ChainComplexOperations

Macaulay2 package ChainComplexOperations -- Symmetric and exterior squares of a complex and the 2nd Adams operation. This package implements the constructions used in Mark Walker’s November 2016 proof of the (weak) Buchsbaum-Eisenbud-Horrocks conjecture, which states: If M is a module of codimension c over a regular local ring S, then the sum of the ranks of the free modules in a free resolution of M is at least 2c. Walker’s proof works for rings where 2 is invertible, and in this package we work over a field of characteristic ≠2. The main new (to Eisenbud) tool in Walker’s proof was the function chi2. Explicitly, if F is a ChainComplex of free S-modules with finite length homology, then chi2 F is the Euler characteristic of sym2 F minus that of wedge2 F. The function chi2 should be regarded as the Euler characteristic of the 2nd Adams operation, applied to F. It has two properties relevant for the proof: 1) Like the Euler characteristic of F, chi2 F is additive on short exact sequences of complexes. 2) If S is a regular local ring of dimension d with residue field k, then chi2 res k = 2d.

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