The 61/2-functor formalism of Cartier modules
Abstract
Let p be a prime. We construct a coherent formalism of geometric operations on Cartier modules with coefficients in ind-coherent sheaves on noetherian derived 𝔽p-schemes. For such a scheme X, we consider the stable presentable category Cart(X) of ind-coherent sheaves M equipped with a structural morphism F*,XM → M, where FX denotes the Frobenius endomorphism. We construct pushforward functors for arbitrary morphisms and exceptional pullback functors for separated morphisms almost of finite type, together with coherent composition and base-change equivalences. We single out conditions under which these functors admit further adjoints. On derived schemes separated and almost of finite type over 𝔽p, we construct compatible closed symmetric monoidal structures given by shriek tensor products, and establish which functors are symmetric monoidal. The constructions arise from a general procedure for forming functor formalisms of Cartier objects, using their description as lax limits and the universal properties of span $2$-categories developed by Cnossen--Lenz--Linskens [arXiv:2505.19192].
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BibTeX
@misc{mattis2026612functor,
title = {{The $6\frac12$-functor formalism of Cartier modules}},
author = {Klaus Mattis and Timo Weiß},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40200},
eprint = {2610.40200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40200},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The 61/2-functor formalism of Cartier modules
AU - Klaus Mattis
AU - Timo Weiß
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40200
UR - https://arxiv.org/abs/2610.40200
AB - Let p be a prime. We construct a coherent formalism of geometric operations on Cartier modules with coefficients in ind-coherent sheaves on noetherian derived 𝔽_p-schemes. For such a scheme X, we consider the stable presentable category Cart(X) of ind-coherent sheaves M equipped with a structural morphism F_*,XM → M, where F_X denotes the Frobenius endomorphism. We construct pushforward functors for arbitrary morphisms and exceptional pullback functors for separated morphisms almost of finite type, together with coherent composition and base-change equivalences. We single out conditions under which these functors admit further adjoints. On derived schemes separated and almost of finite type over 𝔽_p, we construct compatible closed symmetric monoidal structures given by shriek tensor products, and establish which functors are symmetric monoidal. The constructions arise from a general procedure for forming functor formalisms of Cartier objects, using their description as lax limits and the universal properties of span $2$-categories developed by Cnossen--Lenz--Linskens [arXiv:2505.19192].
ER -