arXiv subjects

Daniel Caro

Publications and source records attributed to Daniel Caro.

28 records · Page 2Linked to original sources

arXiv2009

Full faithfulness without Frobenius structure and partially overcoherent isocrystals

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field $k$. Let $X$ be a variety over $k$, $Y$ be an open of $X$, $Y'$ be an open of $Y$ dense in $X$. We extend Kedlaya's full faithfulness as follows (we do not suppose $Y$ to be smooth): the canonical functor $F\text{-}\mathrm{Isoc} ^† (Y,X/K) \to F\text{-}\mathrm{Isoc} ^† (Y,Y/K) $ is fully faithfull. Suppose now $Y$ smooth. We construct the category of partially overcoherent isocrystals over $(Y,X)$ denoted by $\mathrm{Isoc} ^{††} (Y,X/K) $ whose objects are some particular arithmetic $\D$-modules. Furthermore, we check the equivalence of categories $\sp_{(Y,X),+} : \mathrm{Isoc} ^† (Y,X/K) \cong \mathrm{Isoc} ^{††} (Y,X/K)$.

math.AG↗

arXiv2009

Stability of holonomicity over quasi-projective varieties

Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category of complexes of arithmetic $\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations.

math.AG↗

arXiv2009

Sur la compatibilité à Frobenius de l'isomorphisme de dualité relative

Let $\V$ be a mixed characteristic complete discrete valuation ring, let $\X$ and $\Y$ be two smooth formal $\V$-schemes, let $f_0$ : $X \to Y$ be a projective morphism between their special fibers, let $T$ be a divisor of $Y$ such that $T_X := f_0 ^{-1} (T) $ is a divisor of $X$ and let $\M \in D ^\mathrm{b}_\mathrm{coh} (\D ^†_{\X} (\hdag T_X)_{\Q})$. We construct the relative duality isomorphism $ f_{0T +} \circ \DD_{\X, T_X} (\M) \riso \DD_{\Y, T} \circ f_{0T +} (\M)$. This generalizes the known case when there exists a lifting $f : \X \to \Y$ of $f_{0}$. Moreover, when $f_0$ is a closed immersion, we prove that this isomorphism commutes with Frobenius.

math.AG↗

arXiv2008

Overholonomicity of overconvergent $F$-isocrystals over smooth varieties

We prove the overholonomicity of overconvergent $F$-isocrystals over smooth varieties. This implies that the notions of overholonomicity and devissability in overconvergent $F$-isocrystals are equivalent. Then the overholonomicity is stable under tensor products. So, the overholonomicity gives a $p$-adic cohomology stable under Grothendieck's cohomological operations.

math.AG↗

arXiv2008

A characterization of the overcoherence

Let $\mathcal{P}$ be a proper smooth formal $\mathcal{V}$-scheme, $X$ a closed subscheme of the special fiber of $\mathcal{P}$, $\mathcal{E} \in F\text{-}D ^\mathrm{b}_\mathrm{coh} (\D ^†_{\mathcal{P},\mathbb{Q}})$ with support in $X$. We check that $\mathcal{E}$ is $\D ^†_{\mathcal{P},\mathbb{Q}}$-overcoherent if and only if, for any morphism $f : \mathcal{P}' \to \mathcal{P}$ of smooth formal $\mathcal{V}$-schemes, $f ^! (\mathcal{E}) $ is $\D ^†_{\mathcal{P}', \mathbb{Q}}$-coherent.

math.AG↗

arXiv2008

Overholonomic arithmetical D-modules

Let $k$ be a perfect field of characteristic $p >0$, $U$ be a variety over $k$ and $F$ be a power of Frobenius. We construct the category of overholonomic arithmetical ($F$-)$\D$-modules over $U$ and the category of overholonomic ($F$-)complexes over $U$. We prove that overholonomic complexes over $U$ are stables by direct images, inverse images, extraordinary inverse images, extraordinary direct images, dual functors. Moreover, in the smooth case, we check that unit-root overconvergent $F$-isocrystals are overholonomic. In particular, they are holonomic.

math.AG↗

arXiv2007

Overconvergent log isocrystals and holonomicity

Let V be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let X be smooth separated formal V-scheme, Z a strict normal crossing divisor of X and T a divisor of the special fiber of X. We study in this work isocrystals on the smooth log formal V-scheme associated to (X,Z) overconvergent along T.

math.AG↗

arXiv2006

Overconvergent F-isocrystals and differential overcoherence

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $k$ its residual field, $\mathcal{P}$ a proper smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $T$ a divisor of $P$, $U:=P\setminus T$, $Y$ a smooth closed subscheme of $U$. We prove that the category of overconvergent $F$-isocrystals on $Y$ is equivalent to the category of overcoherent $F$-isocrystals on $Y$. More generally, we prove such an equivalence by gluing for any smooth variety $Y$ over $k$. Moreover, we check that overcoherent $F$-complexes of arithmetic $\mathcal{D}$-modules split in overconvergent $F$-isocrystals.

math.AG↗

arXiv2005

D-modules arithmétiques associés aux isocristaux surconvergents. Cas lisse

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $\mathcal{P}$ a separated smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $X$ a smooth closed subscheme of $P$, $T$ a divisor in $P$ such that $T_X = T \cap X$ is a divisor in $X$ and $\smash{\D}^†_{\mathcal{P}}(\hdag T)$ the weak completion of the sheaf of differential operators on $\mathcal{P}$ with overconvergent singularities along $T$. We construct a fully faithful functor denoted by $ \sp_{X \hookrightarrow \mathcal{P},T,+}$ from the category of isocrystal on $X \setminus T_X$ overconvergent along $T_X$ into the category of coherent $\smash{\D}^†_{\mathcal{P}}(\hdag T) \otimes_\mathbb{Z} \mathbb{Q} $-modules with support in $X$. Next, we prove the commutation of $ \sp_{X \hookrightarrow \mathcal{P},T,+}$ with (extraordinary) inverse images and dual functors. These properties are compatible with Frobenius.

math.AG↗

arXiv2005

Dévissages des F-complexes de D-modules arithmétiques en F-isocristaux surconvergents

Nous définissons et étudions les dévissages des $F$-complexes de $\mathcal{D}$-modules arithmétiques en $F$-isocristaux surconvergents. Nous prouvons que les $F$-complexes surholonomes sont dévissables en $F$-isocristaux surconvergents. On établit ensuite une formule cohomologique, étendant celle d'Étesse et Le Stum, des fonctions $L$ associées aux duaux des $F$-complexes de $\mathcal{D}$-modules arithmétiques dévissables en $F$-isocristaux surconvergents. Puis, nous obtenons un analogue $p$-adique de Weil II étendant celui de Kedlaya.

math.AG↗