Warning! This function is very rough currently. It workes if one uses it in the intended manner, as in the example below. But it should be much more general, handling other rings with grace, and also it should handle arbitrary (graded) chain complexes.
i1 : R = QQ[a..d] o1 = R o1 : PolynomialRing |
i2 : I = ideal(a^3, b^3, c^3, d^3, (a+3*b+7*c-4*d)^3)
3 3 3 3 3 2 2 3 2
o2 = ideal (a , b , c , d , a + 9a b + 27a*b + 27b + 21a c + 126a*b*c +
------------------------------------------------------------------------
2 2 2 3 2 2
189b c + 147a*c + 441b*c + 343c - 12a d - 72a*b*d - 108b d - 168a*c*d
------------------------------------------------------------------------
2 2 2 2 3
- 504b*c*d - 588c d + 48a*d + 144b*d + 336c*d - 64d )
o2 : Ideal of R
|
i3 : C = res(ideal gens gb I, Strategy=>4.1)
1 9 25 31 18 4
o3 = R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5
o3 : ChainComplex
|
i4 : betti C
0 1 2 3 4 5
o4 = total: 1 9 25 31 18 4
0: 1 . . . . .
1: . . . . . .
2: . 5 1 . . .
3: . 1 3 1 . .
4: . 3 17 13 4 .
5: . . 4 13 10 3
6: . . . 4 3 1
7: . . . . 1 .
o4 : BettiTally
|
i5 : CR = constantStrand(C, RR_53, 3)
5
o5 = 0 <-- RR <-- 0 <-- 0 <-- 0 <-- 0
53
0 2 3 4 5
1
o5 : ChainComplex
|
i6 : CR.dd_2
o6 = 0
5
o6 : Matrix RR <--- 0
53
|
i7 : CR2 = constantStrand(C, RR_1000, 3)
5
o7 = 0 <-- RR <-- 0 <-- 0 <-- 0 <-- 0
1000
0 2 3 4 5
1
o7 : ChainComplex
|
i8 : CR2.dd_2
o8 = 0
5
o8 : Matrix RR <--- 0
1000
|
i9 : kk1 = ZZ/32003 o9 = kk1 o9 : QuotientRing |
i10 : kk2 = ZZ/1073741909 o10 = kk2 o10 : QuotientRing |
i11 : Cp1 = constantStrand(C, kk1, 3)
5
o11 = 0 <-- kk1 <-- 0 <-- 0 <-- 0 <-- 0
0 1 2 3 4 5
o11 : ChainComplex
|
i12 : Cp2 = constantStrand(C, kk2, 3)
5
o12 = 0 <-- kk2 <-- 0 <-- 0 <-- 0 <-- 0
0 1 2 3 4 5
o12 : ChainComplex
|
i13 : netList {{CR.dd_4, CR2.dd_4}, {Cp1.dd_4, Cp2.dd_4}}
+-+-+
o13 = |0|0|
+-+-+
|0|0|
+-+-+
|
i14 : (clean(1e-14,CR)).dd_4 o14 = 0 o14 : Matrix 0 <--- 0 |
i15 : netList {(clean(1e-14,CR)).dd_4}==netList {(clean(1e-299,CR2)).dd_4}
o15 = true
|
Setting the input ring to be the integers, although a hack, sets each entry to the number of multiplications used to create this number. Warning: the result is almost certainly not a complex! This part of this function is experimental, and will likely change in later versions.
i16 : CZ = constantStrand(C, ZZ, 8)
13 4
o16 = 0 <-- 0 <-- 0 <-- ZZ <-- ZZ <-- 0
0 1 2 3 4 5
o16 : ChainComplex
|
i17 : CZ.dd_4
o17 = | 0 0 0 0 |
| 3 0 0 0 |
| 4 3 2 0 |
| 5 0 3 2 |
| 0 3 0 0 |
| 5 2 3 0 |
| 6 5 4 0 |
| 5 4 3 0 |
| 6 0 4 3 |
| 7 6 5 4 |
| 5 4 3 2 |
| 6 0 4 3 |
| 7 6 5 4 |
13 4
o17 : Matrix ZZ <--- ZZ
|
This function should be defined for any graded chain complex, not just ones created using res(I, Strategy=>4.1). Currently, it is used to extract information from the not yet implemented ring QQhybrid, whose elements, coming from QQ, are stored as real number approximations (as doubles, and as 1000 bit floating numbers), together with its remainders under a couple of primes, together with information about how many multiplications were performed to obtain this number.