The homology complex $H$ is defined by ker dd^C/image dd^C. The differential of the homology complex is the zero map.
The first example is the complex associated to a triangulation of the real projective plane, having 6 vertices, 15 edges, and 10 triangles.
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To see that the first homology group has torsion, we compute a minimal presentation of the homology.
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By dualizing the minimal free resolution of a monomial ideal, we get a free complex with non-trivial homology. This particular complex is related to the local cohomology supported at the monomial ideal.
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