arXiv subjects

Tobias Schedlmeier

Publications and source records attributed to Tobias Schedlmeier.

3 recordsLinked to original sources

arXiv2018

A Riemann-Hilbert correspondence for Cartier crystals

For a variety $X$ separated over a perfect field of characteristic $p>0$ which admits an embedding into a smooth variety, we establish an anti-equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb Z/p \mathbb Z$-sheaves on the étale site $X_{\text{ét}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal O_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent to Gabber's category of perverse sheaves in $D_c^b(X_{\text{ét}},\mathbb Z/p \mathbb Z)$.

math.AG↗

arXiv2018

Grothendieck duality for non-proper morphisms

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism and let $Z' \subseteq X$ and $Z \subseteq Y$ be closed subsets such that $f$ restricts to a proper morphism $f'\colon Z'\to Z$ of $f$. Then the functor $Rf_*$ is left adjoint to $RΓ_{Z'}f^!$ when considered as functors between complexes supported on $Z'$ or $Z$.

math.AG↗

arXiv2018

Cartier crystals and perverse constructible étale $p$-torsion sheaves

For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the étale site $X_{\text{ét}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{O}_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent to Gabber's category of perverse sheaves in $D_c^b(X_{\text{ét}},\mathbb{Z}/p\mathbb{Z})$. Furthermore, we define intermediate extensions for Cartier crystals and show that our equivalence between Cartier crystals and perverse constructible étale sheaves commutes with the intermediate extension functor.

math.AG↗