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Parameters and lim CM sequences: the Q-sequence conjecture and Lech's formula

  • Mel Hochster
  • Olivia Strahan
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Abstract

We prove that if a local ring has a weakly lim Cohen-Macaulay sequence of modules, then it satisfies the conjecture that every system of parameters is a Q-sequence. We also give a generalization of Lech's formula for the multiplicity of a local ring (R,m) with respect to a system of parameters that allows a very large class of sequences of m-primary ideals generated by monomials in the parameters to be used in the calculation. The cases where the sequence of ideals consists of the powers of the ideal generated by parameters, or where the ideals in the sequence are generated by varying powers of the individual parameters all of which tend to infinity, which is Lech's original formula, are included.

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Cite

BibTeXRIS
BibTeX
@misc{hochster2026parameters,
  title = {{Parameters and lim CM sequences: the Q-sequence conjecture and Lech's formula}},
  author = {Mel Hochster and Olivia Strahan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01439},
  eprint = {2611.01439},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01439},
}
RIS
TY  - UNPB
TI  - Parameters and lim CM sequences: the Q-sequence conjecture and Lech's formula
AU  - Mel Hochster
AU  - Olivia Strahan
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01439
UR  - https://arxiv.org/abs/2611.01439
AB  - We prove that if a local ring has a weakly lim Cohen-Macaulay sequence of modules, then it satisfies the conjecture that every system of parameters is a Q-sequence. We also give a generalization of Lech's formula for the multiplicity of a local ring (R,m) with respect to a system of parameters that allows a very large class of sequences of m-primary ideals generated by monomials in the parameters to be used in the calculation. The cases where the sequence of ideals consists of the powers of the ideal generated by parameters, or where the ideals in the sequence are generated by varying powers of the individual parameters all of which tend to infinity, which is Lech's original formula, are included.
ER  -