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Cartier Crystals

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arxiv 1309.1035 v1 pith:JIGC5R2Z submitted 2013-09-04 math.AG math.AC

classification math.AGmath.AC
keywords cartiercrystalscohomologicaldefinedfinitemodulesoperationstheory
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abstract

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we will explain how Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory of $\tau$-crystals as developed by Pink and the second author.

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  1. The derived $\infty$-category of Frobenius modules

    math.AG 2025-10 conditional novelty 6.0 of 10

    For any quasi-compact F_p-scheme with affine diagonal, the derived ∞-category of Frobenius modules is t-exactly equivalent to Frobenius modules on the derived ∞-category, and both satisfy Zariski descent.

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