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The Serre-Grothendieck Finiteness Theorem for the cohomology of coherent sheaves
Abstract
We prove the converse of the Serre-Grothendieck finiteness theorem for the cohomology of coherent sheaves, originally due to Lipman. This result characterizes proper morphisms as those morphisms whose derived pushforward preserves pseudo-coherent complexes. Although the derived pushforward does not reflect pseudo-coherence, we prove that it does after twisting by all perfect complexes. This provides a new characterization of properness.
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BibTeX
@misc{sanders2026serregrothendieck,
title = {{The Serre-Grothendieck Finiteness Theorem for the cohomology of coherent sheaves}},
author = {Beren Sanders},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28945},
eprint = {2609.28945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28945},
}
RIS
TY - UNPB
TI - The Serre-Grothendieck Finiteness Theorem for the cohomology of coherent sheaves
AU - Beren Sanders
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28945
UR - https://arxiv.org/abs/2609.28945
AB - We prove the converse of the Serre-Grothendieck finiteness theorem for the cohomology of coherent sheaves, originally due to Lipman. This result characterizes proper morphisms as those morphisms whose derived pushforward preserves pseudo-coherent complexes. Although the derived pushforward does not reflect pseudo-coherence, we prove that it does after twisting by all perfect complexes. This provides a new characterization of properness.
ER -