PreprintCC BY 4.0arXiv
Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings
Abstract
Let (A,m) be a Cohen-Macaulay local ring of dimension d, residue field k and let I be an m-primary ideal. Assume k is infinite. Let N be a perfect A-module of dimension t ≥ 1. Let M be a MCM A-module. The function n → ℓ(TorAj(M, N/In+1N)) is of polynomial type and let rI, Nj(M) be its degree. In general we have rI, Nj(M) ≤ rI,N2(SyzdA(k)) for j ≥ d + 2. If A is also dominant in the sense of Takahashi and M is non-free then we show limsupj → ∞ rI, Nj(M) = rI,N2(SyzdA(k)). We prove an analogous result when V is a Cohen-Macaulay module of finite injective dimension and the polynomial type function n → ℓ(ExtAj(M, V/In+1V))
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BibTeX
@misc{puthenpurakal2026derived,
title = {{Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings}},
author = {Tony J. Puthenpurakal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12120},
eprint = {2610.12120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12120},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings
AU - Tony J. Puthenpurakal
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12120
UR - https://arxiv.org/abs/2610.12120
AB - Let (A,m) be a Cohen-Macaulay local ring of dimension d, residue field k and let I be an m-primary ideal. Assume k is infinite. Let N be a perfect A-module of dimension t ≥ 1. Let M be a MCM A-module. The function n → ℓ(Tor^A_j(M, N/I^n+1N)) is of polynomial type and let r_I, N^j(M) be its degree. In general we have r_I, N^j(M) ≤ r_I,N^2(Syz^d_A(k)) for j ≥ d + 2. If A is also dominant in the sense of Takahashi and M is non-free then we show limsup_j → ∞ r_I, N^j(M) = r_I,N^2(Syz^d_A(k)). We prove an analogous result when V is a Cohen-Macaulay module of finite injective dimension and the polynomial type function n → ℓ(Ext_A^j(M, V/I^n+1V))
ER -