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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ChainComplexOperations.m2:308:0.