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A smooth non-compactly generated category
Abstract
Let A=ℚ[t1/2n:n≥ 0] and consider A[u,u−1;σ], a skew Laurent polynomial ring, where σ(tq)=t2q. We study the kernel of derived extension of scalars along $ A[u,u^{-1};σ]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by mA[u,u−1;σ], with m=(t1/2n:n≥ 0). A finite free resolution of mA[u,u−1;σ] over $ A[u,u^{-1};σ]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};σ]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.
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BibTeX
@misc{frolov2026smooth,
title = {{A smooth non-compactly generated category}},
author = {Alexander Frolov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14874},
eprint = {2610.14874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14874},
}
RIS
TY - UNPB
TI - A smooth non-compactly generated category
AU - Alexander Frolov
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14874
UR - https://arxiv.org/abs/2610.14874
AB - Let A=ℚ[t^1/2^n:n≥ 0] and consider A[u,u^−1;σ], a skew Laurent polynomial ring, where σ(t^q)=t^2q. We study the kernel of derived extension of scalars along $ A[u,u^{-1};σ]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by mA[u,u^−1;σ], with m=(t^1/2^n:n≥ 0). A finite free resolution of mA[u,u^−1;σ] over $ A[u,u^{-1};σ]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};σ]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.
ER -