On the n-perfectness under split nilpotent extensions
Abstract
Let π:S↠ R be a split ring epimorphism with nilpotent kernel J, and let R↪ S be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if X is a left S-module satisfying ToriS(R,X)=0 for every i>0, then pdS X=pdR(R⊗S X) in ℕ∪{0,∞}. In particular, for every flat left S-module F, pdSF=pdR(F/JF), and F/JF is flat over R. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over S and over R. Consequently, left n-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring R and an arbitrary R-module M, this applies to the idealization S=Rltimes M without any projectivity, flatness, finite generation, or finite flat-dimension assumption on M. We obtain a list of equivalent characterizations of the condition that Rltimes M is n-perfect, together with cotD(Rltimes M)=cotD(R). We further derive exact preservation of the flat-projective-dimension spectrum, strict n-perfectness, pointwise localization, arbitrary iterated idealizations, n-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.
Related
- Molnár's Theorem for split C*-by-nilpotent algebras
- Counting Nilpotent Galois extensions with prescribed genus number
- Higher Cotorsion Pairs on Image-Nilpotent η-Extensions
- Nilpotent Jacobian maps in dimension three and stable tameness in block extensions
- A characterization of central extensions relative to n-nilpotent objects
BibTeX
@misc{qi2026nperfectness,
title = {{On the n-perfectness under split nilpotent extensions}},
author = {Wei Qi and Xiaolei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22853},
eprint = {2609.22853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22853},
}
RIS
TY - UNPB
TI - On the n-perfectness under split nilpotent extensions
AU - Wei Qi
AU - Xiaolei Zhang
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22853
UR - https://arxiv.org/abs/2609.22853
AB - Let π:S↠ R be a split ring epimorphism with nilpotent kernel J, and let R↪ S be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if X is a left S-module satisfying Tor_i^S(R,X)=0 for every i>0, then pd_S X=pd_R(R⊗_S X) in ℕ∪{0,∞}. In particular, for every flat left S-module F, pd_SF=pd_R(F/JF), and F/JF is flat over R. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over S and over R. Consequently, left n-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring R and an arbitrary R-module M, this applies to the idealization S=Rltimes M without any projectivity, flatness, finite generation, or finite flat-dimension assumption on M. We obtain a list of equivalent characterizations of the condition that Rltimes M is n-perfect, together with cotD(Rltimes M)=cotD(R). We further derive exact preservation of the flat-projective-dimension spectrum, strict n-perfectness, pointwise localization, arbitrary iterated idealizations, n-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.
ER -