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Poset embeddings of Hilbert functions

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Abstract

For a standard graded algebra \(R\), we consider embeddings of the poset of Hilbert functions of \(R\)-ideals into the poset of \(R\)-ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements–Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.

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Acknowledgments

We thank A. Conca and the referee for helpful comments. The computer algebra system Macaulay2 [13] provided valuable assistance in studying examples.

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Author notes

  1. Manoj Kummini

    Present address: Chennai Mathematical Institute, Siruseri, Tamilnadu, 603103, India

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Giulio Caviglia & Manoj Kummini

Authors
  1. Giulio Caviglia
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  2. Manoj Kummini
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Correspondence to Manoj Kummini.

Additional information

The work of the first author was supported by a grant from the Simons Foundation (209661 to G. C.)

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Caviglia, G., Kummini, M. Poset embeddings of Hilbert functions. Math. Z. 274, 805–819 (2013). https://doi.org/10.1007/s00209-012-1097-6

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  • DOI: https://doi.org/10.1007/s00209-012-1097-6

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