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Mathematics Seminar Date: Wednesday, 12 March 2025 Time: 11:50 AM - 12:50 PM Venue: Seminar Hall Square-free powers of Cohen-Macaulay simplicial forests Kamalesh Saha Chennai Mathematical Institute. 12-03-25 Abstract Let $ I(\Delta)^{[k]} $ denote the $ k^{\text{th}} $ square-free power of the facet ideal of a simplicial complex $ \Delta $ in a polynomial ring $ R $. Square-free powers are intimately related to the 'Matching Theory' and 'Ordinary Powers'. In this article, we show that if $ \Delta $ is a Cohen-Macaulay simplicial forest, then $ R/I(\Delta)^{[k]} $ is Cohen-Macaulay for all $ k\geq 1 $. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $ \depth(R/I(\Delta)^{[k]}) $ for all $ k\geq 1 $, where $\Delta$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.
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