Derived products and completion defects of inverse limits
Abstract
We compute derived products and inverse limits in Grothendieck categories by exact coresolutions. For the product of r doubled affine lines over B[s], where B is any nonzero commutative ring,the exact dimensions of products, epimorphic countable limits, and arbitrary countable limits are r, r, and r+1. We identify the top derived functors for every B-module of coefficients. Both obstruction modules contain the countable reduced product as a natural B-module direct summand, and a formal-series map relates them by an explicit cokernel sequence. A relative cofree coresolution gives degree shifts over an arbitrary Grothendieck category and computes the localization defect for arbitrary locally nilpotent diagrams. We also construct transfinite scalar extensions preserving a prescribed kernel and its derived limit;these give nonzero completion defects beyond any fixed ordinal.The general epimorphic top-degree vanishing question is not settled by these constructions.
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BibTeX
@misc{zeng2026derived,
title = {{Derived products and completion defects of inverse limits}},
author = {Rui Zeng and Guocheng Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34022},
eprint = {2610.34022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34022},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Derived products and completion defects of inverse limits
AU - Rui Zeng
AU - Guocheng Dai
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34022
UR - https://arxiv.org/abs/2610.34022
AB - We compute derived products and inverse limits in Grothendieck categories by exact coresolutions. For the product of r doubled affine lines over B[s], where B is any nonzero commutative ring,the exact dimensions of products, epimorphic countable limits, and arbitrary countable limits are r, r, and r+1. We identify the top derived functors for every B-module of coefficients. Both obstruction modules contain the countable reduced product as a natural B-module direct summand, and a formal-series map relates them by an explicit cokernel sequence. A relative cofree coresolution gives degree shifts over an arbitrary Grothendieck category and computes the localization defect for arbitrary locally nilpotent diagrams. We also construct transfinite scalar extensions preserving a prescribed kernel and its derived limit;these give nonzero completion defects beyond any fixed ordinal.The general epimorphic top-degree vanishing question is not settled by these constructions.
ER -