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Almost regular sequences and Betti numbers
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 122, Number 4, August 2000
- pp. 689-719
- 10.1353/ajm.2000.0025
- Article
- Additional Information
Let Δ be a simplicial complex, and Δs the shifted simplicial complex of Δ, as defined by Kalai. It is shown that the Stanley-Reisner ideals IΔ and IΔs have the same extremal Betti numbers. This theorem is the squarefree analogue of the corresponding result by Bayer, Charalambous and Popescu which asserts that a graded ideal and its generic ideal have the same extremal Betti numbers. Our theorem implies well-known results by Kalai according to which Δ and Δs have the same simplicial homology, and Δ is Cohen-Macaulay if and only if its shifted complex is Cohen-Macaulay.