Shamash -- Computes the Shamash Complex

Synopsis

• Usage:
FF = Shamash(ff,F,len)
FF = Shamash(Rbar,F,len)
• Inputs:
• ff, , 1 x 1 Matrix over ring F.
• Rbar, a ring, ring F mod ideal ff
• F, , defined over ring ff
• len, an integer,
• Outputs:
• FF, , chain complex over (ring F)/(ideal ff)

Description

Let R = ring F = ring ff, and Rbar = R/(ideal f), where ff = matrix{{f}} is a 1x1 matrix whose entry is a nonzerodivisor in R. The complex F should admit a system of higher homotopies for the entry of ff, returned by the call makeHomotopies(ff,F).

The complex FF has terms

FF_{2*i} = Rbar**(F_0 ++ F_2 ++ .. ++ F_i)

FF_{2*i+1} = Rbar**(F_1 ++ F_3 ++..++F_{2*i+1})

and maps made from the higher homotopies.

For the case of a complete intersection of higher codimension, or to see the components of the resolution as summands of FF_j, use the routine EisenbudShamash instead.

 i1 : S = ZZ/101[x,y,z] o1 = S o1 : PolynomialRing i2 : R = S/ideal"x3,y3" o2 = R o2 : QuotientRing i3 : M = R^1/ideal(x,y,z) o3 = cokernel | x y z | 1 o3 : R-module, quotient of R i4 : F = res M 1 3 5 7 9 o4 = R <-- R <-- R <-- R <-- R 0 1 2 3 4 o4 : ChainComplex i5 : ff = matrix{{z^3}} o5 = | z3 | 1 1 o5 : Matrix R <--- R i6 : R1 = R/ideal ff o6 = R1 o6 : QuotientRing i7 : betti F 0 1 2 3 4 o7 = total: 1 3 5 7 9 0: 1 3 3 1 . 1: . . 2 6 6 2: . . . . 3 o7 : BettiTally i8 : FF = Shamash(ff,F,4) / R\1 / R\3 / R\6 / R\10 / R\15 o8 = |--| <-- |--| <-- |--| <-- |--| <-- |--| | 3| | 3| | 3| | 3| | 3| \z / \z / \z / \z / \z / 0 1 2 3 4 o8 : ChainComplex i9 : GG = Shamash(R1,F,4) 1 3 6 10 15 o9 = R1 <-- R1 <-- R1 <-- R1 <-- R1 0 1 2 3 4 o9 : ChainComplex i10 : betti FF 0 1 2 3 4 o10 = total: 1 3 6 10 15 0: 1 3 3 1 . 1: . . 3 9 9 2: . . . . 6 o10 : BettiTally i11 : betti GG 0 1 2 3 4 o11 = total: 1 3 6 10 15 0: 1 3 3 1 . 1: . . 3 9 9 2: . . . . 6 o11 : BettiTally i12 : ring GG o12 = R1 o12 : QuotientRing i13 : apply(length GG, i->prune HH_i FF) o13 = {cokernel | z y x |, 0, 0, 0} o13 : List

Caveat

F is assumed to be a homological complex starting from F_0. The matrix ff must be 1x1.