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ChainComplexOperations :: chi2

chi2 -- Euler characteristic of the 2nd Adams operation applied to a complex

Synopsis

Description

The definition:

chi2 F := eulerCharacteristic sym2 F - eulerCharacteristic wedge2 F.

Walker’s proof that the sum of the Betti numbers is at least 2{codim M), illustrated:

i1 : S = ZZ/101[a,b,c]

o1 = S

o1 : PolynomialRing
i2 : mm = ideal vars S

o2 = ideal (a, b, c)

o2 : Ideal of S
i3 : M = S^1/(mm^2)

o3 = cokernel | a2 ab ac b2 bc c2 |

                            1
o3 : S-module, quotient of S
i4 : F = res M

      1      6      8      3
o4 = S  <-- S  <-- S  <-- S  <-- 0
                                  
     0      1      2      3      4

o4 : ChainComplex
i5 : sumBetti = sum(4,i->rank F_i)

o5 = 18
i6 : sumTor = sum(4,i->length(Tor_i(M,M)))

o6 = 50
i7 : chi2 F == eulerCharacteristic sym2 F-eulerCharacteristic wedge2 F

o7 = true
i8 : 2^(codim M)*(length M) == chi2 F

o8 = false
i9 : chi2 F <= sumTor

o9 = true
i10 : sumTor <= sumBetti*(length M)

o10 = true

Caveat

Returns an error if any homology has infinite length

Ways to use chi2 :