Sur la compatibilit\'e \`a Frobenius de l'isomorphisme de dualit\'e relative

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 $\smash{\D}^\dag _{\mathcal{P}}(\hdag T)$ the weak completion of the sheaf of differential operators on $\mathcal{P}$ with overconvergent singularities along $T$. We construct a fully faithful functor d...

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $u\colon \mathcal{Z} \hookrightarrow \mathfrak{X}$ be a closed immersion of smooth, quasi-compact, separated formal schemes over $\mathcal{V}$, $T$ be a divisor of $X$ such that $U:= T \cap Z$ is a divisor of $Z$, $\mathfrak{D}$ a strict normal crossing divisor of $\mathfrak{X}$ such that $u ^{-1} (\mathfrak{D})$ is a strict normal crossing divisor of $\mathcal{Z}$. W...

The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincar\'{e} duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead...

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\sD_X$ is the sheaf of differential operators on $X$ [EGAIV], it is well known that giving an action of $\sD_X$ on an $\sO_X$-module $\sE$ is equivalent to giving an infinite sequence of $\sO_X$-modules descending $\sE$ via the iterates of the Frobenius endomorphism of $X$. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characterist...

This paper is Part III of the series of work by the first named author on duality theories for p-primary etale cohomology, whose Parts I and II were published in 1986 and 1987, respectively. In this Part III, we study a duality for p-primary etale nearby cycles on smooth schemes over henselian discrete valuation rings of mixed characteristic (0, p) whose residue field is not necessarily perfect.

For a commutative Noetherian ring $R$ of prime characteristic, denote by $^{f}R$ the ring $R$ with the left structure given by the Frobenius map. We develop Thomas Marley's work on the property of the Frobenius functor $\F(-) = - \otimes_R {^f}R$ and show the interplay between $\F$ and its dual $\widetilde{\F}(-) = \Hom_R({}^{f}R, -)$ which is introduced by J\"{u}rgen Herzog.

This article is the second one of a series of three articles devoted to direct images of isocrystals: here we consider convergent isocrystals with Frobenius structure. Let V be a complete discrete valuation ring, with residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Firstly we characterize convergent F-isocrystals on a smooth affine k-scheme. Secondly, for perfect k and after a detailed exposition of the Teichm\^uller liftings, especiall...

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.

Let K be a complete discretely valued field with residue field k of characteristic p>0. There is a duality theory for cohomology with coefficients in commutative finite K-group schemes in the following cases : char(K)=0 and k finite (Tate), char(K)=p and k finite (Shatz), char(K)=0 and k algebraically closed (B\'egueri). In this paper, we settle the case where char(K)=p and k is algebraically closed.

This work concerns maps of commutative noetherian local rings containing a field of positive characteristic. Given such a map $\varphi$ of finite flat dimension, the results relate homological properties of the relative Frobenius of $\varphi$ to those of the fibers of $\varphi$. The focus is on the complete intersection property and the Gorenstein property.