isExact(...,LengthLimit=>...) -- Option to check exactness only up to a particular homological degree

Synopsis

• Usage:

  t = isExact(F, LengthLimit => n)

• Inputs:
  • F, a chain complex,
  • n, an integer,
• Outputs:
  • t, a Boolean value,

Description

i1 : kk= ZZ/101

o1 = kk

o1 : QuotientRing

i2 : S = kk[a,b,c]

o2 = S

o2 : PolynomialRing

i3 : R = S/ideal(a^3)

o3 = R

o3 : QuotientRing

i4 : M = R^1/ideal(a)

o4 = cokernel | a |

                            1
o4 : R-module, quotient of R

i5 : C = chainComplex{map(M,R^0,0)}

o5 = M <-- 0
            
     0     1

o5 : ChainComplex

i6 : n =resolutionOfChainComplex (C, LengthLimit => 10)

          1             1
o6 = 0 : R  <--------- R  : 0
               | 1 |

                   1
     1 : 0 <----- R  : 1
              0

o6 : ChainComplexMap

i7 : isExact(cone n, LengthLimit=> 10)

o7 = true

i8 : isExact(cone n, LengthLimit=> 12)

o8 = false

Further information

• Default value: infinity
• Function: isExact -- Test to see if the ChainComplex is exact.
• Option key: LengthLimit -- stop when the resolution reaches this length

See also

• isQuasiIsomorphism -- Test to see if the ChainComplexMap is a quasi-isomorphism.

Functions with optional argument named LengthLimit:

• cartanEilenbergResolution(...,LengthLimit=>...) -- How many steps to compute
• isExact(...,LengthLimit=>...) -- Option to check exactness only up to a particular homological degree
• isQuasiIsomorphism(...,LengthLimit=>...) -- Option to check quasi-isomorphism only up to a certain point
• koszulComplex(...,LengthLimit=>...) -- How many steps to compute
• minimalBetti(...,LengthLimit=>...) -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
• resolution(...,LengthLimit=>...) -- stop when the resolution reaches this length
• resolutionOfChainComplex(...,LengthLimit=>...) -- How many steps to compute
• taylorResolution(...,LengthLimit=>...) -- How many steps to compute