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Let R = k [x_{1},… ,x_{n}] be a polynomial ring in n variables over a field k and L a finite multigraded R-module. Bayer, Peeva and Sturmfels use the chain complex of a simplicial complex associated to a monomial ideal I (called the Scarf complex) to construct a free resolution of L = R/I, which is minimal for a certain class of monomial ideals.
Charalambous and Tchernev construct a Scarf complex for a general R-module L using the linear algebra properties of L, which also gives the minimal free resolution for a certain class of modules. However, no combinatorial analogue is used in this general setting.
In a more general construction of Tchernev, a resolution (not generally minimal) employs a matroid to construct a free resolution of L. Although this construction uses a combinatorial model, the matroid, it is only used as an indexing tool. The free modules and maps are defined again using the linear algebra of the matroid.
We show that the free modules in the free resolution of Tchernev are given by the homology of certain simplicial complexes determined by the matroid. The techniques used will be completely combinatorial and do not rely on the linear algebra structure. We will also give evidence to suggest that this matroid will also determine the maps in a free resolution of a multigraded module.
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