Geodesic
Enriched homology and cohomology modules of simplicial complexes
Journal of Algebraic Combinatorics, Tome 25 (2007) no. 3, pp. 285-307.

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Summary: For a simplicial complex $\Delta $on ${1, 2,\cdots , n}$ we define enriched homology and cohomology modules. They are graded modules over $k[ x _{1},\cdots , x _{ n }]$ whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, $l$-Cohen-Macaulay, Buchsbaum, and Gorenstein ^* complexes $\Delta $, and also orientable homology manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex.
Keywords: keywords simplicial complex, homology, girth, Cohen-Macaulay, simplicial complex, block design, homology manifold, Steiner system 
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     author = {Fl{\o}ystad, Gunnar},
     title = {Enriched homology and cohomology modules of simplicial complexes},
     journal = {Journal of Algebraic Combinatorics},
     pages = {285--307},
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     number = {3},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2007__25_3_a2/}
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Fløystad, Gunnar. Enriched homology and cohomology modules of simplicial complexes. Journal of Algebraic Combinatorics, Tome 25 (2007) no. 3, pp. 285-307. http://geodesic.mathdoc.fr/item/JAC_2007__25_3_a2/