If M,N are S-modules annihilated by the elements of the matrix ff = (f_1..f_c), and k is the residue field of S, then the script exteriorTorModule(f,M) returns Tor^S(M, k) as a module over an exterior algebra k<e_1,...,e_c>, where the e_i have degree 1, while exteriorTorModule(f,M,N) returns Tor^S(M,N) as a module over a bigraded ring SE = S<e_1,..,e_c>, where the e_i have degrees {d_i,1}, where d_i is the degree of f_i. The module structure, in either case, is defined by the homotopies for the f_i on the resolution of M, computed by the script makeHomotopies1.
The scripts call makeModule to compute a (non-minimal) presentation of this module.
From the description by matrix factorizations and the paper "Tor as a module over an exterior algebra" of Eisenbud, Peeva and Schreyer it follows that when M is a high syzygy and F is its resolution, then the presentation of Tor(M,S^1/mm) always has generators in degrees 0,1, corresponding to the targets and sources of the stack of maps B(i), and that the resolution is componentwise linear in a suitable sense. In the following example, these facts are verified. The Tor module does NOT split into the direct sum of the submodules generated in degrees 0 and 1, however.
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The object exteriorTorModule is a method function.