Abstract
Let Δ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex Δs of Δ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, …, xn] of the symmetric algebraic shifted complex, exterior algebraic shifted complex and combinatorial shifted complex of Δ are equal.
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