36 : Ideal of S
i37 : F = freeResolution comodule I
1 2 4 4 1
o37 = S <-- S <-- S <-- S <-- S
0 1 2 3 4
o37 : Complex
i38 : assert isWellDefined F
i39 : dd^F
1 2
o39 = 0 : S <--------------- S : 1
| xDy yDx |
2
1 : S <--------------------------------------------------
{2} | yDx^2Dy+2Dx^2 xyDxDy+2xDx-yDy-2 y2Dx^2
{2} | -xDxDy^2-2Dy^2 -x2Dy^2 -xyDxDy+x
4 4
2 : S <--------------------------- S : 3
{6} | x y 0 0 |
{6} | -Dx 0 y 0 |
{6} | 0 -Dy 0 x |
{6} | 0 0 -Dy -Dx |
4 1
3 : S <--------------- S : 4
{7} | y |
{7} | -x |
{7} | Dx |
{7} | -Dy |
o39 : ComplexMap
i40 :
------------------------------------------------------------------------
)
y2/packages/Complexes/ChainComplexDoc.m2:1115:0
1])
+ e e , e e
3 5 1 2
-------------
4
----------------------- S : 2
xy2Dx-y2 |
Dx-2yDy+2 -x2yDy+x2 |
o1 : Sequence
|