excess -- Difference between the sum of the lengths of Tor_i(M,M) and the Walker bound 2^d*length(M)

Synopsis

Usage:

exs = excess F

exs = excess M
Inputs:
- F, a chain complex, with finite length homology
- M, a module, of finite length
Outputs:
- exs, a sequence, (excess1a, excess1b, excess2)

Description

The three positive summands that make up the difference (sum Betti numbers M) and 2^{codim M} in Walker's proof of the weak Buchsbaum-Eisenbud-Horrocks conjecture:

excess1a = 2*oddHomologyLength sym2 F;

excess1b = 2*evenHomologyLength wedge2 F;

The difference between the sum of the lengths of Tor(M,M) and chi2 F is excess1a+excess1b.

excess2 = (sum of the betti numbers of M)*length M - sum(length Tor_i(M,M))

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 : sumTor - chi2 F

o9 = 56

i10 : sumBetti*(length M) - sumTor

o10 = 22

i11 : excess M

o11 = (6, 12, 22)

o11 : Sequence

Caveat

Returns an error if any homology has infinite length

Ways to use excess:

excess(ChainComplex)
excess(Module)

For the programmer

The object excess is a method function.