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Combinatorial invariance of Stanley-Reisner rings. (English) Zbl 0872.55018
The authors prove the following result: Let \(k\) be a field and \(\Delta\), \(\Delta'\) be two abstract simplicial complexes defined at the vertex sets \(V=\{v_1,\dots,v_n\}\) and \(U=\{u_1,\dots,u_n\}\), respectively. Suppose \(k[\Delta]\) and \(k[\Delta']\), the associated Stanley-Reisner rings, are isomorphic as \(k\)-algebras. Then there exists a bijective mapping \(\psi:V\to U\) which induces an isomorphism between \(\Delta\) and \(\Delta'\).

MSC:
55U05 Abstract complexes in algebraic topology
13G05 Integral domains
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References:
[1] W. Bruns and J. Herzog, Cohen-Macaulay Rings.Cambridge Studies in Advanced Mathematics 39,Cambridge University Press,Cambridge, 1993.
[2] T. Hibi, Algebraic Combinatorics on Convex Polytopes.Carslaw Publications, 1992. · Zbl 0772.52008
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