Derived products and top-degree inverse limits
Abstract
We compute top derived products and inverse limits on products of affine lines with doubled origin over B[s], for any nonzero commutative ring B. For every B-module M, the two top obstruction modules fit into an explicit exact sequence and contain Mℕ/M(ℕ)as a natural B-module direct summand.This gives exact product dimensions, sharp vanishing bounds for inverse sequences, and faithfulness in the coefficient module. For a Grothendieck category of finite product dimension, we also obtain degree shifts under adjoining commuting locally nilpotent endomorphisms and prove that this construction preserves top-degree vanishing for epimorphic inverse sequences. A fixed Gabriel--Popescu presentation then describes the general top obstruction through actual Ext sequences and separates two module-theoretic terms.
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BibTeX
@misc{zeng2026derived,
title = {{Derived products and top-degree inverse limits}},
author = {Rui Zeng and Guocheng Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41117},
eprint = {2610.41117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41117},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Derived products and top-degree inverse limits
AU - Rui Zeng
AU - Guocheng Dai
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41117
UR - https://arxiv.org/abs/2610.41117
AB - We compute top derived products and inverse limits on products of affine lines with doubled origin over B[s], for any nonzero commutative ring B. For every B-module M, the two top obstruction modules fit into an explicit exact sequence and contain M^ℕ/M^(ℕ)as a natural B-module direct summand.This gives exact product dimensions, sharp vanishing bounds for inverse sequences, and faithfulness in the coefficient module. For a Grothendieck category of finite product dimension, we also obtain degree shifts under adjoining commuting locally nilpotent endomorphisms and prove that this construction preserves top-degree vanishing for epimorphic inverse sequences. A fixed Gabriel--Popescu presentation then describes the general top obstruction through actual Ext sequences and separates two module-theoretic terms.
ER -