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Filtration theorems, Hilbert functions, and generic freeness

  • Mel Hochster
  • Yuanyuan Chen
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Abstract

Let M be a finitely generated module over a ring R that is a finitely generated algebra over a Noetherian ring A by r elements. We prove that there are finitely many primes P1, …, Pk of A such that M has a countable ascending filtration by finitely generated A-modules with the property that every factor Fn has a filtration by cyclic modules of the form A/Pi, 1 ≤ i ≤ k such that the number of copies Hi(n) of A/Pi that occur in the filtration of Fn is an eventually polynomial function of n. We prove a number of bounds on the degrees of these eventually polynomial functions: r−1 is always a bound, and the dimension of dim(MP)−1 is a bound under mild hypotheses. One application is an immediate proof Grothedieck's classic theorem on generic freeness. Several other applications are given.

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Cite

BibTeXRIS
BibTeX
@misc{hochster2026filtration,
  title = {{Filtration theorems, Hilbert functions, and generic freeness}},
  author = {Mel Hochster and Yuanyuan Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41200},
  eprint = {2610.41200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41200},
}
RIS
TY  - UNPB
TI  - Filtration theorems, Hilbert functions, and generic freeness
AU  - Mel Hochster
AU  - Yuanyuan Chen
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41200
UR  - https://arxiv.org/abs/2610.41200
AB  - Let M be a finitely generated module over a ring R that is a finitely generated algebra over a Noetherian ring A by r elements. We prove that there are finitely many primes P_1, …, P_k of A such that M has a countable ascending filtration by finitely generated A-modules with the property that every factor F_n has a filtration by cyclic modules of the form A/P_i, 1 ≤ i ≤ k such that the number of copies H_i(n) of A/P_i that occur in the filtration of F_n is an eventually polynomial function of n. We prove a number of bounds on the degrees of these eventually polynomial functions: r−1 is always a bound, and the dimension of dim(M_P)−1 is a bound under mild hypotheses. One application is an immediate proof Grothedieck's classic theorem on generic freeness. Several other applications are given.
ER  -