constantStrand -- a constant strand of a chain complex

Synopsis

• Usage:
Cd = constantStrand(C, kk, deg)
• Inputs:
• C, , a chain complex created using res(I, Strategy=>4.1)
• kk, a ring, if the coefficient ring of the ring of C is QQ, then this should be either: RR_{53}, RR_{1000},ZZ/32003, or ZZ/1073741909.
• deg, an integer, the degree that one wants to choose.
• Outputs:
• Cd, , a chain complex over kk, consisting of the submatrices of C of degree deg

Description

Warning! This function is very rough currently. It works 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

Caveat

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.