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Working with associated chain complexes -- information about the chain complexes and their homogenizations

Each abstract simplicial complex $\Delta$ determines a chain complex of free modules over its coefficient ring. For all integers $i$, the $i$-th term in this chain complex has a basis corresponding to the $i$-dimensional faces in the simplicial complex $\Delta$. The reduced homology of $\Delta$ is, by definition, the homology of this chain complex. Similarly, the reduced cohomology of $\Delta$ is obtained by first using the $\operatorname{Hom}$ functor and then taking homology.

Basic operations arising of

Constructing abstract simplicial complexes from monomial ideals

Monomial ideals produce a few different abstract simplicial complexes, where the vertices are in bijection with the unique minimal set of monomial generators.

Free chain complexes arising from abstract simplicial complexes

By labelling or identifying the vertices in an abstract simplicial complex with distinct monomials (and each face with the least common multiple of its vertices), one transforms a chain complex of vector spaces into chain complex of free modules over a polynomial ring. This approach allows one to understand the minimal resolutions of some monomial ideals.

See also