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.

- chainComplex(SimplicialComplex) -- create the chain complex associated to a simplicial complex.
- boundaryMap(ZZ,SimplicialComplex) -- make a boundary map between the oriented faces of an abstract simplicial complex
- HH_ZZ SimplicialComplex -- compute the reduced homology of an abstract simplicial complex
- HH^ZZ SimplicialComplex -- compute the reduced cohomology of an abstract simplicial complex

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

- buchbergerSimplicialComplex(MonomialIdeal,Ring) -- make the Buchberger complex of a monomial ideal
- lyubeznikSimplicialComplex(MonomialIdeal,Ring) -- create a simplicial complex supporting a Lyubeznik resolution of a monomial ideal
- scarfSimplicialComplex(MonomialIdeal,Ring) -- create the Scarf simplicial complex for a list of monomials

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.

- chainComplex(SimplicialComplex,Labels=>...) -- create the chain complex associated to a simplicial complex.
- buchbergerResolution(MonomialIdeal) -- make a Buchberger resolution of a monomial ideal
- lyubeznikResolution(MonomialIdeal) -- create the Lyubeznik resolution of an ordered set of monomials.
- scarfChainComplex(MonomialIdeal) -- create the Scarf chain complex for a list of monomials.
- taylorResolution(MonomialIdeal) -- create the Taylor resolution of a monomial ideal

- Making an abstract simplicial complex -- information about the basic constructors
- Finding attributes and properties -- information about accessing features of an abstract simplicial complex
- Working with simplicial maps -- information about simplicial maps and the induced operations