# 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, , with finite length homology
• M, , of finite length
• Outputs:
• exs, , (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 .